Higher algebra in $t$-structured tensor triangulated $\infty$-categories

Higher algebra in t -structured tensor triangulated ∞ -categories Jiac heng Liang Abstract W e generalize fundamen tal notions of higher algebra, traditionally dev elop ed within the ∞ - category of sp ectra, to the broader setting of t -structured tensor triangulated ∞ -categories ( ttt - ∞ - categories). Under a natural structural condition, which w e call “pro jective rigidit y”, we establish higher categorical analogues of Lazard’s theorem and pro ve the existence and univ ersal property of Cohn lo calizations. F urthermore, we generalize higher almost ring theory to the ttt - ∞ -categorical setting, sho wing that π 0 -epimorphic idemp otent algebras are in natural bijection with idempotent ideals. By exploiting deformation theory , we establish a general étale rigidit y theorem, proving that the ∞ -category of étale algebras ov er a ﬁxed connectiv e base is completely determined by its discrete counterpart. Finally , we c haracterize the moduli of such pro jectively rigid ttt - ∞ -categories, demonstrating that the presheaf ∞ -category on the 1 -dimensional framed cob ordism ∞ -category serv es as the universal pro jectively rigid ttt - ∞ -category . Con ten ts In tro duction 2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Con ven tions and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 A ckno wledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1 Preliminaries 8 1.1 Prestable ∞ -categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Algebra theory in Grothendieck ab elian categories . . . . . . . . . . . . . . . . . . . . . 13 2 Flatness and faithful ﬂatness 17 2.1 t -structure on the category of mo dules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Flat mo dules and algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Pro jectiv e mo dules 27 3.1 Basic prop erties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Pro jectiv e rigidity and Lazard’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 Mo dules o ver discrete algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4 Cohn lo calizations of E ∞ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.5 Cohn lo calizations in an ab elian base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Date: March 29, 2026 1 4 Finiteness prop erties 40 4.1 P erfect and almost p erfect mo dules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2 Finite presen tation and almost of ﬁnite presentation . . . . . . . . . . . . . . . . . . . . 44 5 Descendable algebras and idemp otent algebras 46 5.1 F aithful algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.2 Descendable algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.3 Almost algebra th eory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6 Deformation theory and étale rigidit y 57 6.1 The cotangen t complex formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.2 L-étale algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.3 Étale rigidit y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7 The ∞ -category of pro jectiv ely rigid ttt - ∞ -categories 72 7.1 The univ ersal example via 1-dimensional cob ordism . . . . . . . . . . . . . . . . . . . . . 73 7.2 Algebraic functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 8 Questions and future directions 77 A Duality 78 A.1 Dualizable ob jects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 A.2 Dualit y of Bimo dules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 B Ind(Pro)-completion of large ∞ -categories 81 References 83 In tro duction Ov er the past few decades, the integration of homotop y theory and higher categories in to deriv ed geometry has fundamentally reshaped mo dern mathematics. Classical algebraic geometry and ring theory , while incredibly p ow erful, often struggle to gracefully handle deriv ed phenomena such as non-transverse in tersections, higher T or-groups, and the subtleties of deformation theory . T o resolve these limitations, p eople turned to the derived categories of rings D ( R ) . How ever, the structural deﬁciencies of classical triangulated categories, such as the lac k of functorial mapping cones and well-behav ed limits, even tually necessitated a more robust and coherent framework. This culminated in the developmen t of higher algebr a [ HA ], whic h systematically replaces discrete comm utativ e rings with E ∞ -rings and classical mo dule categories with the stable ∞ -categories of mo dules ov er E ∞ -rings. In this stable ∞ -categorical setting, the classical to ols of comm utativ e algebra (including tensor pro ducts, localizations, ﬂat descen t, étale morphisms, and the cotangent complex) are not merely reco vered but are v astly generalized and computationally enriched. The resulting machinery o ver the stable ∞ -category of sp ectra Sp has b een highly successful, pro viding foundational techniques for derived algebraic geometry , c hromatic homotopy theory , and mo dern arithmetic geometry . By natively enco ding deriv ed phenomena, it pro vides a rigorous homotopical foundation for geometric constructions. 2 Ho wev er, the geometric and algebraic b ehaviors of mo dule sp ectra are frequently gov erned not b y their top ological nature, but by their interaction with the underlying t -structure. A key fact motiv ating this pap er is that the ﬂatness of a module ov er a connective E ∞ -ring can b e entirely c haracterized by the t -exactness of the tensor pro duct functor, as observ ed in [ HA , §7.2.2]. This suggests that the machinery of higher algebra, including (faithfully) ﬂat morphisms, ﬁnitely presen ted morphisms, and étale morphisms, do es not strictly dep end on the sp eciﬁc ∞ -category of sp ectra. Instead, these notions can b e generalized to the setting of an arbitrary t -structured tensor triangulated ∞ -category ( ttt - ∞ -category). A dopting this generalized framework yields a uniﬁed approac h to higher algebra across diverse underlying geometries, such as equiv arian t, motivic, and analytic settings. The foundation of this approac h is the structural interpla y b etw een an accessible t -structure and a compatible symmetric monoidal structure, which provides the essen tial categorical scaﬀolding required to support these op erations. Similar philosophies hav e recently surfaced in the context of derived (analytic) geometry [ KM25 ; BKK24 ; M an24 ; KKM22 ; Rak20 ]. The ubiquit y of this framew ork is p erhaps b est reﬂected by the rich v ariety of examples that naturally inhabit it. A central fo cus of this pap er is the notion of pr oje ctive rigidity , a structural condition under whic h the dualizable ob jects of the connectiv e part coincide with its compact pro jectiv es. This condition is remark ably widespread. Key examples of pro jectiv ely rigid ttt - ∞ -categories include the following (for a comprehensiv e list, see Example 3.15 ): • Sp ectra and ﬁltered v ariants: The standard ∞ -category of sp ectra Sp , graded sp ectra Gr(Sp) , and ﬁltered spec tra Fil(Sp) equipp ed with either the p oint wise or the homotopy t -structure. • Equiv ariant and motivic contexts: The ∞ -category of genuine G -sp ectra Sp G for a ﬁnite group G equipp ed with the p oint wise t -structure, the Artin–T ate motivic sp ectra SH ( k ) A T o ver a p erfect ﬁeld k [ BHS20 ], and the V oevodsky ∞ -category DM ( k , Z [1 /p ]) equipp ed with the Chow t -structure. • Geometric and sheaf-theoretic examples: (Derived) quasi-coherent sheav es QCoh ( X ) on an aﬃne quotien t stac k, and the ∞ -category of shea ves Sh v( X , Sp) ⊗ on a s tone space. In this w ork, we systematically develop the theory of higher algebra internal to ttt - ∞ -categories. Under the assumption of pro jectiv e rigidit y , we establish generalized analogues of Lazard’s theorem and étale rigidit y . F urthermore, we connect our framework with recent adv ances in higher almost ring theory [ HS24 ] and lo cally rigid ∞ -categories [ Ram24b ]. Ultimately , we show that the class of pro jectiv ely rigid ttt - ∞ -categories is gov erned by a univ ersal geometric example related to the 1 -dimensional cob ordism category . Main results Con v en tion 0.1. Throughout this pap er, we ﬁx a t -structured tensor triangulated ∞ -category ( A ⊗ , A ≥ 0 , A ≤ 0 ) . (W e will refer to such data as a ttt - ∞ -category). It consists of a presen tably stable symmetric monoidal ∞ -category A ⊗ ∈ CAlg ( P r L st ) equipp ed with an accessible t -structure 1 ( A ≥ 0 , A ≤ 0 ) that is compatible with the monoidal structure in the following sense: 1 An “accessible” t -structure here refers to a t -structure such that A ≥ 0 is presentable. 3 (1) The unit 1 ∈ A ≥ 0 ; (2) F or any tw o connective ob jects X, Y ∈ A ≥ 0 , w e ha ve X ⊗ Y ∈ A ≥ 0 . W e list our main results as follows. W e b egin with the structure of ﬂat and almost p erfect mo dules. Under the assumption of pro jectiv e rigidit y , w e establish a higher categorical analogue of Lazard’s theorem, demonstrating that ﬂat mo dules are entirely controlled by the ﬁltered colimits of compact pro jectiv e mo dules: Theorem 0.2 ( Theorem 3.20 , and Prop osition 4.8 ) . Assume that A ≥ 0 is pr oje ctively gener ate d. L et R ∈ Alg ( A ≥ 0 ) . Then the fol lowing hold: (1) (L azar d’s The or em.) If A ⊗ ≥ 0 is pr oje ctively rigid (se e Deﬁnition 3.12 ), then the ∞ -c ate gory of ﬂat R -mo dules c an b e identiﬁe d with: LMo d R ( A ) f l ≃ Ind(LMo d R ( A ≥ 0 ) cpro j ) . (2) The inclusion LMo d R ( A ≥ 0 ) cpro j  → LMo d R ( A ≥ 0 ) aperf of c omp act pr oje ctive R -mo dules into almost p erfe ct R -mo dules induc es an e quivalenc e LMo d R ( A ≥ 0 ) aperf ≃ P ⊔ , ∆ op ⊔ (LMo d R ( A ≥ 0 ) cpro j ) , wher e the left-hand side is the r elative c o c ompletion obtaine d by formal ly adding ge ometric r e aliza- tions and ﬁnite c opr o ducts while pr eserving existing ﬁnite c opr o ducts. A lternatively, we have: LMo d R ( A ≥ 0 ) aperf ≃ P ∆ op ∅ (LMo d R ( A ≥ 0 ) cpro j ) . This tight control ov er mo dules via pro jective rigidity allows us to systematically lift discrete algebraic structures from the heart A ♡ to the ambien t category A . Sp eciﬁcally , we show that the deriv ed category of a d iscrete ring in the heart provides a complete mo del for its mo dule category in A : Theorem 0.3 ( Theorem 3.21 ) . Supp ose that the ttt - ∞ -c ate gory ( A ⊗ , A ≥ 0 ) is pr oje ctively rigid. Then the fol lowing hold: (1) F or any discr ete R ∈ Alg ( A ♡ ) ther e exists a (unique up to c ontr actible choic es) e quivalenc e in P r t - rex st D (LMo d π 0 R ( A ♡ )) ∼ − → LMo d R ( A ) which induc es the identity functor on the he art. (2) F or any discr ete commutative algebr a R ∈ CAlg ( A ♡ ) ther e exists a (unique up to c ontr actible choic es) e quivalenc e in CAlg( P r t - rex st ) D (Mo d π 0 R ( A ♡ )) ⊗ ∼ − → Mo d R ( A ) ⊗ which induc es the identity functor on the he art, wher e the symmetric monoidal structur e on the left-hand side is induc e d by the pr oje ctive mo del with the tensor pr o duct of chain c omplexes. Because our working examples t ypically admit enough pro jectiv es but lack free generators, classical lo calization techniques must b e carefully adapted. W e address this b y pro ving the existence and univ ersal prop ert y of Cohn lo calizations, which allow us to formally inv ert morphisms b etw een compact pro jectiv e mo dules: 4 Theorem 0.4 ( Theorem 3.25 , Cohn lo calization) . Supp ose that the ttt - ∞ -c ate gory ( A ⊗ , A ≥ 0 ) is pr oje c- tively rigid. L et R ∈ CAlg ( A ≥ 0 ) , and let S b e a set of morphisms b etwe en c omp act pr oje ctive R -mo dules. Then ther e exists a Cohn lo c alization, a map R → R [ S − 1 ] in CAlg ( A ≥ 0 ) , satisfying the fol lowing universal pr op erty: F or any B ∈ CAlg ( A ) , the map Map CAlg( A ) ( R [ S − 1 ] , B ) → Map CAlg( A ) ( R, B ) , induc e d by pr e c omp osition, is ( − 1) -trunc ate d. F urthermor e, the image of the induc e d map on c onne cte d c omp onents c onsists pr e cisely of those (homotopy classes of ) maps R → B such that for e ach morphism f ∈ S , the map B ⊗ R f is an e quivalenc e of B -mo dules. Bey ond lo calizations, another fundamental result ab out (higher) idemp otent algebras is the theory of (higher) almost mathematics. Generalizing recent work of Hebestreit and Sc holze [ HS24 ] to the ttt - ∞ -categorical setting, we demonstrate that epimorphic idemp oten t algebras can b e classiﬁed en tirely in terms of classical idemp otent ideals in the discrete heart: Theorem 0.5 ( Theorem 5.18 ) . Assume that A is b oth right and left c omplete. L et R ∈ CAlg ( A ≥ 0 ) . Consider the ful l sub c ate gory LQ R of CAlg ( A ≥ 0 ) R/ sp anne d by the maps ϕ : R → S for which (1) the multiplic ation S ⊗ R S → S is an e quivalenc e, i.e., ϕ is idemp otent, (2) π 0 ( ϕ ) : π 0 R → π 0 S is epimorphic in A ♡ . Then the functor LQ R − →  I ⊂ π 0 R | I 2 = I  , ϕ 7− → Ker( π 0 ϕ ) is an e quivalenc e of c ate gories, wher e we again r e gar d the tar get as a p oset via the inclusion or dering. The inverse image of some I ⊂ π 0 ( R ) c an b e describ e d mor e dir e ctly as R/I ∞ , wher e I ∞ = lim ← − n ∈ N op J ⊗ R n I with J I → R the ﬁbr e of the c anonic al map R → H ( π 0 ( R ) /I ) . F urthermor e, this inverse system stabilizes on π i for n > i + 1 . The image of the ful ly faithful r estriction functor Mo d R/I ∞ ( A ) → Mo d R ( A ) c onsists exactly of those mo dules whose homotopy is kil le d by I , as desir e d. This higher algebraic behavior naturally leads to questions of deformation theory . Utilizing the cotangen t complex formalism ov er a ttt - ∞ -base, we establish an étale rigidit y theorem. Generalizing Lurie’s result for sp ectra [ HA , §7.5], we prov e that the étale top ology is insensitive to higher categorical nilp otence and is entirely gov erned b y its π 0 -truncation: Theorem 0.6 ( Theorem 6.26 , Étale rigidity) . Assume that A is Gr othendie ck (se e Deﬁnition 1.15 ) and left c omplete. L et A ∈ CAlg( A ) . Then: (1) L et CAlg ( A ) ﬂ ,L - et A/ denote the ful l sub c ate gory of CAlg ( A ) A/ sp anne d by the ﬂat L-étale maps A → B . Supp ose that A is c onne ctive. Then the functor π 0 induc es an e quivalenc e CAlg( A ) ﬂ ,L - et A/ ∼ − → CAlg( A ♡ ) ﬂ ,L - et π 0 A/ with the (1-)c ate gory of the discr ete ﬂat L-étale c ommutative π 0 A -algebr as. 5 (2) Supp ose that A ⊗ ≥ 0 is pr oje ctively rigid. L et CAlg ( A ) et A/ denote the ful l sub c ate gory of CAlg ( A ) A/ sp anne d by the étale maps A → B . Then the functor π 0 induc es an e quivalenc e CAlg( A ) et A/ ∼ − → CAlg( A ♡ ) et π 0 A/ with the (1-)c ate gory of the discr ete étale c ommutative π 0 A -algebr as. Finally , having developed this comprehensive algebraic machinery in ternal to a ﬁxed ttt - ∞ -category , w e zo om out to study the global mo duli of suc h categories. W e prov e that the entire landscap e of pro jectiv ely rigid ttt - ∞ -categories is gov erned by a remark able universal geometric example: Theorem 0.7 ( Theorem 7.11 , Univ ersal pro jectiv ely rigid ttt - ∞ -category) . CAlg rig , at Sp ≥ 0 admits a c omp act gener ator F un( Cob op 1 , Sp ≥ 0 ) ⊗ wher e the symmetric monoidal structur e is given by Day c onvolution and Cob 1 denotes the 1 -dimensional fr ame d c ob or dism ( ∞ , 1) -c ate gory. Outline The pap er is organized as follows. • Section 1 (Preliminaries): W e review the theory of prestable ∞ -categories and presentable Grothendiec k categories, establishing the foundational relationship b et ween a presentable prestable ∞ -category and its stabilization. • Section 2 (Flatness and F aithful Flatness): W e redeﬁne ﬂatness via the t -exactness of the relativ e tensor pro duct. W e establish the stabilit y of ﬂat mo dules under base c hange and ﬁltered colimits. • Section 3 (Pro jective Mo dules): W e introduce the core condition of pr oje ctive rigidity . Under this h yp othesis, we prov e a generalized L azar d’s The or em , showing that ﬂat mo dules are precisely ﬁltered colimits of compact pro jectiv es, and establish the universal prop erty of Cohn lo c alizations . • Section 4 (Finiteness Prop erties): W e deﬁne p erfect and almost p erfect mo dules, utilizing T or-amplitude b ounds to control homological b ehavior in a ttt - ∞ -category . • Section 5 (F aithful and Descendable Algebras): W e study descen t in higher algebra. Generalizing a theorem of Heb estreit and Scholze [ HS24 ], we classify epimorphic idemp otent algebras via id emp oten t ideals. • Section 6 (Deformation Theory and Étale Rigidit y): Utilizing the cotangent complex formalism, we establish étale rigidity , proving that the étale top ology of an E ∞ -algebra is gov erned en tirely by its π 0 -truncation, generalizing a theorem of Lurie [ HA , §7.5]. • Section 7 (The ∞ -category of pro jectiv ely rigid ttt - ∞ -categories): W e prov e that the presheaf ∞ -category on the 1 -dimensional framed cobordism category serv es as the univ ersal pro jectiv ely rigid ttt - ∞ -category . • Section 8 (Questions and F uture Directions): W e conclude by discussing several remaining questions and future directions. 6 Con v en tions and notations This pap er makes heavy use of the theory of ∞ -categories and related notions, as developed in Jacob Lurie’s foundational references [ HTT ; HA ; SAG ]. Con v en tion 0.8. (1) Since a t -structure is determined b y its connectiv e part, we will simply denote a ttt - ∞ -category by ( A ⊗ , A ≥ 0 ) rather than ( A ⊗ , A ≥ 0 , A ≤ 0 ) . (2) Some references refer to a presen tably stable symmetric monoidal ∞ -category as a “big” tt - ∞ - category . Ho wev er, we do not deal with small ones in this pap er, so we will omit the preﬁx “big”; th us, when we use the terms tt - ∞ -category or ttt - ∞ -category , they are implicitly assumed to b e big. (3) W e will often say A satisﬁes some prop erty if the t -structured stable ∞ -category ( A , A ≥ 0 ) satisﬁes prop ert y P . F or example, we will say that A is right complete if its t -structure is right complete. W e will say that A ⊗ satisﬁes some prop ert y if this prop ert y in volv es the monoidal structure. (4) W e often denote the tensor pro duct in the truncated ∞ -category A [0 ,n ] b y −⊗− . Notation 0.9. F or the reader’s conv enience, we record b elow the recurring notions and notation used throughout. (a) In this paper, ∞ -categories refer to ( ∞ , 1) -categories. (b) S denotes the ∞ -category of spaces (i.e., ∞ -group oids). Cat ∞ denotes the ∞ -category of small ∞ -categories, functors, and natural transformations. Sp denotes the ∞ -category of sp ectra. (c) F or n ∈ N , S ≤ n is the full sub category of S spanned b y n -truncated spaces. F or example, S ≤− 1 is the full subcategory spanned by ∅ and ∗ , and S ≤− 2 is the full sub category spanned by ∗ . (d) F or functor categories F un ( C , D ) , we use sup erscripts lex , rex , ex , lim , and colim to indicate the full sub categories of functors preserving ﬁnite limits, ﬁnite colimits, ﬁnite limits and ﬁnite colimits (exact functors), small limits, and small colimits, resp ectively . (e) W e use the sup erscript C ⊗ to denote the ∞ -category C equipp ed with a certain ( E k -)monoidal structure for some 0 ⩽ k ⩽ ∞ . (f ) Let C ⊗ b e a monoidal ∞ -category . W e denote b y Alg ( C ) the ∞ -category of E 1 -algebras in C . If C ⊗ is E k -monoidal for some 0 ⩽ k ⩽ ∞ , we denote by Alg E k ( C ) the ∞ -category of E k -algebras in C . In particular when C ⊗ is symmetric monoidal, w e denote by CAlg ( C ) the ∞ -category of E ∞ -algebras (i.e. commutativ e algebra ob jects) in C . (g) W e work relative to a chain of strongly inaccessible cardinals δ 0 < δ 1 < δ 2 . Then V δ i is a Grothendiec k universe, and elements of V δ 0 , V δ 1 , V δ 2 are called small, large, and very large, resp ec- tiv ely . A hat b · indicates largeness of the ob jects in the category , e.g., b S or d Cat ∞ for the (very large) ∞ -categories of large spaces or large ∞ -categories. (h) P r L denotes the ∞ -category of presentable ∞ -categories with left adjoin t functors. Its default symmetric monoidal structure is the Lurie tensor pro duct, whic h corepresents functors out of the Cartesian pro duct that are colimit-preserving in each v ariable. F or a cardinal κ , P r L κ ⊂ P r L denotes the sub category of κ -accessible presentable ∞ -categories with functors that preserve κ -compact ob jects. 7 (i) W e denote P r L st to b e the ∞ -category of presen table stable ∞ -categories. W e denote P r L ad to b e the ∞ -category of presentable additive ∞ -categories. W e denote P r L ad , 1 to b e the ∞ -category of presen table additive 1 -categories. (j) W e use the notation Map to indicate the enrichmen t if the enriched context is clear. A c kno wledgmen ts I w ould like to express my sincere appreciation to Germán Stefanic h for sharing his work [ Ste23 ], which has laid the groundwork for man y foundational results closely tied to this pap er. I am also grateful to Jac k Davies for highlighting the p ossibility of étale rigidity in m y context, and to Maxime Ramzi for emphasizing the equiv alence betw een our notion of “pro jectiv ely rigid” and the atomically generated rigid Sp ≥ 0 -algebras. Additionally , I wish to conv ey my deep gratitude to my advisor, David Gepner, for his un wa vering supp ort and inspiration throughout this endeav or. I would also like to thank Ko Aoki, T obias Barthel, T om Bac hmann, Tim Campion, Marc Hoy ois, Xiansheng Li, Marius Nielsen, Arp on Raksit, Xiangrui Shen, V ov a Sosnilo, Yifei Zhu, and Changhan Zou for their helpful conv ersations at v arious stages of this pro ject. F urthermore, muc h of this work was carried out during my visit to the Max Planck Institute for Mathematics (MPIM), and I gratefully ackno wledge the hospitality extended to me by the institute. 1 Preliminaries The goal of this section is to gather basic preliminaries ab out prestable ∞ -categories and ab elian categories. 1.1 Prestable ∞ -categories Before developing homological algebra ov er arbitrary bases, we m ust establish the formal prop erties of the connective part underlying our ttt - ∞ -categories. In this subsection, w e recall the theory of prestable ∞ -categories as developed in [ SAG ], fo cusing heavily on Grothendieck prestable ∞ -categories. These categories naturally serv e as the connectiv e parts of Grothendiec k t -structures. Deﬁnition 1.1 (See [ SAG ] C.1.2.2) . A prestable ∞ -category is an ∞ -category C satisfying the following prop erties: (1) The initial and ﬁnal ob jects of C agree (that is, C is p ointed). (2) Ev ery coﬁb er sequence in C is also a ﬁb er sequence. (3) Ev ery map in C of the form f : X → Σ( Y ) is the coﬁb er of its ﬁb er. Moreo ver, we sa y C is a Grothendieck prestable ∞ -category if it further satisﬁes that it is presen table and that ﬁltered colimits and ﬁnite limits commute in C . W e let Groth ∞ ⊂ P r L denote the full sub category whose ob jects are Grothendieck prestable ∞ -categories. Example 1.2. (1) Any stable ∞ -category is prestable. (2) Let C b e a stable ∞ -category equipp ed with a t -structure ( C ≥ 0 , C ≤ 0 ) . Then the full sub category C ≥ 0 ⊂ C is prestable. 8 Prestable ∞ -categories hav e a quite c lose relation to t -structured stable ∞ -categories. Prop osition 1.3. L et C b e an ∞ -c ate gory. The fol lowing c onditions ar e e quivalent: (1) The ∞ -c ate gory C is pr estable and admits ﬁnite limits. (2) The ∞ -c ate gory C is p ointe d and admits ﬁnite c olimits, and the c anonic al map ρ : C → SW ( C ) is ful ly faithful. Mor e over, the stable ∞ -c ate gory SW ( C ) admits a t -structur e (SW( C ) ≥ 0 , SW( C ) ≤ 0 ) wher e SW ( C ) ≥ 0 is the essential image of ρ . (3) Ther e exists a stable ∞ -c ate gory D e quipp e d with a t -structur e ( D ≥ 0 , D ≤ 0 ) and an e quivalenc e of ∞ -c ate gories C ≃ D ≥ 0 . wher e SW ( − ) denotes the Sp anier-Whitehe ad c onstruction. Prop osition 1.4 (See [ SAG ] C.1.4.1) . L et C b e a pr esentable ∞ -c ate gory. The fol lowing c onditions ar e e quivalent: (a) The ∞ -c ate gory C is pr estable and ﬁlter e d c olimits in C ar e left exact (see [ HTT , Deﬁnition 7.3.4.2]) . (b) The ∞ -c ate gory C is pr estable and the functor Ω : C → C c ommutes with ﬁlter e d c olimits. (c) The ∞ -c ate gory C is pr estable and the functor Ω ∞ : Sp( C ) → C c ommutes with ﬁlter e d c olimits. (d) Ther e exists a pr esentable stable ∞ -c ate gory D , a t -structur e ( D ≥ 0 , D ≤ 0 ) on D which is c omp atible with ﬁlter e d c olimits, and an e quivalenc e C ≃ D ≥ 0 . (e) The susp ension functor Σ + : C → Sp ( C ) is ful ly faithful and its essential image Sp ( C ) ≥ 0 is the c onne ctive p art of a t -structur e on Sp( C ) which is c omp atible with ﬁlter e d c olimits. Deﬁnition 1.5. W e say a presentable prestable ∞ -category is Grothendieck if it satisﬁes the ab ov e equiv alent conditions. Theorem 1.6 (See [ SA G ] C.4.2.1) . The ful l sub c ate gory Groth ∞ ⊂ P r L ad c ontains the unit obje ct of Sp ≥ 0 and is close d under Lurie tensor pr o ducts. Conse quently, Groth ∞ inherits a symmetric monoidal structur e for which the inclusion Groth ∞  → P r L ad is symmetric monoidal. Deﬁnition 1.7 (See [ SA G ] C.3.1.3) . Let C b e a presentable stable ∞ -category . W e deﬁne a full sub category C ≥ 0 ⊂ C to b e a core if it is closed under small colimits and extensions. W e will refer to P r + st as the ∞ -category of cored s table ∞ -categories. The ob jects of P r + st are pairs ( C , C ≥ 0 ) , where C is a presentable stable ∞ -category and C ≥ 0 ⊂ C is a core. A morphism from ( C , C ≥ 0 ) to ( D , D ≥ 0 ) is given by a colimit-preserving functor f : C → D satisfying f ( C ≥ 0 ) ⊂ D ≥ 0 . W arning 1.8. C ≥ 0 in this deﬁnition do es not necessarily form a t -structure, but if C ≥ 0 is presentable, then it does. In that case, w e call the pair ( C , C ≥ 0 ) a pr esentably t -structur e d stable ∞ -c ate gory . Remark 1.9. (1) Our P r + st actually refers to Groth + ∞ in [ SA G , Remark C.3.1.3]. (2) In fact, we only care about the full subcategory P r t -rex st ⊂ P r + st spanned b y those presentably t -structured stable ∞ -categories with right t -exact functors. How ev er, the technical adv an tage of P r + st is that it admits go o d colimits and limits [ SAG , Remark C.3.1.7]. Deﬁnition 1.10. There is a natural symmetric monoidal structure on P r + st giv en by the construction: ( C , C ≥ 0 ) ⊗ ( D , D ≥ 0 ) = ( C ⊗ D , m ! ( C ≥ 0 , D ≥ 0 )) 9 where C ⊗ D is the Lurie tensor pro duct and m ! ( C ≥ 0 , D ≥ 0 ) is the smallest full sub category of C ⊗ D whic h is closed under colimits and extensions and contains the ob jects m ( C, D ) for each C ∈ C ≥ 0 and D ∈ D ≥ 0 . Remark 1.11. (1) The full sub category P r t -rex st ⊂ P r + st is closed under tensor pro ducts, since m ! ( C ≥ 0 , D ≥ 0 ) is presen table if b oth C ≥ 0 and D ≥ 0 are presen table. (2) An ob ject in CAlg( P r t -rex st ) can b e identiﬁed with a ttt - ∞ -category . Prop osition 1.12 (See [ SAG ] C.3.1.1) . L et C and D b e Gr othendie ck pr estable ∞ -c ate gories. Then the c anonic al map θ : LF un( C , D ) → LF un(Sp( C ) , Sp( D )) is a ful ly faithful emb e dding, whose essential image c onsists of those functors Sp ( C ) → Sp ( D ) which pr eserve smal l c olimits and ar e right t -exact (with r esp e ct to the c anonic al t -structur e). Prop osition 1.13 (See [ SAG ] C.3.2.1) . L et C and D b e Gr othendie ck pr estable ∞ -c ate gories and let f : C → D b e a c olimit-pr eserving functor. Then the fol lowing c onditions ar e e quivalent: (1) The functor f is left exact. (2) The functor f c arries 0-trunc ate d obje cts of C to 0-trunc ate d obje cts of D . (3) The induc e d map F : Sp( C ) → Sp( D ) is left t -exact. Corollary 1.14. (1) The c onstruction C 7→ (Sp( C ) , Sp( C ) ≥ 0 ) determines a ful ly faithful emb e dding Groth ∞  → P r + st fr om the ∞ -c ate gory of Gr othendie ck pr estable ∞ -c ate gories to the ∞ -c ate gory of c or e d stable ∞ -c ate gories. (2) A p air ( C , C ≥ 0 ) b elongs to the essential image of Groth ∞  → P r + st if and only if it forms an ac c essible t -structur e ( C ≥ 0 , C ≤ 0 ) which is c omp atible with ﬁlter e d c olimits and is right c omplete. (3) F urthermor e, the emb e dding Groth ∞  → P r + st is symmetric monoidal, henc e induc es a ful ly faithful emb e dding CAlg(Groth ∞ )  → CAlg( P r + st ) . Deﬁnition 1.15 (Grothendieck t -structured stable ∞ -categories) . W e sa y that a presen tably t -structured stable ∞ -category ( C , C ≥ 0 ) ∈ P r t -rex st is Grothendiec k if it lies in the essential image of the embedding Groth ∞  → P r t -rex st , or equiv alently , if the t -structure on C is right complete and compatible with ﬁltered colimits (see [ SA G , Remark C.3.1.5]). Deﬁnition 1.16 (See [ SA G ] C.1.2.12) . Let C b e a prestable ∞ -category which admits ﬁnite limits. W e sa y that an ob ject X ∈ C is ∞ -connective if τ ≤ n X ≃ 0 for every integer n . W e say C is separated if every ∞ -connective ob ject of C is a zero ob ject. W e say C is left complete if it is a homotopy limit of the to wer of ∞ -categories: · · · → τ ≤ 2 C τ ≤ 1 − − → τ ≤ 1 C τ ≤ 0 − − → τ ≤ 0 C = C ♡ . In other w ords, C is left complete if it is Postnik o v complete. 10 Remark 1.17. (1) If a prestable ∞ -category C is left complete, then it is separated. (2) Let C b e a stable ∞ -category . Then C is separated if and only if C ≃ ∗ . Prop osition 1.18. L et C b e a pr estable ∞ -c ate gory with ﬁnite limits. Then: (1) The c anonic al t -structur e (Sp( C ) ≥ 0 , Sp( C ) ≤ 0 ) is hyp er c omplete if and only if C is sep ar ate d. (2) The c anonic al t -structur e (Sp( C ) ≥ 0 , Sp( C ) ≤ 0 ) is left c omplete if and only if C is left c omplete. Pr o of. (1) The “only if ” direction is ob vious. No w assume that C is separated and that X ∈ Sp ( C ) satisﬁes τ ≤ i X = 0 for every integer i . W e need to show that X = 0 . Since Sp ( C ) is right complete, we ha ve X ≃ lim − → τ ≥− n X . Ho w ever, each Σ n τ ≥− n X is ∞ -connectiv e as an ob ject in C , so τ ≥− n X = 0 by assumption, therefore X = 0 . (2) The “only if ” direction is obvious. Now assume that C is left complete. W e wish to show the diagram . . . Sp( C ) ≤ 1 Sp( C ) Sp( C ) ≤ 0 is a limit diagram of ∞ -categories. Since Sp( C ) is right complete, we hav e that Sp( C ) lim ← − τ ≥− n − − − − − − → lim ← − Sp( C ) ≥− n is an equiv alence of ∞ -categories. How ev er, the diagram . . . Sp( C ) [ − n, 1] Sp( C ) ≥− n Sp( C ) [ − n, 0] is a limit diagram of ∞ -categories by the left completeness of C , so by [ HTT , Lemma 5.5.2.3] we are done. Deﬁnition 1.19. Let C b e a prestable ∞ -category with ﬁnite limits. W e will sa y that a to wer X • : ( Z ▷ ≥ 0 ) op → C is highly connected if, for ev ery n ≥ 0 , there exists an integer k suc h that the induced map τ ≤ n X ∞ → τ ≤ n X k ′ is an equiv alence for k ′ ≥ k . W e will say that a preto wer Y • : Z op ≥ 0 → C 11 is highly connected if, for every n ≥ 0 , there exists an in teger k suc h that the map τ ≤ n Y k ′′ → τ ≤ n Y k ′ is an equiv alence for k ′′ ≥ k ′ ≥ k . It is clear that every Postnik o v (pre)tow er is highly connected. Prop osition 1.20. Assume C is a left c omplete pr estable ∞ -c ate gory with c ountable limits. L et Y • : Z op ≥ 0 → C b e a highly c onne cte d pr etower. Then: (1) F or any n ≥ 0 , ther e exists an inte ger k such that for any l ≥ k the natur al pr oje ction lim ← − Y • → Y l has n -c onne ctive c oﬁb er. (2) Supp ose further that C admits a monoidal structur e C ⊗ which is c omp atible with ﬁnite c olimits. F or any obje ct X ∈ C , the natur al maps X ⊗ lim ← − Y • → lim ← − ( X ⊗ Y • ) and lim ← − Y • ⊗ X → lim ← − ( Y • ⊗ X ) ar e e quivalenc es. Pr o of. (1) Let us analyze the Postnik o v to w ers of the ob jects Y m . F or each integer d , let τ ≤ d denote the truncation functor. By the deﬁnition of a highly connected pretow er, the connectivity of the coﬁb er of the transition maps Y m → Y l tends to inﬁnit y . This means that for an y ﬁxed degree d ≥ 0 , there exists an in teger K ( d ) suc h that for all m > l ≥ K ( d ) , the coﬁb er of Y m → Y l is ( d + 1) -connectiv e. Consequently , applying the truncation functor yields an equiv alence τ ≤ d Y m ∼ − → τ ≤ d Y l for all m > l ≥ K ( d ) . In other w ords, for an y ﬁxed d , the tow er { τ ≤ d Y m } m is essentially constant (pro-constant), and its limit is ac hiev ed at the ﬁnite stage K ( d ) . Since C is left complete, ev ery ob ject is the limit of its Postnik o v tow er. W e can therefore form a bito wer and compute the limit of Y • b y commuting the limits: lim ← − m Y m ≃ lim ← − m lim ← − d τ ≤ d Y m ≃ lim ← − d lim ← − m τ ≤ d Y m ! Let X = lim ← − m Y m . Because the to wer { τ ≤ d Y m } m stabilizes, the limit ov er m comm utes with the truncation τ ≤ d . Thus, we obtain a canonical equiv alence for the truncations: τ ≤ d X ≃ lim ← − m τ ≤ d Y m ≃ τ ≤ d Y l for an y l ≥ K ( d ) . No w, for any given n ≥ 0 , let k = K ( n − 1) . F or any l ≥ k , the natural pro jection X → Y l induces an equiv alence on their ( n − 1) -truncations: τ ≤ n − 1 X ∼ − → τ ≤ n − 1 Y l This precisely means that the coﬁb er of the pro jection lim ← − Y • → Y l has a v anishing ( n − 1) -truncation, i.e., the coﬁb er is n -connective. (2) Let C b e the coﬁb er of the natural map X ⊗ lim ← − Y • → lim ← − ( X ⊗ Y • ) . Since C is left complete, to show C ≃ 0 (and thus the map is an equiv alence), it suﬃces to show that C is m -connectiv e for 12 arbitrarily large m . F or any index l , consider the following commutativ e diagram: X ⊗ lim ← − Y • lim ← − ( X ⊗ Y • ) X ⊗ Y l p l q l Giv en any integer m ≥ 0 , by (1), we can ﬁnd a suﬃcien tly large l suc h that the coﬁb er of lim ← − Y • → Y l is m -connectiv e. Since X ∈ C is alwa ys ( 0 -)connective, and taking the tensor pro duct with a connective ob ject preserves m -connectiv e ob jects, the coﬁb er of p l is m -connectiv e. Similarly , since X is connective, the tow er X ⊗ Y • is also a highly connected pretow er. Applying (1) to this new tow er, for l large enough, the coﬁb er of q l is also m -connective. The commutativ e triangle induces a ﬁb er sequence of coﬁb ers: C → coﬁb( p l ) → coﬁb( q l ) Since b oth coﬁb ( p l ) and coﬁb ( q l ) can b e made m -connectiv e by choosing l large enough, their ﬁb er C is at least ( m − 1) -connectiv e. Since m w as arbitrary , C is m -connectiv e for all m . By left completeness (h yp ercompleteness suﬃces here), w e conclude C ≃ 0 , hence the arrow is an equiv alence. The argumen t for lim ← − Y • ⊗ X → lim ← − ( Y • ⊗ X ) is similar. 1.2 Algebra theory in Grothendieck ab elian categories A standard and p ow erful technique for studying deriv ed algebraic geometry is to reduce structural questions to their discrete comp onents via the π 0 functor. In this subsection, we establish the 1- categorical foundations of algebra within a symmetric monoidal Grothendieck ab elian category A ⊗ . W e introduce the notion of 1-pro jectiv e rigidity and provide 1-categorical analogues for ﬂatness, ﬁnitely presen ted maps, and Lazard’s theorem, whic h will serv e as the discrete shadows for our higher categorical generalizations. Some of them hav e b een developed in [ TV09 ], [ Ban12 ] and [ Ban17 ]. Deﬁnition 1.21. Let Groth 1 ⊂ P r L ad , 1 denote the full sub category of presentable additive 1-categories spanned b y those ab elian categories such that ﬁltered colimits commute with ﬁnite limits in it. W e call an ob ject in Groth 1 a Grothendiec k ab elian category . Con v en tion 1.22. Throughout Section 1.2 , we ﬁx a symmetric monoidal Grothendieck ab elian category A ⊗ ∈ CAlg(Groth 1 ) . Remark 1.23. F or a Grothendieck prestable ∞ -category C ∈ Groth ∞ , the heart C ♡ is a Grothendiec k Ab elian category . Remark 1.24. Groth 1 is closed under the Lurie tensor pro duct on P r L b y [ SAG , Theorem C.5.4.16]. W e call an ob ject in CAlg (Groth 1 ) a symmetric monoidal Grothendieck ab elian category . Deﬁnition 1.25. Let R ∈ Alg ( A ) . (1) W e say a left R -mo dule M is ﬁnitely presented if M is a compact ob ject in Mo d R ( A ) . (2) W e say a left R -mo dule M is (faithfully) 1-ﬂat if the relativ e tensor product functor ( − ) ⊗ R M : RMo d R ( A ) → A is (conserv ative) left exact. (3) W e deﬁne a left ideal I of R as a left R -submo dule of R . 13 Prop osition 1.26. L et f : A → B ∈ CAlg( A ) b e a faithful ly 1-ﬂat map. Then (1) F or any A -mo dule M , the map M ≃ M ⊗ A A → M ⊗ A B is monomorphic. In p articular, A → B is monomorphic. (2) Cok er( f ) is a 1-ﬂat A -mo dule. Pr o of. (1) Let N b e the k ernel of M → M ⊗ A B . Considering the follo wing diagram. N N ⊗ A B M M ⊗ A B i i ⊗ A B Then b y the base change adjoint we conclude that the map N ⊗ A B → M ⊗ A B is zero. The faithful 1-ﬂatness implies i = 0 and hence N = 0 . (2) It follows immediately from (1) and snake lemma. Deﬁnition 1.27. Let B ∈ Groth 1 b e a Grothendiec k ab elian category . (1) W e say an ob ject X ∈ B is 1-pro jectiv e if it is a pro jectiv e ob ject in the ordinary sense of an ab elian category . (2) W e sa y B is 1-pro jectiv ely generated if B is generated by a small set of compact 1-pro jectiv e ob jects under small colimits. (3) If B ∈ CAlg ( Groth 1 ) is a symmetric monoidal Grothendieck ab elian category , then w e say B ⊗ is 1-pro jectiv ely rigid if B is 1-pro jectiv ely generated and the dualizable ob jects in B coincide with compact 1-pro jectiv e ob jects. Remark 1.28. W e use the terminology “1-pro jectiv e” to distinguish it from the “pro jective” in the sense of [ HTT , Deﬁnition 5.5.8.18] for an ∞ -category . Generally they do not agree in an ab elian 1-category [see HTT , Example 5.5.8.21]. Prop osition 1.29. Supp ose that A is 1-pr oje ctively gener ate d and R ∈ CAlg ( A ) . Then: (1) A ny R -mo dule M c an b e written as a pushout in Mo d R ( A ) as the fol lowing form P 1 0 P 0 M wher e P 1 , P 0 ∈ Mo d R ( A ) ar e 1-pr oje ctive R -mo dules. (2) A n R -mo dule M is ﬁnitely pr esente d if and only if P 1 , P 0 c an b e pr omote d to c omp act 1-pr oje ctive R -mo dules. Pr o of. (1) Let C = Mo d R ( A ) . W e ﬁrst pro v e that any R -mo dule M can b e written as the desired pushout. Since C has enough 1-pro jectiv e ob jects, for any given R -mo dule M , there exists a 1-pro jectiv e R -mo dule P 0 and an eﬀectiv e epimorphism p : P 0 ↠ M . Consider the kernel of p , whic h is given by the ﬁb er pro duct: K ≃ P 0 × M 0 . 14 Again, since C has enough 1-pro jectiv e ob jects, for the ob ject K , there exists a 1-pro jectiv e R -mo dule P 1 and an eﬀective epimorphism q : P 1 ↠ K . By comp osing q with the canonical map K → P 0 , w e obtain the follo wing pushout square: P 1 0 P 0 M (2) Next, we prov e that an R -mo dule M is ﬁnitely presented if and only if P 1 , P 0 can b e chosen to b e compact 1-pro jectiv e R -mo dules. ( ⇐ = ) Assume the pushout square exists where P 1 and P 0 are compact 1-pro jectiv e R -mo dules. In any presen table category , the full sub category of compact ob jects is closed under ﬁnite colimits. A pushout is a ﬁnite colimit. Since 0 , P 1 , and P 0 are all compact ob jects, the ob ject M formed by their pushout m ust also b e a compact ob ject. ( = ⇒ ) Assume M is ﬁnitely presented (i.e., M is a compact ob ject in C ). W e need to sho w that P 1 and P 0 can b e c hosen to b e compact. First, we construct P 0 . Since C is 1-pro jectiv ely generated, we can ﬁnd an epimorphism ` i ∈ I G i ↠ M , where each G i is a compact 1-pro jectiv e ob ject. Since M is compact, the iden tity map id M factoring through this ﬁltered colimit (the ﬁltered p oset of ﬁnite sub-copro ducts) must factor through a ﬁnite stage. That is, there exists a ﬁnite subset F ⊂ I such that the map factors as: a i ∈ F G i ↠ M . Let P 0 = ` i ∈ F G i . Being a ﬁnite copro duct of compact 1-pro jectiv e ob jects, P 0 is itself a compact 1-pro jectiv e R -mo dule. Next, we construct P 1 . Consider the kernel K ≃ P 0 × M 0 . Because C is generated by compact 1-pro jectiv e ob jects, w e can express K as a ﬁltered colimit of ﬁnitely generated submo dules { Q α } , where “ﬁnitely generated” means Q α admits an epimorphism from a compact 1- pro jectiv e mo dule. F or eac h α , deﬁne M α as the cokernel of Q α  → P 0 . Since colimits commute with colimits, w e ha v e lim − → M α ≃ M . Using the compactness of M again, the identit y morphism id M : M → M m ust factor through some ﬁnite stage M k in this ﬁltered system. This forces the generating relations to b e entirely captured at this ﬁnite stage. Consequently , K = Q k for some k . By construction, Q k admits an epimorphic cov er P 1 → Q k from a compact 1-pro jective R -mo dule. Thus, we hav e successfully constructed the righ t exact sequence P 1 → P 0 → M → 0 . Prop osition 1.30. Supp ose that A ⊗ is 1-pr oje ctively rigid. L et R b e in CAlg ( A ≥ 0 ) . Then Mo d R ( A ) ⊗ is 1-pr oje ctively rigid to o. Pr o of. Since the symmetric monoidal functor A ⊗ R ⊗ ( − ) − − − − → Mo d R ( A ) ⊗ preserv es compact 1-pro jectiv e ob jects and dualizable ob jects, we conclude that (1) The unit R is dualizable in Mo d R ( A ) . (2) If P ∈ A is compact 1-pro jectiv e, then R ⊗ P is dualizable in Mo d R ( A ) . 15 So the full sub category of dualizable ob jects Mo d R ( A ) d con tains { R ⊗ X | X ∈ A 1 − cpro j } . Then combining Lemma 2.1 (2) and Prop osition A.11 (2)(3), we get Mo d R ( A ) 1 − cpro j ⊂ Mo d R ( A ) d . Finally , the fact that the unit R is compact 1-pro jectiv e implies the equality Mo d R ( A ) 1 − cpro j = Mo d R ( A ) d . Lazard’s theorem ov er commutativ e algebras app eared in [ Ste23 , Prop. 2.2.22]. W e ﬁnd the argument there also w orks in the non-comm utativ e setting. Theorem 1.31 (Lazard’s theorem) . Supp ose that A ⊗ is 1-pr oje ctively rigid. L et R ∈ Alg ( A ) and M b e a left R -mo dule of A . Then: (1) If M is 1-pr oje ctive, then M is 1-ﬂat. (2) M is c omp act 1-pr oje ctive if and only if it is left dualizable in LMo d R ( A ) . (3) M is 1-ﬂat if and only if it is a ﬁlter e d c olimit of c omp act 1-pr oje ctive left R -mo dules. Pr o of. (1) Since 1-ﬂat mo dules are closed under small copro ducts and retractions, we reduce to the case M = R ⊗ P where P ∈ A 1 − cpro j is compact 1-pro jective. It b ecomes easy b ecause ( − ) ⊗ R ( R ⊗ P ) ≃ ( − ) ⊗ P reduces to the case R = 1 , whic h follo ws from the dualizabilit y of P in A . (2) By Corollary A.16 , we see that left dualizable ob jects are closed under ﬁnite copro ducts and retracts. W e observe that ev ery R ⊗ P is left dualizable (given by P ∨ ⊗ R ), whic h prov es “only if ” direction. F or the “if ” direction, if M is left dualizable, then it follows from Map LMod R ( A ) ( M , − ) ≃ Map A ( 1 , ∨ M ⊗ R − ) and compact 1-pro jectivit y of the unit. (3) It i s a parallel argument with Theorem 3.20 . Corollary 1.32. Supp ose that A ⊗ is 1-pr oje ctively rigid. L et R ∈ Alg ( A ) and let M b e a left R -mo dule. Then the fol lowing ar e e quivalent: (1) M is a c omp act 1-pr oje ctive left R -mo dule. (2) M is a ﬁnitely pr esente d and 1-ﬂat left R -mo dule. Pr o of. The direction (1) ⇒ (2) is obvious. F or (2) ⇒ (1) , by Theorem 1.31 M can b e written as the ﬁltered colimit of a set of compact 1-pro jectiv e left R -mo dules. Then the compactness implies M is the retraction of some compact 1-pro jective mo dule, and hence compact 1-pro jectiv e to o. Deﬁnition 1.33. (1) W e say a map R → S ∈ CAlg ( A ) is of ﬁnite presen tation if S is a compact ob ject in CAlg( A ) R/ . Prop osition 1.34. Supp ose that A is c omp actly gener ate d. Then a map R → S ∈ CAlg ( A ) is of ﬁnite pr esentation if and only if S c an b e written as a pushout in CAlg ( A ) as the fol lowing form Sym ∗ R ( N ) R Sym ∗ R ( M ) S α wher e M , N ∈ Mo d R ( A ) ω ar e c omp act R -mo dules and α is the natur al augmentation. (Note that φ her e is not ne c essarily induc e d by a map of N → M .) F urthermor e, if A is 1-pr oje ctively gener ate d, then M , N c an b e chosen as c omp act 1-pr oje ctive R -mo dules. 16 Pr o of. The pro of is parallel with the pro of of Prop osition 4.17 . Prop osition 1.35. L et R → S ∈ CAlg ( A ) b e an epimorphism of c ommutative algebr as. Then the for getful functor Mo d S ( A ) → Mo d R ( A ) is ful ly faithful. Pr o of. Since R → S is an epimorphism is equiv alen t to that S is an idemp otent commutativ e R -algebra, the result follo ws immediately from [ HA , Prop. 4.8.2.10]. Prop osition 1.36. A faithful ly 1-ﬂat epimorphism in CAlg( A ) is an isomorphism. Pr o of. The epimorphism implies the map R ⊗ R S → S ⊗ R S is an isomorphism, so by the fully faithful 1-ﬂatness, R → S is an isomorphism. 2 Flatness and faithful ﬂatness The main goal of this section is to study the ﬂatness in the setting of ttt - ∞ -categories. W e are inspired b y equiv alen t conditions of the ﬂatness ov er structured ring sp ectra app eared in [ HA , Theorem 7.2.2.15]. 2.1 t -structure on the category of mo dules Giv en a ttt - ∞ -category A and a connective algebra R ∈ Alg ( A ≥ 0 ) , the ∞ -category of left R -mo dules naturally inherits an asso ciated t -structure. In this subsection, we demonstrate that LMo d R ( A ) inherits the completeness and pro jectiv e generation prop erties of the base category A . W e establish the pre- cise compatibilities b etw een the symmetric monoidal structure, the mo dule structure, and P ostniko v truncations, setting the stage for robust homological algebra ov er R . Lemma 2.1. L et C F ⇄ G D b e an adjoint p air of ∞ -c ate gories. (1) Assume κ is a r e gular c ar dinal, C is κ -pr esentable, and D is lo c al ly smal l and admits smal l c olimits. If G is c onservative and pr eserves smal l κ -ﬁlter e d c olimits, then D is κ -pr esentable. F urthermor e, D κ is the smal lest ful l sub c ate gory gener ate d by F ( C κ ) under κ -smal l c olimits and r etr actions. Conse quently, D is gener ate d by the image of F under smal l c olimits. (2) (See [ HA , Corollary 4.7.3.18]). Assume D admits smal l ﬁlter e d c olimits and ge ometric r e alizations, and G pr eserves b oth. Also, assume C is pr oje ctively gener ate d 2 If the functor G is c onservative, then D is pr oje ctively gener ate d. An obje ct D ∈ D is c omp act and pr oje ctive if and only if ther e exists a c omp act pr oje ctive obje ct C ∈ C such that D is a r etr act of F ( C ) . Henc e, D is gener ate d by the image of F under smal l c olimits. Pr o of. W e ﬁrst pro v e (1). Let D 0 ⊂ D b e the smallest full sub category generated by F ( C κ ) under ﬁnite colimits and retractions. Then the inclusion D 0 ⊂ D extends to a fully faithful embedding F 2 : Ind κ ( D 0 )  → D (b y [ HTT , p. 5.3.5.10]). Since F preserv es small colimits, it admits a right adjoin t H ([ HTT , Prop osition 5.5.1.9]). Th us, we hav e the following factorization of adjoint pairs: C F 1 ⇄ G 1 Ind κ ( D 0 ) F 2 ⇄ G 2 D 2 That means C ≃ P Σ ( C cpro j ) ; see [ HTT , Deﬁnition 5.5.8.23]. 17 It will therefore suﬃce to show that the functor G 2 is conserv ativ e. Let α : X → Y b e a morphism in D suc h that G 2 ( α ) is an equiv alence. W e aim to show that α is an equiv alence. F or this, since C is κ -compactly generated, it will suﬃce to show that α induces a homotopy equiv alence θ : Map C ( C, G ( X )) → Map C ( C, G ( Y )) for ev ery κ -compact ob ject C ∈ C . This map can b e iden tiﬁed with θ : Map Ind κ ( D 0 ) ( F 1 ( C ) , G 2 ( X )) → Map Ind κ ( D 0 ) ( F 1 ( C ) , G 2 ( Y )) Our assumption that G 2 ( α ) is an equiv alence guarantees that θ is a homotopy equiv alence, as desired. F or (2), the argument is entirely parallel. See also [ HA , Cor. 4.7.3.18]. Remark 2.2. The condition in (2) guarantees that D is locally small b ecause it is monadic ov er C b y the Barr-Bec k-Lurie theorem (see [ HA , Thm. 4.7.3.5]). Corollary 2.3. L et R ∈ Alg ( A ) . Applying Lemma 2.1 to the adjoint p air A R ⊗− ⇄ LMo d R ( A ) , we obtain: (1) If A is κ -pr esentable for some r e gular c ar dinal κ , then so is LMo d R ( A ) . (2) LMo d R ( A ) is pr esentable. (3) LMo d R ( A ) is gener ate d by { R ⊗ X | X ∈ A } under smal l c olimits. Remark 2.4. By [ HA , Prop osition 7.1.1.4], LMo d R ( A ) is stable for any R ∈ Alg( A ) . Deﬁnition 2.5. Let C b e a stable ∞ -category equipp ed with a t -structure. W e say that it is hypercom- plete, if for an ob ject X ∈ C , the condition τ ≤ n X = 0 for every integer n implies X = 0 . Example 2.6. Let X b e a hypercomplete ∞ -top os. Then the natural t -structure (Sh v( X , Sp) ⊗ , Shv( X , Sp) ≥ 0 ) dev elop ed in [ SAG , Prop osition 1.3.2.7] is h yp ercomplete b y [ SAG , Prop osition 1.3.3.3]. Prop osition 2.7. L et R b e in Alg ( A ≥ 0 ) . Then LMo d R ( A ) is a pr esentable stable ∞ -c ate gory which admits a natur al ac c essible t -structur e ( LMo d R ( A ) ≥ 0 , LMo d R ( A ) ≤ 0 ) satisfying the fol lowing pr op erties 3 : (1) LMo d R ( A ) ≥ 0 and LMo d R ( A ) ≤ 0 ar e the inverse images of A ≥ 0 and A ≤ 0 under the pr oje ction θ : LMo d R ( A ) → A . (2) The natur al inclusion LMo d R ( A ≥ 0 )  → LMo d R ( A ) induc es an e quivalenc e LMo d R ( A ≥ 0 ) ∼ − → LMo d R ( A ) ≥ 0 . (3) The functor τ ≤ n : A ≥ 0 → A [0 ,n ] induc es an e quivalenc e LMo d R ( A ) [0 ,n ] ∼ − → LMo d τ ≤ n R ( A [0 ,n ] ) . In p articular, the π 0 functor induc es an e quivalenc e LMo d R ( A ) ♡ ∼ − → LMo d π 0 R ( A ♡ ) . (4) If A is left (r esp. right, r esp. hyp er) c omplete, then so is LMo d R ( A ) . (5) If the t -structur e on A is c omp atible with ﬁlter e d c olimits, me aning A ≤ 0 ⊂ A is close d under ﬁlter e d c olimits, then so is the induc e d t -structur e on LMo d R ( A ) . (6) If A ≥ 0 is pr oje ctively gener ate d, then so is LMo d R ( A ) ≥ 0 . 3 See also a similar discussion in [ AN21 , App endix]. 18 Pr o of. W e ﬁrst prov e (1). It follows immediately from the deﬁnitions that the full sub category LMo d R ( A ) ≥ 0 ⊂ LMo d R ( A ) is closed under small colimits and extensions. Also, note that LMo d R ( A ) ≥ 0 is presen table since the follo wing is a pullbac k square in P r L : LMo d R ( A ) ≥ 0 LMo d R ( A ) A ≥ 0 A ⌜ θ Using [ HA , Prop. 1.4.4.11], we deduce the existence of an accessible t -structure (LMo d R ( A ) ≥ 0 , LMo d R ( A ) ′ ) on LMo d R ( A ) . T o complete the pro of, it will suﬃce to show that LMo d R ( A ) ′ = LMo d R ( A ) ≤ 0 . Supp ose ﬁrst that N ∈ LMo d R ( A ) ′ . Then the mapping space Map LMod R ( A ) ( M , N ) is discrete for ev ery ob ject M ∈ LMo d R ( A ) ≥ 0 . In particular, for ev ery connective ob ject X ∈ A ≥ 0 , the mapping space Map LMod R ( A ) ( R ⊗ X , N ) ≃ Map A ( X, θ ( N )) is discrete, so that θ ( N ) ∈ A ≤ 0 , and therefore N ∈ LMod R ( A ) ≤ 0 . Con versely , supp ose that N ∈ LMo d R ( A ) ≤ 0 . W e wish to prov e that N ∈ LMo d R ( A ) ′ . Let C denote the full sub category of LMo d R ( A ) spanned by those ob jects M ∈ LMo d R ( A ) for which the mapping space Map LMod R ( A ) ( M , N ) is discrete. W e wish to prov e that C con tains LMo d R ( A ) ≥ 0 . Firstly , w e ha ve that θ induces a functor LMo d R ( A ) ≥ 0 → A ≥ 0 whic h is conserv ative and preserves small colimits; moreo ver, this functor has a left adjoint F r , given informally by the formula F r ( X ) ≃ R ⊗ X . Using Lemma 2.1 , we conclude that LMo d R ( A ) ≥ 0 is generated under small colimits by the essential image of F r . Since C is stable under colimits, it will suﬃce to show that C con tains the essential image of F r . Un winding the deﬁnitions, w e are reduced to pro ving that the mapping space Map LMod R ( A ) ( F ( X ) , N ) ≃ Map A ( X, θ ( N )) is discrete for every connective ob ject X in A ≥ 0 , which is equiv alen t to our assumption that N ∈ LMo d R ( A ) ≤ 0 . This completes the pro of of (1). F or (2), the pro of follows directly from the deﬁnition. F or (3), we observe that we hav e a natural factorization: LMo d R ( A ) [0 ,n ] LMo d R ( A ≥ 0 ) LMo d τ ≤ n R ( A [0 ,n ] ) F 0 F It suﬃces to prov e that F 0 is fully faithful and essentially surjective. It is easy to see that F and F 0 preserv e colimits. W e wish to prov e that, for a ﬁxed N ∈ LMo d R ( A ) [0 ,n ] , the full sub category D of LMo d R ( A ) ≥ 0 spanned b y those ob jects M for which the map Map LMod R ( A ) ( M , N ) → Map LMod τ ≤ n R ( A [0 ,n ] ) ( F ( M ) , F ( N )) is an equiv alence. It is easy to see that D is stable under colimits and con tains R ⊗ X for all X ∈ A ≥ 0 . Lemma 2.1 sho ws that D = LMo d R ( A ) ≥ 0 . In particular, F 0 is fully faithful. 19 It remains to show that F 0 is essen tially surjective. Since F 0 is fully faithful and preserves small colimits, the essential image of F 0 is closed under small colimits. By applying Lemma Lemma 2.1 to A [0 ,n ] ⇄ LMo d τ ≤ n R ( A [0 ,n ] ) , it will therefore suﬃce to show that ev ery free left τ ≤ n R -mo dule τ ≤ n R ⊗ Y where Y ∈ A [0 ,n ] b elongs to the essential image of F 0 , where ⊗ denotes the tensor pro duct in A [0 ,n ] . W e no w conclude b y observing that F ( R ⊗ X ) ≃ τ ≤ n R ⊗ τ ≤ n X . (4) and (5) are concluded by the fact that θ : LMo d R ( A ) → A is t -exact, conserv ative, and preserves small colimits and limits. (6) is concluded by Lemma 2.1 (2) applied to the adjoint pair A ≥ 0 F ⇄ G LMo d R ( A ) ≥ 0 . Corollary 2.8. By Pr op osition 2.7 , we se e that if the pr esentably t -structur e d stable ∞ -c ate gory ( A , A ≥ 0 ) is Gr othendie ck, then so is (Mo d R ( A ) , Mo d R ( A ) ≥ 0 ) for any R ∈ CAlg( A ≥ 0 ) . Corollary 2.9. L et R ∈ Alg E k +1 ( A ≥ 0 ) b e a c onne ctive E k +1 -algebr a wher e 1 ≤ k ≤ ∞ . Then the pr esentably E k -monoidal c ate gory LMo d R ( A ) ⊗ → E ⊗ k satisﬁes: (1) The natur al t -structur e (LMo d R ( A ) ≥ 0 , LMo d R ( A ) ≤ 0 ) is c omp atible with the monoidal structur e. (2) The natur al inclusion LMo d R ( A ≥ 0 )  → LMo d R ( A ) is an E k -monoidal functor which induc es an e quivalenc e LMo d R ( A ≥ 0 ) ⊗ ∼ − → LMo d R ( A ) ⊗ ≥ 0 of E k -monoidal c ate gories. (3) The symmetric monoidal functor τ ⊗ ≤ n : A ⊗ ≥ 0 → A ⊗ [0 ,n ] induc es an e quivalenc e LMo d R ( A ) ⊗ [0 ,n ] ∼ − → LMo d τ ≤ n R ( A [0 ,n ] ) ⊗ of E k -monoidal ( n + 1) -c ate gories. In p articular, the π 0 functor induc es an e quivalenc e of E k -monoidal 1-c ate gories LMo d R ( A ) ♡ , ⊗ ∼ − → LMo d π 0 R ( A ♡ ) ⊗ . Note that when k > 1 , E k -algebr as in A ♡ ar e E ∞ -algebr as, so LMo d π 0 R ( A ♡ ) ⊗ is symmetric monoidal in this c ase. Remark 2.10. (1) When R ∈ CAlg ( A ≥ 0 ) is a connective E ∞ -algebra, the symmetric monoidal ∞ - category of left mo dules LMo d R ( A ) ⊗ with the induced t -structure forms a new ttt - ∞ -category (LMo d R ( A ) ⊗ , LMo d R ( A ) ≤ 0 ) . (2) Prop osition 2.7 also holds for the right mo dule category RMo d R ( A ) and the bimo dule category R BMo d S ( A ) when R , S are connective. Con v en tion 2.11. In the case where R ∈ CAlg ( A ) is commutativ e, we will simply denote LMo d R ( A ) ⊗ b y Mo d R ( A ) ⊗ . 2.2 Flat mo dules and algebras Motiv ated by the classical equiv alen t conditions of ﬂatness o ver structured ring sp ectra, we deﬁne ﬂat mo dules o ver an algebra R ∈ Alg ( A ) via the t -exactness of the relativ e tensor pro duct. This subsection explores the stability of ﬂat mo dules under base c hange, comp osition, etc. Crucially , we extend these deﬁnitions from the connective to the non-connective case, demonstrating that the ﬂatness can b e reliably trac ked through connectiv e cov ers. W e ﬁrst consider the connective case. Deﬁnition 2.12 (The connective case) . Let R ∈ Alg ( A ≥ 0 ) b e a connective E 1 -algebra. (1) W e say a left R -mo dule M is ﬂat if the relative tensor pro duct functor ( − ) ⊗ R M : RMo d R ( A ) → A is t -exact. (2) W e say a left R -mo dule M is faithfully ﬂat if the relative tensor pro duct functor ( − ) ⊗ R M : RMo d R ( A ) → A is t -exact and conserv ativ e. 20 (3) If R is E ∞ and f : R → S is a morphism in CAlg ( A ) , we say f is (faithfully) ﬂat if S is a (faithfully) ﬂat R -mo dule. Remark 2.13. If M is ﬂat on a connective E 1 -algebra R ∈ Alg ( A ≥ 0 ) , then M ≃ R ⊗ R M itself is connectiv e. Remark 2.14. If A is Grothendieck, then a connective left R -mo dule M for some R ∈ Alg ( A ≥ 0 ) is ﬂat if and only if the tensor pro duct functor ( − ) ⊗ R M : RMo d R ( A ≥ 0 ) → A ≥ 0 is left exact b y Prop osition 1.13 . Prop osition 2.15. Assume that A is Gr othendie ck. L et R ∈ Alg ( A ≥ 0 ) b e a c onne ctive E 1 -algebr a and M b e a c onne ctive left R -mo dule. Then the fol lowing c onditions ar e e quivalent: (1) M is ﬂat. (2) The tensor pr o duct functor ( − ) ⊗ R M is left t -exact (me aning that it sends the c o c onne ctive p art to the c o c onne ctive p art). (3) The tensor pr o duct functor ( − ) ⊗ R M sends discr ete obje cts to discr ete obje cts. Pr o of. The direction (1) ⇔ (2) and (2) ⇒ (3) are ob vious. Now we claim that (3) ⇒ (2) . Giv en a co connective right R -mo dule M ∈ RMo d R ( A ) ≤ 0 , w e wish to show that N ⊗ R M ∈ A ≤ 0 . Since A is right complete, w e hav e that M ≃ lim − → τ ≥− i M . Now we will prov e that N ⊗ R τ ≥− n M ∈ A [ − n, 0] inductiv ely . The case n = 0 is true by the assumption. No w assume that for n − 1 ≥ 0 it is true, we need to show that N ⊗ R τ ≥− n M ∈ A [ − n, 0] , which is by observing that the ﬁrst and third items in the follo wing exact sequence N ⊗ R τ ≥− ( n − 1) M → N ⊗ R τ ≥− n M → N ⊗ R π − n M b elong to A [ − n, 0] . Since the t -structure is compatible with ﬁltered colimits, w e hav e that N ⊗ R M ≃ lim − → N ⊗ R τ ≥− i M b elong to A ≤ 0 . Prop osition 2.16. L et R ∈ Alg ( A ≥ 0 ) b e a c onne ctive E 1 -algebr a. Then: (1) The ful l sub c ate gory of ﬂat mo dules LMo d R ( A ) f l ⊂ LMo d R ( A ) is close d under ﬁnite c opr o ducts, r etr actions, and extensions. If the t -structur e on A is c omp atible with ﬁlter e d c olimits, then LMo d R ( A ) f l ⊂ LMo d R ( A ) is furthermor e close d under ﬁlter e d c olimits. (2) If R ∈ CAlg ( A ≥ 0 ) is E ∞ , then the ful l sub c ate gory of ﬂat mo dules Mo d R ( A ) f l ⊂ Mo d R ( A ) c ontains the unit and is close d under tensor pr o duct, and henc e forms a symmetric monoidal ful l sub c ate gory. (3) If R ∈ CAlg ( A ≥ 0 ) is E ∞ and M ∈ Mo d R ( A ) is a dualizable R -mo dule, then M is ﬂat if and only if b oth M and the dual M ∨ ar e c onne ctive R -mo dules. Pr o of. (1) and (2) are obvious by deﬁnition of ﬂatness. (3) Assume that M is ﬂat, then M is connectiv e by the remark ab o ve. Since Map Mod R ( A ) ( M ∨ , N ) ≃ Map Mod R ( A ) ( R, M ⊗ R N ) is con tractible for an y ( − 1) -truncated N , M ∨ is connectiv e to o. No w assume b oth M and the dual M ∨ are connectiv e. Since M is connectiv e, the tensor product ( − ) ⊗ R M is right t -exact. So it suﬃces to chec k the left t -exactness of ( − ) ⊗ R M . Let Q b e a connective R -mo dule and N is b e a ( − 1) -truncated R -mo dule. Then Map Mod R ( A ) ( Q ⊗ R M ∨ , N ) ≃ Map Mod R ( A ) ( Q, M ⊗ R N ) 21 is con tractible. So the functor ( − ) ⊗ R M is indeed left t -exact. Prop osition 2.17. If R → S ∈ Alg( A ≥ 0 ) b e a morphism of c onne ctive E 1 -algebr as, then (1) The r elative tensor pr o duct S ⊗ R ( − ) : LMo d R ( A ) → LMo d S ( A ) sends (faithful ly) ﬂat mo dules to (faithful ly) ﬂat mo dules. (2) If S is ﬂat as a left R -mo dule, then the for getful functor θ : LMo d S ( A ) → LMo d R ( A ) sends ﬂat mo dules to ﬂat mo dules. If furthermor e S is faithful ly ﬂat as a left R -mo dule, then the for getful functor θ : LMo d S ( A ) → LMo d R ( A ) pr eserves faithful ly ﬂat mo dules. Pr o of. (1) Let M ∈ LMo d R ( A ) . W e observ e that ( − ) ⊗ S ( S ⊗ R M ) ≃ ( − ) ⊗ R M . (2) Giv en N ∈ LMo d S ( A ) , w e observ e that ( − ) ⊗ R θ ( N ) ≃ ( − ) ⊗ R S ⊗ S N . W e now start to inv estigate the non-connective ﬂatness. Deﬁnition 2.18 (The non-connective case) . Let R ∈ Alg ( A ) and θ : LMo d R ( A ) → LMo d τ ≥ 0 R ( A ) b e the forgetful functor. (1) W e say a left R -mo dule M is ﬂat if the counit map R ⊗ τ ≥ 0 R τ ≥ 0 θ ( M ) → M with resp ect to the follo wing comp osite adjunction LMo d τ ≥ 0 R ( A ) ≥ 0 ⇄ τ ≥ 0 LMo d τ ≥ 0 R ( A ) ⇄ θ LMo d R ( A ) is an equiv alence and τ ≥ 0 θ ( M ) is ﬂat ov er τ ≥ 0 R . (2) W e say a left R -mo dule M is faithfully ﬂat if it is ﬂat ov er R and τ ≥ 0 θ ( M ) is faithfully ﬂat o ver τ ≥ 0 R . (3) If f : R → S is a morphism in CAlg ( A ) , we say that f is a (faithfully) ﬂat morphism if S is a (faithfully) ﬂat R -mo dule. Remark 2.19. Let R ∈ Alg ( A ) b e an E 1 -algebra. Then (1) The full sub category of ﬂat mo dules LMo d R ( A ) f l ⊂ LMo d R ( A ) is closed under ﬁnite copro ducts, retractions. Note that, unlike the connectiv e case, it is not closed under extensions in general. (2) If the t -structure on A is compatible with ﬁltered colimits, then LMo d R ( A ) f l ⊂ LMo d R ( A ) is closed under ﬁltered colimits. (3) If M is a ﬂat left R -mo dule, then M is faithfully ﬂat implies that the tensor product functor ( − ) ⊗ R M is conserv ativ e. Note that, unlike the connective case, the conv erse do es not hold in general. Prop osition 2.20. L et R → S b e a map in Alg( A ) . (1) If τ ≥ 0 R → τ ≥ 0 S is an e quivalenc e, then the r elative tensor pr o duct S ⊗ R ( − ) : LMo d R ( A ) → LMo d S ( A ) r estricts to an e quivalenc e LMo d R ( A ) f l ∼ − → LMo d S ( A ) f l b etwe en the ful l sub c ate gories of ﬂat mo dules. (See [ HA , Prop. 7.2.2.16] for the case of sp ectra.) 22 (2) If R ∈ CAlg ( A ) , then the ful l sub c ate gory of ﬂat mo dules Mo d R ( A ) f l ⊂ Mo d R ( A ) c ontains the unit and is close d under tensor pr o duct, and henc e forms a symmetric monoidal ful l sub c ate gory. (3) If R → S is a morphism in CAlg ( A ) such that τ ≥ 0 R → τ ≥ 0 S is an e quivalenc e, then the b ase change induc es an e quivalenc e of symmetric monoidal c ate gories Mo d R ( A ) f l, ⊗ ∼ − → Mo d S ( A ) f l, ⊗ . Pr o of. The (2), (3) are conclusions of (1). So it suﬃces to prov e (1) in the case when R = τ ≥ 0 S . Since R is connective, the ∞ -category LMo d R ( A ) admits a t -structure. Let F ′ denote the comp osite functor LMo d R ( A ) ≥ 0 ⊂ LMo d R ( A ) F − → LMo d S ( A ) Then F ′ has a right adjoint, giv en by the comp osition G ′ = τ ≥ 0 ◦ G . Giv en M is a ﬂat left R -mo dule, w e observ e that M → G ′ F ′ ( M ) is equiv alen t by the ﬂatness of M . Now w e wish to prov e F ′ preserv es ﬂatness, i.e. F ′ G ′ F ′ ( M ) → F ′ ( M ) is equiv alen t, which is ob vious. Then we wish to prov e G ′ preserv es ﬂatness to o, which is b y deﬁnition of ﬂatness in the non-connective case. Consequently , F ′ and G ′ induce adjoin t functors LMo d f l R ( A ) F ′ ⇆ G ′ LMo d f l S ( A ) It no w suﬃces to show that the unit and counit of the adjunction are equiv alences. In other words, we m ust show: (i) F or ev ery ﬂat left R -mo dule M , the unit map M → G ′ F ′ ( M ) is an equiv alence, whic h has b een done b y the argumen t ab ov e. (ii) F or every ﬂat left S -mo dule N , the counit map F ′ G ′ ( N ) → N is an equiv alence, which is by deﬁnition of ﬂatness in the non-connective case. Prop osition 2.21. If R → S ∈ Alg( A ) is a morphism of (non-c onne ctive) E 1 -algebr as, then (1) The r elative tensor pr o duct S ⊗ R ( − ) : LMo d R ( A ) → LMo d S ( A ) sends (faithful ly) ﬂat mo dules to (faithful ly) ﬂat mo dules. (2) If S is ﬂat as a left R -mo dule, then the for getful functor θ : LMo d S ( A ) → LMo d R ( A ) sends ﬂat mo dules to ﬂat mo dules. If furthermor e S is faithful ly ﬂat as a left R -mo dule, then the for getful functor θ : LMo d S ( A ) → LMo d R ( A ) pr eserves faithful ly ﬂat mo dules. Pr o of. The (1) is deduced b y combination of Prop osition 2.17 and Prop osition 2.20 . F or (2), we claim the following diagram is right adjointable, LMo d τ ≥ 0 R ( A ) LMo d τ ≥ 0 S ( A ) LMo d R ( A ) LMo d S ( A ) b ecause S ⊗ τ ≥ 0 S ( − ) ≃ R ⊗ τ ≥ 0 R τ ≥ 0 S ⊗ τ ≥ 0 S ( − ) by ﬂatness of S o ver R . Then it reduces to the connectiv e case, whic h is Prop osition 2.17 . 23 Prop osition 2.22. (1) If f : R → S is a morphism in CAlg ( A ) , then f is ﬂat if and only if f ≥ 0 : τ ≥ 0 R → τ ≥ 0 S is ﬂat and the fol lowing diagr am τ ≥ 0 R τ ≥ 0 S R S is a pushout diagr am in CAlg ( A ) . (2) If f : R → S is a ﬂat map in CAlg ( A ≥ 0 ) , then the fol lowing diagr am is a pushout diagr am in CAlg( A ≥ 0 ) . R S τ ≤ n R τ ≤ n S Henc e τ ≤ n R → τ ≤ n S is also ﬂat for any n ≥ 0 . (3) L et f : R → S b e a (faithful) ﬂat map in CAlg ( A ) and R → A b e another map in CAlg ( A ) . Then the map A → A ⊗ R S given by the fol lowing pushout diagr am is (faithful) ﬂat. R S A A ⊗ R S Pr o of. (1) If f is ﬂat, then we ha ve S ≃ R ⊗ τ ≥ 0 R τ ≥ 0 S b y the ﬂatness of S o ver R . If the conv erse is true, then R → S is ﬂat b y Prop osition 2.21 (1). (2) Since ( − ) ⊗ R S is t -exact, the following diagram is a pushout diagram in CAlg ( A ) . R S τ ≤ n R τ ≤ n S So τ ≤ n R → τ ≤ n S is ﬂat by Prop osition 2.21 (1). (3) It follows immediately from Prop osition 2.21 . Prop osition 2.23. Given a diagr am A B C f h g in CAlg ( A ) . (1) wher e f , g ar e ﬂat morphisms, then so is the c omp osition h . (2) If h is ﬂat and g is faithful ly ﬂat, then f is ﬂat. 24 Pr o of. (1) It follows immediately from deﬁnition. (2) Considering the following diagram τ ≥ 0 A τ ≥ 0 B τ ≥ 0 C A B C τ ≥ 0 f τ ≥ 0 g f g in CAlg ( A ) . Then the right square and outer square are pushouts by Prop osition 2.22 . The faithful ﬂatness of g implies the tensor pro duct functor ( − ) ⊗ τ ≥ 0 B τ ≥ 0 C is conserv ativ e, whic h implies the left square is also a pushout. So w e reduce to the case when A, B , C are connective. No w given a co connective A -mo dule M ∈ Mod A ( A ) ≤ 0 , we wish to show that N = B ⊗ A M is also co connectiv e. How ev er C ⊗ B N is co connective by assumption, so N is co connective by faithful ﬂatness of g . Corollary 2.24. Given a pushout diagr am in CAlg ( A ) A ′ A B ′ B ϕ ψ ϕ ′ ψ ′ wher e ψ is faithful ly ﬂat. If B is ﬂat over A , then B ′ is ﬂat over A ′ . Pr o of. Since ψ is faithfully ﬂat, the morphism ψ ′ is also faithfully ﬂat. By virtue of Prop osition 2.23 (2), it will suﬃce to show that the comp osition ψ ′ ◦ φ ≃ φ ◦ ψ is ﬂat. This also follows from Prop osition 2.23 , since ψ and φ are b oth ﬂat. W e now study the relation with discrete ﬂatness. Prop osition 2.25. L et R ∈ Alg ( A ) . Then: (1) L et M ∈ LMo d R ( A ) . If M is (faithful ly) ﬂat over R , then π 0 M ∈ LMo d π 0 R ( A ♡ ) is (faithful ly) 1-ﬂat over π 0 R in the sense of Deﬁnition 1.25 . (2) Assume that A is hyp er c omplete. L et f : M → N b e a map b etwe en ﬂat left R -mo dules. If π 0 f : π 0 M → π 0 N is an e quivalenc e, then f : M → N is an e quivalenc e. (3) L et M ∈ LMo d R ( A ) b e a ﬂat left R -mo dule. Then for any n ∈ Z , the natur al map π n ( R ) ⊗ π 0 R π 0 M → π n M is an e quivalenc e in LMo d π 0 R ( A ♡ ) . (4) Assume that A is Gr othendie ck. If f : R → S is a faithful ly ﬂat morphism in CAlg ( A ) , then coﬁb( f ) is a ﬂat R -mo dule; the c onverse holds pr ovide d further that A is hyp er c omplete. Pr o of. F or (1), w e hav e that τ ≥ 0 M is ﬂat ov er τ ≥ 0 R , so it suﬃces to show the case when R, M are 25 connectiv e. Therefore by t -exactness we hav e the following factorization RMo d τ ≥ 0 R ( A ) A RMo d τ ≥ 0 R ( A ) ≥ 0 A ≥ 0 τ ≥ 0 ( − ) ⊗ R M τ ≥ 0 ( − ) ⊗ R M whic h implies that ( − ) ⊗ R M : RMo d R ( A ≥ 0 ) → A ≥ 0 is left exact. Now since π 0 : A ⊗ ≥ 0 → ( A ♡ ) ⊗ is a symmetric monoidal functor preserving geometric realizations, we hav e the comm utative diagram of relativ e tensor pro duct functors. RMo d R ( A ≥ 0 ) A ≥ 0 RMo d π 0 R ( A ♡ ) A ♡ ( − ) ⊗ R M ( − ) ⊗ π 0 R π 0 M So we conclude that ( − ) ⊗ π 0 R π 0 M is left exact since the ab ov e horizontal functor preserves discrete ob jects. F or the faithfully ﬂat case, it follows directly from deﬁnition. (2) By deﬁnition of non-connective ﬂatness, without loss of generalization we can assume that R , M and N are connectiv e. Since b oth M and N are ﬂat and A is h yp ercomplete we may reduce to proving that π n M ≃ π n R ⊗ R M π n R ⊗ R f − − − − − − → π n R ⊗ R N ≃ π n N is an equiv alence for all n ≥ 0 . How ev er, this map agrees with π n R ⊗ π 0 R π 0 M π n R ⊗ π 0 R π 0 f − − − − − − − − − → π n R ⊗ π 0 R π 0 N , whic h is an equiv alence by virtue of the fact that π 0 ( f ) is an equiv alence. (3) By deﬁnition we hav e that τ ≥ 0 M is ﬂat o ver τ ≥ 0 R and R ⊗ τ ≥ 0 R τ ≥ 0 M ≃ M . Then it follo ws b y combining the t -exactness of ( − ) ⊗ τ ≥ 0 R τ ≥ 0 M and the natural equiv alence π n ( R ) ⊗ τ ≥ 0 R τ ≥ 0 M ≃ π n ( R ) ⊗ π 0 R π 0 M . (4) Due to Prop osition 2.22 (1), we may reduce to the case where R and S are connectiv e. Let C denote coﬁb( f ) . No w assume that C is ﬂat ov er R . By Prop osition 2.15 , it suﬃces to show that C ⊗ R M is discrete for any discrete R -mo dule M . Since R ⊗ R M and S ⊗ R M are discrete, we hav e that C ⊗ R M ∈ A ≤ 1 . Therefore it suﬃces to show that π i ( C ⊗ R M ) = 0 when i  = 0 , whic h is deduced by combining the Prop osition 2.25 (1), Prop osition 1.26 and the long exact sequence asso ciated with R ⊗ R M → S ⊗ R M → C ⊗ R M . F or the con v erse implication, to see that S is ﬂat ov er R , it suﬃces to show that S ⊗ R M is discrete for an y discrete R -mo dule M , which is obvious by the coﬁb er sequence R ⊗ R M → S ⊗ R M → C ⊗ R M with the ﬁrst and third terms discrete. T o see the faithfulness, due to the hypercompleteness, it is reduced to pro ving M = 0 if M is discrete and S ⊗ R M = 0 , which is again by the coﬁb er sequence ab ov e. W arning 2.26. Suppose that R ∈ Alg ( A ) is discrete. Let M ∈ LMo d R ( A ) b e a discrete left R -mo dule. Bew are that, if π 0 M is 1-ﬂat ov er π 0 R in the sense of Deﬁnition 1.25 , this do es not imply that M is ﬂat 26 o ver R in the (derived) sense Deﬁnition 2.18 . How ev er, under the pro jectiv e rigidity assumption, this implication do es hold (see Prop osition 3.19 ). 3 Pro jectiv e mo dules 3.1 Basic prop erties Notation 3.1. Throughout Section 3.1 , we denote the following condition b y (*): (*) A is right complete and A ≥ 0 is pro jectiv ely generated, meaning A ≥ 0 ≃ P Σ ( A cpro j ≥ 0 ) . Remark 3.2. Under the condition (*), the following hold: (1) A is Grothendieck. (2) A is left complete by combining Prop osition 1.18 and [ SAG , Remark C.1.5.9]. (3) F or any R ∈ Alg ( A ≥ 0 ) , LMo d R ( A ) ≥ 0 satisﬁes the condition (*) to o. Prop osition 3.3. Supp ose that A satisﬁes the c ondition (*). L et R ∈ Alg ( A ≥ 0 ) , and let C b e the smal lest idemp otent c omplete stable sub c ate gory of LMo d R ( A ) which c ontains al l c omp act pr oje ctive left mo dules. Then C = LMo d R ( A ) c is the ful l sub c ate gory of c omp act mo dules. Pr o of. Since LMo d R ( A ) is right complete, the collection of connective cov er functors { τ ≥− n | n ≥ 0 } is join tly conserv ativ e. Therefore by Lemma 2.1 LMo d R ( A ) is generated b y { R ⊗ Σ − n P | n ≥ 0 , P ∈ A cpro j ≥ 0 } under small colimits. Then the compact generation of A implies C = LMo d R ( A ) c . Remark 3.4. As we will show in Prop osition 4.14 , the “idemp oten t-complete” condition ab ov e can b e remo ved under the stronger assumption of pro jective rigidity . W e b egin our study of pro jectiv e mo dules, by examining their b ehavior under truncations and geometric realizations. Deﬁnition 3.5. Supp ose that A satisﬁes the condition (*). Let R b e in Alg ( A ≥ 0 ) . W e sa y P ∈ LMo d R ( A ) ≥ 0 is a pro jectiv e left R -mo dule if it is a pro jectiv e ob ject in LMo d R ( A ) ≥ 0 , meaning that the corepresen table functor Map LMod R ( A ) ≥ 0 ( P , − ) : LMo d R ( A ) ≥ 0 → S preserv es geometric realizations. No w we in tro duce the following lemma, whic h is a sligh t strengthening of [ HA , Lemma 1.3.3.11(2)] 4 . Lemma 3.6. L et C and C ′ b e stable ∞ -c ate gories e quipp e d with t -structur es. Then: (1) If F : C ≥ 0 → C ′ ≥ 0 is a functor pr eserves ﬁnite c olimits, then τ ≤ n ◦ F ∼ − → τ ≤ n ◦ F ◦ τ ≤ n is a natur al e quivalenc e in F un( C ≥ 0 , C ′ [0 ,n ] ) for any n ≥ 0 . (2) If C ≥ 0 admits ge ometric r e alizations and C ′ is left c omplete, then a functor F : C ≥ 0 → C ′ ≥ 0 pr eserves ﬁnite c olimits if and only if it pr eserves ﬁnite c opr o ducts and ge ometric r e alizations. 4 The statement there assumes that b oth C and C ′ are left complete; w e show that the assumption on C is unnecessary . 27 Pr o of. (1) Since F is right exact, it preserv es susp ension. Given X ∈ C ≥ 0 , then we hav e that the sequence F ( τ ≥ n +1 X ) → F ( X ) → F ( τ ≤ n X ) is a coﬁb er sequence in C ′ ≥ 0 and that F ( τ ≥ n +1 X ) ∈ C ′ ≥ n +1 . T aking τ ≤ n , w e get the natural equiv alence τ ≤ n F ( X ) ∼ − → τ ≤ n F ( τ ≤ n X ) . (2) If F preserv es ﬁnite copro ducts and geometric realizations of simplicial ob jects, then F is right exact [ HA , Lemma 1.3.3.10]. Con versely , supp ose that F is right exact; we wish to prov e that F preserv es geometric realizations of simplicial ob jects. It will suﬃce to show that each comp osition C ≥ 0 F − → C ′ ≥ 0 τ ≤ n − − →  C ′ ≥ 0  ≤ n By the (1), in virtue of the right exactness of F , this functor is equiv alen t to the comp osition C ≥ 0 τ ≤ n − − → ( C ≥ 0 ) ≤ n τ ≤ n ◦ F − − − − →  C ′ ≥ 0  ≤ n . It will therefore suﬃce to prov e that τ ≤ n ◦ F preserv es geometric realizations of simplicial ob jects, which follo ws from [ HA , Lemma 1.3.3.10] since b oth the source and target are equiv alen t to n -categories. W e also introduce the following strengthening lemma of [ HA , Prop. 7.2.2.6] 5 . Prop osition 3.7. L et C b e a stable ∞ -c ate gory with a t -structur e such that C ≥ 0 admits ge ometric r e alizations. Given P ∈ C ≥ 0 , then the fol lowing c onditions ar e e quivalent: (1) P is pr oje ctive in C ≥ 0 . (2) F or any Q ∈ C ≥ 0 , the ab elian gr oup Ext 1 ( P , Q ) = 0 . (3) F or any Q ∈ C ≥ 0 , the ab elian gr oup Ext i ( P , Q ) = 0 when i > 0 . (4) The mapping sp e ctrum functor Map C ( P , − ) : C → Sp is t -exact. Pr o of. The implications (3) ⇒ (2) and (3) ⇔ (4) are obvious. The implication (2) ⇒ (3) follows by replacing Q by Q [ i − 1] . W e ﬁrst sho w that (1) ⇒ (2) . Let f : C → S b e the functor corepresented by P . Let M • b e a Čech nerv e for the morphism 0 → Q [1] , so that M n ≃ Q n ∈ C ≥ 0 . Then Q [1] can be iden tiﬁed with the geometric realization | M • | . Since P is pro jectiv e, f ( Q [1]) is equiv alen t to the geometric realization | f ( M • ) | . W e ha ve a surjectiv e map ∗ ≃ π 0 f ( M 0 ) → π 0 | f ( M • ) | , so that π 0 f ( Q [1]) = Ext 1 C ( P , Q ) = 0 . W e now show that (3) ⇒ (1) . That C is stable implies that f is homotopic to a comp osition C F − → Sp Ω ∞ − − → S , where F is an exact functor. Applying (3), we deduce that F is right t -exact (see [ HA , Deﬁnition 1.3.3.1]). The Lemma 3.6 implies that the induced map C ≥ 0 → Sp ≥ 0 preserv es geometric realizations of simplicial ob jects. Applying [ HA , Prop osition 1.4.3.9] that Sp Ω ∞ − − → S preserv es small sifted colimits, we conclude that f | C ≥ 0 preserv es geometric realizations as w ell. 5 The statement there assumes that C is left complete; we show that such assumption is unnecessary . 28 No w w e show that the π 0 -truncation induces an equiv alence b etw een the homotopy categories of the (compact) pro jectiv e ob jects in the connectiv e part and the (compact) 1-pro jectiv e ob jects in the heart. Prop osition 3.8 (See [ Ste23 ] Prop osition 2.4.8) . L et C b e a pr oje ctively gener ate d Gr othendie ck pr estable ∞ -c ate gory 6 . Then (1) The trunc ation functor H 0 : C → C ♡ sends pr oje ctive obje cts to 1-pr oje ctive obje cts and c omp act obje cts to c omp act obje cts. (2) The 0 -trunc ations of the c omp act pr oje ctive obje cts of C pr ovide a family of c omp act 1-pr oje ctive gener ators for C ♡ . (3) The functor h ( π 0 ) : h ( C ) → C ♡ induc e d at the level of homotopy c ate gories r estricts to an e quivalenc e b etwe en the ful l sub c ate gories of (c omp act) pr oje ctive obje cts and (c omp act) 1-pr oje ctive obje cts. Pr o of. W e ﬁrst prov e (1). The fact that π 0 sends compact ob jects to compact ob jects follows directly from the fact that the inclusion C ♡ → C preserv es ﬁltered colimits. The fact that π 0 sends pro jective ob jects to 1-pro jectiv e ob jects follows from Prop osition 3.7 . Item (2) follows directly from (1) together with the fact that π 0 is a lo calization. It remains to establish (3). W e ﬁrst prov e fully faithfulness. Let X, Y b e a pair of pro jectiv e ob jects of C . Then the map Map C ( X, Y ) → Map C ♡ ( π 0 ( X ) , π 0 ( Y )) induced by π 0 is equiv alent to the map η ∗ : Map C ( X, Y ) → Map C ( X, π 0 ( Y )) of comp osition with the unit η : Y → π 0 ( Y ) . The fact that X is pro jectiv e and η induces an equiv alence on π 0 implies that η ∗ is an eﬀective epimorphism. Its ﬁb er is given by Map C ( X, τ ≥ 1 ( Y )) whic h is connected since X is pro jectiv e. W e conclude that η ∗ induces an equiv alence on π 0 , and therefore h( π 0 ) is fully faithful when restricts on the full sub category of pro jectiv e ob jects. It remains to prov e the essential surjectivit y . In other words, we hav e to show that ev ery (compact) 1-pro jectiv e ob ject of C ♡ is the image under π 0 of a (compact) pro jectiv e ob ject of C . W e will establish the case of compact pro jectiv e ob jects, and the pro of in the pro jectiv e case b eing similar. Let Y b e a compact 1-pro jectiv e ob ject of C ♡ . Applying (2) we may ﬁnd a compact pro jectiv e ob ject X in C such that Y is a retract of π 0 ( X ) . Let r : π 0 ( X ) → π 0 ( X ) be the induced retraction. The fully faithfulness part of (3) allows us to lift r to an idemp otent endomorphism ρ of X inside h ( C ) . Let X ′ b e a representativ e in C of the image of ρ . Then X ′ is a direct summand of X (see similar argument in [ HA , Lem. 1.2.4.6]) and therefore it is compact pro jective. The pro of ﬁnishes by observing that π 0 ( X ′ ) = Im( r ) = Y . Prop osition 3.9. Supp ose that A satisﬁes the c ondition (*). L et R b e in Alg ( A ≥ 0 ) and P ∈ LMo d R ( A ) ≥ 0 . Then: (1) P is a pr oje ctive R -mo dule if and only if every map X → P in LMo d R ( A ) ≥ 0 which induc es an epimorphism on π 0 admits a se ction. (2) If P is a pr oje ctive R -mo dule, then π 0 P ∈ LMo d π 0 R ( A ♡ ) is a 1-pr oje ctive discr ete π 0 R -mo dule. Pr o of. (1) This follows from the equiv alen t c haracterization b et ween Prop osition 3.7 (1) and (2). (2) It follo ws b y combining (1) and the equiv alence Map LMod R ( A ) ≥ 0 ( P , M ) ≃ Map LMod π 0 R ( A ♡ ) ( π 0 P , M ) for a dis crete π 0 R -mo dule M . Corollary 3.10. Supp ose that A satisﬁes the c ondition (*). L et R ∈ Alg ( A ≥ 0 ) . Then the he art LMo d π 0 R ( A ♡ ) has enough (1-)pr oje ctives. 6 W e do not use the notation A here, b ecause the prop osition do es not require a monoidal structure. 29 Prop osition 3.11. Supp ose that A satisﬁes the c ondition (*). L et R b e in Alg ( A ≥ 0 ) and P ∈ LMo d R ( A ) ≥ 0 . Then P is pr oje ctive if and only if ther e exists a smal l c ol le ction of c omp act pr oje ctive mo dules { P α } in LMo d R ( A ) ≥ 0 such that P is a r etr action of ⊕ α P α . Pr o of. Supp ose ﬁrst that P is pro jectiv e. By the pro jectiv e generation there exists an equiv alence of left R -mo dules colim α M α ∼ − → P where eac h M α is compact pro jectiv e. Then the induced map ⊕ α π 0 M α → π 0 P is epimorphic. Inv oking Prop osition 3.7 , we deduce that p admits a section (up to homotop y), so that P is a retract of M . T o pro ve the conv erse, we observe that the collection of pro jectiv e left R -mo dules is stable under small copro ducts and retracts by Prop osition 3.7 . 3.2 Pro jectiv e rigidit y and Lazard’s theorem W e now introduce the notion of pr oje ctive rigidity , a structural condition requiring that the dualizable ob jects of A ⊗ ≥ 0 coincide exactly with its compact pro jectiv es. Under such condition, we prov e the Lazard’s Theorem, which classiﬁes ﬂat mo dules as precisely the ﬁltered colimits of compact pro jective mo dules as desired. Deﬁnition 3.12. W e say that a presentably symmetric monoidal additive ∞ -category C ⊗ ∈ CAlg ( P r L ad ) is pro jectiv ely rigid if it satisﬁes the following: (1) C is pro jectively generated (this implies that C is prestable). (2) C d = C cpro j , i.e. the dualizable ob jects coincide with compact pro jectiv e ob jects. Deﬁnition 3.13. W e say that the ttt - ∞ -category ( A ⊗ , A ≥ 0 ) is pro jectiv ely rigid if A is righ t complete and A ⊗ ≥ 0 is pro jectiv ely rigid. Remark 3.14. (1) W arning: In general, ( A ≥ 0 ) d ⊊ A d ∩ A ≥ 0 b ecause the dual of an ob ject X ∈ A d ∩ A ≥ 0 is not necessarily connective! Ho w ev er, that holds exactly when X is ﬂat; see Prop osition 2.16 (3), which claims that ( A ≥ 0 ) d = A d ∩ A f l . (2) Supp ose that A ⊗ ≥ 0 is pro jectiv ely rigid. Then A ⊗ ≥ 0 can b e identiﬁed with the (symmetric monoidal) sifted co completion P Σ ( A cpro j ≥ 0 ) ⊗ . And its heart ( A ♡ ) ⊗ is 1-pro jectiv ely rigid. Example 3.15. W e list several examples of pro jectiv ely rigid ttt - ∞ -categories: (1) The ∞ -category of sp ectra Sp with the standard t -structure. (2) Gr(Sp) ⊗ = F un( Z disc , Sp) ⊗ is the ∞ -category of graded sp ectra with the p oint wise t -structure. (3) The ∞ -category of ﬁltered sp ectra Fil(Sp) with the p oint wise t -structure. (4) The ∞ -category of ﬁltered sp ectra Fil ( Sp ) with the homotopy t -structure. Its connective part is deﬁned b y Fil(Sp) h ≥ 0 def = { X ∗ | X n ∈ Sp ≥ n for eac h n ≥ 0 } . (5) The ∞ -category Sp ⊗ G, ≥ 0 = P Σ ( Span ( Fin G ); Sp ≥ 0 ) of connective gen uine G -sp ectra for a ﬁnite group G . 30 (6) The univ ersal example: cob ordism F un( Cob op 1 , Sp) ⊗ with the point wise t -structure. (7) The ∞ -category Sh v( X , Sp) ⊗ of shea ves on a stone space with the standard t -structure. (8) The ∞ -category SH ( k ) A T of Artin–T ate motivic sp ectra o ver a p erfect ﬁeld k with the standard t -structure; see [ BHS20 ] for a detailed inv estigation. (9) Qcoh ( X ) with the standard t -structure when X is an aﬃne quotient stack, i.e. a stack of the form Sp ec ( R ) /G for a linearly reductive group G acting on Sp ec ( R ) . In this case the compact-pro jectiv e ob jects are generated, under retracts, by pullbacks of G -representations, and the dual is given by the pullbac k of the dual represen tation. (10) The (connective) V oevodsky ∞ -category DM ( k , Z [1 /p ]) (where p is the characteristic of k , or p = 1 if k is a Q -algebra) with the Chow t -structure generated b y smo oth pro jectiv e v arieties and their P 1 -desusp ensions. The mapping sp ectra b etw een smo oth pro jectiv e v arieties are connective, so they form compact pro jectiv e generators, and they are also dualizable within the retract-closed sub category generated b y them. Example 3.16. Here are some examples whose connective parts are pro jectiv ely generated but not pro jectiv ely rigid: (1) The ∞ -category of light condensed sp ectra Cond light (Sp) ⊗ = Sp( P Σ ( Stonean light )) ⊗ . (2) The ∞ -category of light Solid sp ectra Solid light (Sp) ⊗ . (3) F un ( X, Sp ) ⊗ the ∞ -category of parametrized sp ectra on a small ∞ -group oid X , with p oint wise tensor pro duct. Prop osition 3.17. Supp ose that the ttt - ∞ -c ate gory ( A ⊗ , A ≥ 0 ) is pr oje ctively rigid. Then: (1) A ⊗ is c omp actly-rigid ly gener ate d, me aning the c omp act obje cts and dualizable obje cts in it c oincide. Particularly we have A ⊗ ∈ CAlg( P r L st ,ω ) . (2) L et R b e in CAlg ( A ≥ 0 ) . Then Mo d R ( A ≥ 0 ) ⊗ is pr oje ctively rigid to o. (3) L et R b e in CAlg ( A ≥ 0 ) . Then Mo d R ( A ) ⊗ is c omp actly-rigid ly gener ate d. Pr o of. (1) Since A is righ t complete, the collection of connectiv e co ver functors { τ ≥− n | n ≥ 0 } is join tly conserv ative. Therefore by Lemma 2.1 , A is generated b y { Σ − n P | n ≥ 0 , P ∈ A cpro j ≥ 0 } under small colimits. By Prop osition A.11 (4), we hav e { Σ − n P | n ≥ 0 , P ∈ A cpro j ≥ 0 } ⊂ A d and hence A c = A d . (2) Since the symmetric monoidal functor A ⊗ ≥ 0 R ⊗ ( − ) − − − − → Mo d R ( A ≥ 0 ) ⊗ preserv es compact pro jectiv e ob jects and dualizable ob jects, we conclude that (1) The unit R is dualizable in Mo d R ( A ≥ 0 ) . (2) R ⊗ P is dualizable in Mo d R ( A ≥ 0 ) if P ∈ A ≥ 0 is compact pro jectiv e. So the full sub category of dualizable ob jects Mo d R ( A ≥ 0 ) d con tains { R ⊗ X | X ∈ A cpro j ≥ 0 } . Then com bining Lemma 2.1 (2) and Prop osition A.11 (2)(3), we get Mo d R ( A ≥ 0 ) cpro j ⊂ Mo d R ( A ≥ 0 ) d . And that the unit R is compact pro jectiv e implies the equality Mo d R ( A ≥ 0 ) cpro j = Mo d R ( A ≥ 0 ) d . (3) It follows by applying (1) and (2) to Mo d R ( A ) ⊗ . 31 Prop osition 3.18. Supp ose that the ttt - ∞ -c ate gory ( A ⊗ , A ≥ 0 ) is pr oje ctively rigid. L et R ∈ CAlg ( A ≥ 0 ) and M ∈ Mod R ( A ) ≥ 0 . Then M is c omp act pr oje ctive if and only if M is dualizable in the whole stable ∞ -c ate gory Mo d R ( A ) ⊗ and M is a ﬂat R -mo dule. Pr o of. It follows by combining the pro jectiv e rigidit y and Prop osition 2.16 (3). Prop osition 3.19. Supp ose that the ttt - ∞ -c ate gory ( A ⊗ , A ≥ 0 ) is pr oje ctively rigid. L et R ∈ Alg ( A ≥ 0 ) and M b e a c onne ctive left R -mo dule. Then: (1) If M is pr oje ctive, then M is ﬂat. (2) M is c omp act pr oje ctive if and only if it is left dualizable in LMo d R ( A ≥ 0 ) . (3) M is (c omp act) pr oje ctive if and only if it is ﬂat and π 0 M is (c omp act) 1-pr oje ctive in LMo d π 0 R ( A ♡ ) . (4) Supp ose that R ∈ Alg ( A ♡ ) is discr ete. Then M is ﬂat if and only if M is discr ete and 1-ﬂat over π 0 R in the sense of Deﬁnition 1.25 . Pr o of. (1) Since ﬂat mo dules are closed under small copro ducts and retractions, we reduce to the case M = R ⊗ P where P ∈ A cpro j ≥ 0 . That is easy b ecause ( − ) ⊗ R ( R ⊗ P ) ≃ ( − ) ⊗ P reduces to the case R = 1 , which is deduced by Prop osition 3.18 . (2) By Corollary A.16 , we see that left dualizable ob jects are closed under ﬁnite copro ducts and retracts. W e observe that ev ery R ⊗ P is left dualizable (given by P ∨ ⊗ R ), whic h prov es “only if ” direction. F or the “if ” direction, if M is left dualizable, then it follows from Map LMod R ( A ≥ 0 ) ( M , − ) ≃ Map A ≥ 0 ( 1 , ∨ M ⊗ R − ) and compact pro jectivit y of the unit. (3) By the (1) w e ha ve that every pro jectiv e left R -mo dule is ﬂat. Secondly , the fact that π 0 sends pro jectiv e ob jects to 1-pro jectiv e ob jects was already observed in Prop osition 3.9 . This ﬁnishes the pro of of the “only if ” direction. Assume now that M is ﬂat and π 0 M is (compact) 1-pro jectiv e. Applying Prop osition 3.8 we may ﬁnd a (compact) pro jectiv e R -mo dule M ′ and an isomorphism π 0 M ′ = π 0 M . The fact that M ′ is pro jectiv e allo ws us to lift this isomorphism to a map f : M ′ → M . W e observe that f is an equiv alence by Prop osition 2.25 (2). (4) The “only if ” direction follows from (1) and M ≃ R ⊗ R M . F or the “if ” direction, given a discrete righ t R -mo dule N , we wish to show that N ⊗ R M is discrete to o. T ak e a map f : P → N of right R -mo dule such that P is pro jectiv e and f induces an epimorphism on π 0 . Then we hav e exact sequence 0 → π 0 ﬁb( f ) → π 0 P → π 0 N → 0 and hence ﬁb( f ) is discrete. Now tensoring with M , we get an exact sequence 0 → π 0 (ﬁb( f ) ⊗ R M ) → π 0 ( P ⊗ R M ) → π 0 ( N ⊗ R M ) → 0 32 b y 1-ﬂatness and the comm utative diagram of relativ e tensor pro duct functors. RMo d R ( A ≥ 0 ) A ≥ 0 RMo d π 0 R ( A ♡ ) A ♡ π 0 ( − ) ⊗ R M π 0 ( − ) ⊗ π 0 R π 0 M Because P is also ﬂat by (1), w e see that π 1 ( N ⊗ R M ) = 0 . W e actually ha v e pro v ed for any discrete right R -mo dule N has the prop erty π 1 ( N ⊗ R M ) = 0 . Then b y induction on n w e get that π n (ﬁb( f ) ⊗ R M ) = π n +1 ( N ⊗ R M ) = 0 for all n > 0 , which implies N ⊗ R M is discrete. W e now state the main result of this subsection. Theorem 3.20 (Lazard’s Theorem) . Supp ose that the ttt - ∞ -c ate gory ( A ⊗ , A ≥ 0 ) is pr oje ctively rigid. L et R ∈ Alg ( A ≥ 0 ) and M ∈ LMod R ( A ) . Then M is a ﬂat left R -mo dule if and only if it is e quivalent to a ﬁlter e d c olimit of c omp act pr oje ctive left R -mo dules. Pr o of. W e take the strategy in [ Ste23 , Prop. 2.2.22]. The “if ” direction can b e concluded by combining Prop osition 3.18 and Prop osition 2.16 (1). F or the “only if ” direction, assume now that M is ﬂat. Let L M cp denote the full sub category of L M = LMo d R ( A ) ≥ 0 spanned by compact pro jectiv e ob jects and consider the functor F ( − ) : ( L M cp ) op → S represen ted by M . W e wish to show that this functor deﬁnes an ind-ob ject of L M cp . Let ( − ) ∨ : R M cp → ( L M cp ) op b e the dualization equiv alence introduced in Corollary A.15 . W e will prov e that F (( − ) ∨ ) : R M cp → S deﬁnes a pro-ob ject of R M cp . Let p : E → R M b e the left ﬁbration asso ciated to the functor Map A ≥ 0 ( 1 , − ⊗ R M ) : R M → S . Then the base change of p to R M cp is the left ﬁbration classifying F (( − ) ∨ ) . W e ha v e to show that every ﬁnite diagram G : I → E × R M R M cp admits a left cone. The fact that M is ﬂat implies that the functor Map A ≥ 0 ( 1 , − ⊗ R M ) : R M → S is left exact, and therefore E is coﬁltered and G extends to a left cone G ◁ : I ◁ → E . Let N = ( M , ρ : 1 → N ⊗ R M ) b e the v alue of G ◁ at the cone p oint. T o show that G extends to a left cone in E × R M R M cp it is enough to prov e that N receiv es a map from an ob ject in E × R M R M cp . This amounts to sho wing that there exists a map N ′ → N from a compact pro jectiv e right R -mo dule N ′ with the prop ert y that ρ factors through N ′ ⊗ R M . This follows from the fact that 1 is compact pro jectiv e in A ≥ 0 . 3.3 Mo dules ov er discrete algebras Ha ving established Lazard’s theorem, we turn our attention to the b ehavior of discrete algebras within a top ological framework. In this subsection, we prov e that under the assumption of pro jectiv e rigidity , the mo dule category LMo d R ( A ) ov er a discrete algebra R ∈ Alg ( A ♡ ) is naturally (monoidally if R is E ∞ ) equiv alent to the (unbounded) derived category D (LMo d π 0 R ( A ♡ )) . Theorem 3.21. Supp ose that the ttt - ∞ -c ate gory ( A ⊗ , A ≥ 0 ) is pr oje ctively rigid. The fol lowing hold: 33 (1) F or any discr ete R ∈ Alg ( A ♡ ) ther e exists a (unique up to c ontr actible choic es) e quivalenc e in P r t - rex st D (LMo d π 0 R ( A ♡ )) ∼ − → LMo d R ( A ) which induc es the identity functor on the he art. (2) F or any discr ete commutative algebr a R ∈ CAlg ( A ♡ ) ther e exists a (unique up to c ontr actible choic es) e quivalenc e in CAlg( P r t - rex st ) D (Mo d π 0 R ( A ♡ )) ⊗ ∼ − → Mo d R ( A ) ⊗ which induc es the identity functor on the he art, wher e the symmetric monoidal structur e on left-hand side is induc e d by the pr oje ctive mo del with tensor pr o duct of chain c omplexes. Pr o of. (1) Since both are righ t complete and we hav e P Σ ( LMo d R ( A ≥ 0 ) cpro j ) ≃ LMo d R ( A ≥ 0 ) and D ( LMo d π 0 R ( A ♡ )) ≥ 0 ≃ P Σ ( LMo d π 0 R ( A ♡ ) cpro j ) , it suﬃces to show that taking π 0 induces an equiv alence LMo d R ( A ≥ 0 ) cpro j ≃ LMo d π 0 R ( A ♡ ) cpro j . By Prop osition 3.8 it is reduced to showing every compact pro jective R -mo dule P ∈ LMo d R ( A ≥ 0 ) cpro j is discrete, but that follows from Prop osition 3.19 . (2) By remark Remark 3.14 (1), it suﬃces to show that D ( Mo d π 0 R ( A ♡ )) ⊗ ≥ 0 ≃ P Σ ( Mo d π 0 R ( A ♡ ) cpro j ) ⊗ is the symmetric monoidal pro jective sifted co completion [see HA , Prop. 4.8.1.10] of Mo d π 0 R ( A ♡ ) cpro j . That is to show the following: (a) The natural inclusion Mo d π 0 R ( A ♡ ) cpro j  → D ( Mo d π 0 R ( A ♡ )) ≥ 0 is a symmetric monoidal functor whic h preserves ﬁnite copro ducts. (b) D (Mo d π 0 R ( A ♡ )) ⊗ ≥ 0 is presen tably symmetric monoidal. (c) The inclusion induces an equiv alence P Σ (Mo d π 0 R ( A ♡ ) 1 − cpro j ) ≃ D (Mo d π 0 R ( A ♡ )) ≥ 0 . The (a) and (c) follow directly from the construction of pro jective mo del on derived category . The (b) follo ws from [ HA , Prop. 1.3.5.21] and the explicit in ternal hom construction in D (Mo d π 0 R ( A ♡ )) Map D ( M ∗ , N ∗ ) p = Y n ∈ Z Hom M ( M n , N n + p ) for each integer p , where we denote D = D ( Mo d π 0 R ( A ♡ )) and M = Mo d π 0 R ( A ♡ ) . W e view Map D ( M ∗ , N ∗ ) ∗ as a c hain complex with v alues in M , with diﬀerential given by the formula ( d f )( x ) = d ( f ( x )) − ( − 1) p f ( dx ) for f ∈ Map D ( M ∗ , N ∗ ) p . Remark 3.22. In fact, by our argumen t the uniqueness in ab o ve theorem can b e promoted as which induces the iden tit y functor on compact 1-pro jectiv e π 0 R -mo dules in the heart. Example 3.23. Note that if the ttt - ∞ -category ( A ⊗ , A ≥ 0 ) is not pro jectiv ely rigid, then Theorem 3.21 is not true in general. F or instance, considering Sp B S 1 := F un ( B S 1 , Sp ) with point wise symmetric monoidal structure, then H Z with trivial S 1 -action is a discrete commutativ e algebra in it. How ever, Mo d H Z (Sp B S 1 )  = D (Mo d H Z (Sp B S 1 , ♡ )) ≃ D ( Z ) . 34 In this case, the pro jectiv e ob jects are not necessarily ﬂat. F or example, the representable presheaf Σ ∞ + Map B S 1 ( − , ∗ ) is compact pro jectiv e in Sp B S 1 ≥ 0 but not ﬂat in it. 3.4 Cohn lo calizations of E ∞ -algebras While ordinary commutativ e lo calization strictly in verts elements, Cohn lo calization forces more general morphisms b etw een ﬁnitely generated pro jectiv e mo dules to b ecome inv ertible. In this subsection, we generalize Neeman and Schoﬁeld’s derived Cohn lo calization [ Coh71 ; Sch85 ] to the ttt - ∞ -categorical setting. W e establish the existence and universal property of Cohn lo calizations for E ∞ -algebras in a pro jectiv ely rigid base. Let us ﬁrst recall the following classical result of Cohn lo calizations. Theorem 3.24 ([ Sch85 ] Theorem 4.1) . L et A b e an asso ciative ring. L et Σ b e a set of morphisms b etwe en ﬁnitely gener ate d pr oje ctive right A -mo dules. Then ther e ar e a ring A Σ and a morphism of rings f Σ : A − → A Σ , c al le d the universal lo c alisation of A at Σ , such that (1) f Σ is Σ -inverting, i.e. if α : P − → Q b elongs to Σ , then α ⊗ A 1 A Σ : P ⊗ A A Σ − → Q ⊗ A A Σ is an isomorphism of right A Σ -mo dules, and (2) f Σ is universal Σ -inverting, i.e. for any Σ -inverting ring homomorphism ψ : A − → B , ther e is a unique ring homomorphism ψ : A Σ − → B such that ψ f Σ = ψ . Mor e over, the homomorphism f Σ is a ring epimorphism and T or A 1 ( A Σ , A Σ ) = 0 . This theorem also works for comm utative rings 7 , which is the case we mainly care. Neeman constructed the Cohn lo calization in the deriv ed category of a comm utativ e ring in [ NRS06 , §4]. That motiv ates us to giv e a higher categorical corresp ondence. The Cohn lo calization is very useful in our abstract framework b ecause most in teresting cases are only pro jectively generated but not freely generated. The main result in this subsection is the following. Theorem 3.25. Supp ose that the ttt - ∞ -c ate gory ( A ⊗ , A ≥ 0 ) is pr oje ctively rigid. L et R ∈ CAlg ( A ≥ 0 ) and S = { P β f β − → Q β } b e a set of morphisms b etwe en c omp act pr oje ctive R -mo dules. Then ther e exists a Cohn lo c alization R → R [ S − 1 ] ∈ CAlg( A ≥ 0 ) satisfying the fol lowing universal pr op erty: F or any B ∈ CAlg ( A ) , the induc e d map Map CAlg( A ) ( R [ S − 1 ] , B ) → Map CAlg( A ) ( R, B ) is a ( − 1) -trunc ate d map whose image on π 0 c onsists those maps R → B such that for e ach f β ∈ S , B ⊗ R P β → B ⊗ R Q β is an e quivalenc e of B -mo dules. Remark 3.26. (1) See [ Ho y20 , §3] or [ Man24 , §3.4] for a discussion in the case where Q β = 1 for eac h β , which is related to Mo ore ob jects in the general setting. (2) The Cohn localization with resp ect to S is unique up to contractible choices by Theorem 3.25 . Before the p ro of, we recall some useful lemmas. 7 See [ AMŠTV20 ] for a discussion ab out the relation betw een Cohn lo calizations and epimorphisms. 35 Lemma 3.27 (See [ Ara25 ] B.5) . L et C p − → B b e a c o c artesian ﬁbr ation of ∞ -c ate gories. L et { S b | b ∈ B } b e given c ol le ctions of morphisms such that S b ⊂ F un (∆ 1 , C b ) for e ach b ∈ B . W e denote S = S b S b . If for any morphism s → t ∈ B the c o c artesian tr ansformation C s → C t sends S s into S t , then the induc e d functor q : D = C [ S − 1 ] → B fr om the lo c alization of C at S is a c o c artesian ﬁbr ation and c anonic al functor C D B p q pr eserves c o c artesian e dges and exhibits D b ≃ C b [ S − 1 b ] for e ach b ∈ B . And for any c o c artesian ﬁbr ation E → B , the c omp osition induc es a ful ly faithful emb e dding F un coCar / B ( D , E ) → F un coCar / B ( C , E ) whose image c onsists of those c o c artesian functors over B sending S to e quivalenc es in E . Remark 3.28. (1) Note that a co cartesian functor C → E o ver B sends S to equiv alences in E if and only if the induced functor on each ﬁb er C b → E b sends S b to equiv alences in E b . (2) The lemma ab ov e is a generalization of [ HA , Prop. 2.2.1.9]. The statemen t there only gives a construction in the case of reﬂective lo calization. Corollary 3.29. L et C ⊗ b e a symmetric monoidal ∞ -c ate gory and S b e a c ol le ction of morphisms in C satisfying that f ⊗ g ∈ S if b oth f , g ∈ S . Then the lo c alization D = C [ S − 1 ] inherits a natur al symmetric monoidal structur e and the lo calization c an b e pr omote d to a symmetric monoidal functor C ⊗ → D ⊗ satisfying the universal pr op erty that for any symmetric monoidal ∞ -c ate gory E ⊗ the c omp osition induc es a ful ly faithful emb e dding F un ⊗ / N(Fin ∗ ) ( D ⊗ , E ⊗ ) → F un ⊗ / N(Fin ∗ ) ( C ⊗ , E ⊗ ) whose image c onsists of those symmetric monoidal functors sending S to e quivalenc es in E . Lemma 3.30. L et C ⊗ F ⊗ − − → D ⊗ ∈ CAlg ( P r L ) . L et G ⊗ : D ⊗ → C ⊗ b e the r elative right adjoint of F ⊗ . If C is gener ate d by dualizables under smal l c olimits and G is c onservative and smal l-c olimit-pr eserving, then G ⊗ is symmetric monoidal monadic, i.e. ther e exist an R ∈ CAlg ( C ) and a symmetric monoidal e quivalenc e Mo d R ( C ) ⊗ ≃ D ⊗ such that the fol lowing diagr am is c ommutative. Mo d R ( C ) ⊗ D ⊗ C ⊗ ∼ G ⊗ Pr o of. By [ HA , Cor. 4.8.5.21], it suﬃces to show that G satisﬁes the pro jection formula, that is, for ev ery ob ject C ∈ C and D ∈ D , the canonical map C ⊗ G ( D ) → G ( F ( C ) ⊗ D ) is an equiv alence. By the assumption, it suﬃces to verify the case C is dualizable. In this case, for any M ∈ C we hav e Map C ( M , C ⊗ G ( D )) ≃ Map C ( C ∨ ⊗ M , G ( D )) ≃ Map D ( F ( C ∨ ⊗ M ) , D ) ≃ Map D ( C ∨ ⊗ F ( M ) , D ) ≃ Map D ( F ( M ) , C ⊗ D ) ≃ Map C ( M , G ( C ⊗ D )) . 36 That indicates the pro jection formula holds. Pr o of of The or em 3.25 : Let S 1 ⊂ F un(∆ 1 , Mo d R ( A ≥ 0 )) b e the set of morphisms { X α ⊗ R f β | X α ∈ Mo d R ( A ≥ 0 ) cpro j , f β ∈ S } . Then S 1 is small and thereb y generates a strongly saturated class S 1 of small generation (see [ HTT , §5.5.4]). Then S 1 ⊂ F un (∆ 1 , Mo d R ( A ≥ 0 )) satisﬁes conditions in Corollary 3.29 , thereby i t pro duces a symmetric monoidal lo calization Mo d R ( A ≥ 0 ) ⊗ F ⊗ − − → Mo d R ( A ≥ 0 )[ S − 1 1 ] ⊗ = D ⊗ suc h that F ⊗ ∈ CAlg ( P r L ) . Since D ⊂ Mo d R ( A ≥ 0 ) closed under ﬁnite pro ducts, it lies in CAlg ( P r L ad ) and F ⊗ ∈ CAlg( P r L ad ) . No w we wish to show that F ⊗ satisﬁes conditions in Lemma 3.30 . It suﬃces to v erify that D ⊂ Mo d R ( A ≥ 0 ) closed under small colimits, i.e. S 1 -lo cal ob jects are closed under small colimits. Unwinding the deﬁnition, a connective R -mo dule M is S 1 -lo cal if and only if Map Mod R ( A ≥ 0 ) ( X α ⊗ R Q β , M ) → Map Mod R ( A ≥ 0 ) ( X α ⊗ R P β , M ) is equiv alent for any X α ⊗ R f β ∈ S 1 . How ev er, this map can b e identiﬁed with Map Mod R ( A ≥ 0 ) ( X α , Q ∨ β ⊗ R M ) → Map Mod R ( A ≥ 0 ) ( X α , P ∨ β ⊗ R M ) . So by the pro jectiv e generation, M is S 1 -lo cal if and only if f ∨ β ⊗ R M : Q ∨ β ⊗ R M → P ∨ β ⊗ R M is equiv alent for each f β ∈ S . That implies S 1 -lo cal ob jects are closed under small colimits. So there exist an R [ S − 1 ] ∈ CAlg ( A ≥ 0 ) R/ and an equiv alence Mo d R [ S − 1 ] ( A ≥ 0 ) ⊗ ≃ D ⊗ suc h that the follo wing diagram is comm utative. Mo d R [ S − 1 ] ( A ≥ 0 ) ⊗ D ⊗ Mo d R ( A ≥ 0 ) ⊗ ∼ G ⊗ No w given B ∈ CAlg ( A ) , we need to show that the induced map Map CAlg( A ) ( R [ S − 1 ] , B ) → Map CAlg( A ) ( R, B ) is a ( − 1) -truncated map whose image on π 0 consists those maps R → B suc h that for any β ∈ J , B ⊗ R P β → B ⊗ R Q β is an equiv alence of B -mo dules. Without loss of generalit y , w e can assume that B is connectiv e. By [ HA , Cor. 4.8.5.21], we hav e the following Morita embedding, CAlg( A ≥ 0 ) → CAlg( P r L ad ) A ⊗ ≥ 0 / therefore it suﬃces to show that F ⊗ induces a fully faithful embedding F un ⊗ ,L / N(Fin ∗ ) (Mo d R [ S − 1 ] ( A ≥ 0 ) ⊗ , Mo d B ( A ≥ 0 ) ⊗ ) → F un ⊗ ,L / N(Fin ∗ ) (Mo d R ( A ≥ 0 ) ⊗ , Mo d B ( A ≥ 0 ) ⊗ ) whose image consists of those functors sending S to equiv alences in Mo d B ( A ≥ 0 ) , where F un ⊗ ,L / N(Fin ∗ ) denotes symmetric monoidal functors which preserve small colimits. Ho wev er, that is implied b y Corollary 3.29 . 37 Remark 3.31. The argument ab ov e works for a set S ⊂ F un (∆ 1 , C d ) of morphisms b etw een dualizables inside an arbitrary presentably symmetric monoidal ∞ -category C ⊗ whic h is generated by dualizables under small colimits. W e now start to inv estigate the prop erties of Cohn lo calizations. Prop osition 3.32. L et f S : R → R [ S − 1 ] ∈ CAlg ( A ≥ 0 ) b e the Cohn lo c alization at S in The or em 3.25 . Then the map π 0 f S : π 0 R → π 0 ( R [ S − 1 ]) exhibits π 0 ( R [ S − 1 ]) ≃ ( π 0 R )[( π 0 S ) − 1 ] as the Cohn lo c alization of π 0 R at π 0 S in the sense of The or em 3.40 , wher e π 0 S = { π 0 P β π 0 f β − − − → π 0 Q β | f β ∈ S } . Pr o of. It follows immediately from the universal prop erty of the Cohn lo calization. Prop osition 3.33. L et f S : R → R [ S − 1 ] b e the Cohn lo c alization at S in The or em 3.25 . Then R [ S − 1 ] is an idemp otent c ommutative R -algebr a. Pr o of. It suﬃces to show that the following diagram is a pushout in CAlg( A ≥ 0 ) , R R [ S − 1 ] R [ S − 1 ] R [ S − 1 ] i.e. to show that f S is an ( ∞ -categorical) epimorphism in CAlg ( A ≥ 0 ) . That is implied by the description of mapping spaces in Theorem 3.25 . Deﬁnition 3.34. Suppose that the ttt - ∞ -category ( A ⊗ , A ≥ 0 ) is pro jectiv ely rigid. W e sa y a map A → B ∈ CAlg ( A ≥ 0 ) is a (ﬁnitary) Cohn lo calization if there exists a (ﬁnite) set S of morphisms b et ween compact pro jectiv e R -mo dules such that B ≃ A [ S − 1 ] . Remark 3.35. Note that if S = { P i f i − → Q i } is ﬁnite, then A [ S − 1 ] ≃ A [ f − 1 ] is equiv alen t to the Cohn lo calization at the single element f = L i f i . Prop osition 3.36. Supp ose that the ttt - ∞ -c ate gory ( A ⊗ , A ≥ 0 ) is pr oje ctively rigid. L et A → B ∈ CAlg( A ≥ 0 ) b e a ﬁnitary Cohn lo c alization. Then B is ﬁnitely pr esente d over A . Pr o of. By the remark ab ov e, we can assume that S = { f } consists of a single elemen t. Now given a ﬁltered colimit of connective commutativ e A -algebras lim − → α C α = C w e need to show that the natural map lim − → α Map CAlg( A ≥ 0 ) A/ ( B , C α ) → Map CAlg( A ≥ 0 ) A/ ( B , C ) is an equiv alence. By Prop osition 3.33 (1), eac h mapping space ab ov e is empty or a single p oin t. If Map CAlg( A ≥ 0 ) A/ ( B , C ) = ∅ , then nothing needs to pro ve. No w assume that Map CAlg( A ≥ 0 ) A/ ( B , C ) ≃ {∗} , we wish to show that there exists an α suc h that Map CAlg( A ≥ 0 ) A/ ( B , C α ) is not empt y . By assumption, the natural map f ⊗ A C is an equiv alence, thereby coﬁb ( f ) ⊗ A C = 0 . Since coﬁb ( f ) is a compact A -mo dule, there exists an α suc h that the natural map coﬁb( f ) → coﬁb( f ) ⊗ A C α is zero. That implies coﬁb( f ) ⊗ A C α = 0 and we are done. Remark 3.37. Note that, unlike the case of ( E ∞ -)rings, a Cohn lo calization R → R [ S − 1 ] is not necessarily ﬂat in general. See Remark 6.25 for an example of a ﬁnitary Cohn lo calization that fails to b e ﬂat. 38 In fact, we can prov e that an y pro jectiv ely rigid ∞ -category is a smashing lo calization of some presheaf category induced b y a Cohn localization. Prop osition 3.38. L et I ⊗ b e a smal l rigid (se e Deﬁnition 7.7 ) symmetric monoidal ∞ -c ate gory which admits ﬁnite c opr o ducts and whose tensor pr o duct is c omp atible with ﬁnite c opr o ducts. Then the natur al symmetric monoidal functor F un( I op , Sp ≥ 0 ) ⊗ L ⊗ − − → F un × ( I op , Sp ≥ 0 ) ⊗ ≃ P Σ ( I ) ⊗ induc e d by the universal pr op erty of Y one da emb e dding is a smashing lo c alization, that is, ther e exists an idemp otent c ommutative algebr a R ∈ CAlg (F un( I op , Sp ≥ 0 )) such that L ( − ) ≃ R ⊗ ( − ) . Pr o of. By Lemma 3.30 , it only suﬃces to show that the inclusion F un × ( I op , Sp ≥ 0 ) ⊂ F un( I op , Sp ≥ 0 ) is closed under small colimits. That is obvious b e cause b oth sides are additive and the inclusion is closed under ﬁnite produc ts and sifted colimits. Remark 3.39. In fact, the idemp oten t algebra 1 → R ab o ve can b e identiﬁed with the Cohn localization 1 → 1 [ S − 1 ] , where S = n n M i =1 Σ ∞ + h ( x i ) − → Σ ∞ + h ( n a i =1 x i ) | x i ∈ I for eac h i o ⊂ F un  ∆ 1 , F un( I op , Sp ≥ 0 ) cpro j  . 3.5 Cohn lo calizations in an ab elian base T o complement and ground our higher ∞ -categorical construction, this subsection analyzes Cohn lo calization strictly within a 1-pro jectiv ely rigid symmetric monoidal Grothendieck ab elian category . W e demonstrate that ﬁnitary Cohn lo calizations reliably yield ﬁnitely presented algebras, mirroring the higher categorical behavior. Theorem 3.40. L et A ⊗ b e a 1-pr oje ctively rigid symmetric monoidal additive 1-c ate gory (it is automat- ic al ly Gr othendie ck ab elian). L et R ∈ CAlg ( A ) and S = { P β f β − → Q β } b e a set of morphisms b etwe en c omp act 1-pr oje ctive R -mo dules. Then ther e exists a Cohn lo c alization R → R [ S − 1 ] ∈ CAlg( A ) satisfying the fol lowing universal pr op erty: F or any B ∈ CAlg ( A ) , the induc e d map Hom CAlg( A ) ( R [ S − 1 ] , B ) → Hom CAlg( A ) ( R, B ) is an inje ction whose image c onsists those maps R → B such that for e ach f β ∈ S , B ⊗ R P β → B ⊗ R Q β is an e quivalenc e of B -mo dules. Pr o of. The pro of is parallel with the pro of of Theorem 3.25 . Also see Remark 3.31 . W e just need to replace the Morita embedding CAlg ( A ≥ 0 ) → CAlg( P r L ad ) A ⊗ ≥ 0 / in the argumen t b y CAlg( A ) → CAlg ( P r L ad , 1 ) A ⊗ / 39 to adapt the 1-categorical setting. Prop osition 3.41. L et f S : R → R [ S − 1 ] b e the Cohn lo c alization at S in The or em 3.40 . Then R [ S − 1 ] is an idemp otent c ommutative R -algebr a Pr o of. It suﬃces to show that the following diagram is a pushout in CAlg( A ) , R R [ S − 1 ] R [ S − 1 ] R [ S − 1 ] i.e. to show that f S is an epimorphism in CAlg ( A ) . That is implied b y the description of the Hom set in Theorem 3.40 . Deﬁnition 3.42. Let A ⊗ b e a 1-pro jectiv ely rigid symmetric monoidal Grothendieck ab elian category . W e say a map A → B ∈ CAlg ( A ) is a (ﬁnitary) Cohn lo calization if there exists a (ﬁnite) set S = { P β f β − → Q β } of morphisms b etw een compact 1-pro jectiv e R -mo dules such that B ≃ A [ S − 1 ] . Remark 3.43. Note that if S = { P i f i − → Q i } is ﬁnite, then A [ S − 1 ] = A [ f − 1 ] can b e written as the Cohn lo calization at a single element f = L i f i . Prop osition 3.44. L et A ⊗ b e a 1-pr oje ctively rigid symmetric monoidal Gr othendie ck ab elian c ate gory. L et A → B ∈ CAlg ( A ) b e a ﬁnitary Cohn lo c alization. Then B is ﬁnitely pr esente d over A . Pr o of. The pro of is similar to Prop osition 3.36 but we need to take a diﬀerent strategy b ecause the k ernel is not preserv ed by base c hange. Let A ⊗ = D ( A ) ⊗ . By Theorem 3.21 w e ha v e that A ♡ , ⊗ ≃ A ⊗ and A cpro j ≥ 0 = A 1 − cpro j . Let A ′ [ S − 1 ] b e the (higher) Cohn lo calization at S in the sense of Theorem 3.25 . Then A ′ [ S − 1 ] is compact in CAlg ( A ≥ 0 ) A/ b y Proposition 3.36 . Therefore π 0 A ′ [ S − 1 ] is comp act in CAlg ( A ) A/ . Ho w ever by Prop osition 3.32 , A [ S − 1 ] = π 0 A ′ [ S − 1 ] . W e are done. 4 Finiteness prop erties In this section, we inv estigate ﬁnitene ss conditions in the ttt - ∞ -categorical setting. 4.1 P erfect and almost p erfect mo dules In this subsection, we analyze p erfect and almost p erfect mo dules, mapping their relationship directly via T or-amplitude. W e demonstrate that an almost p erfect mo dule is p erfect if and only if it p ossesses a ﬁnite T or-amplitude, systematically extending results in [ HA , §7.2] to arbitrary pro jectively rigid bases. Deﬁnition 4.1. Let R ∈ Alg ( A ) . W e say a left R -mo dule M is p erfect if it is compact in LMo d R ( A ) . Prop osition 4.2. Supp ose that A is right c omplete. L et R ∈ Alg ( A ≥ 0 ) and M b e a left R -mo dule. If M is p erfe ct, then M is b ounde d-b elow. Pr o of. By the right completeness we ha ve M ≃ lim − → τ ≥− n M , then the compactness of M implies that M is a retract of τ ≥− n M for some n . 40 Deﬁnition 4.3. Let C b e a presentable ∞ -category . W e will say an ob ject C ∈ C is almost compact if τ ≤ n C is a compact ob ject of τ ≤ n C for all n ≥ 0 . Remark 4.4. Let C b e a compactly generated ∞ -category . Then every compact ob ject of C is almost compact b y [ HTT , Corollary 5.5.7.4]. Deﬁnition 4.5. Suppose that A ≥ 0 ∈ P r L ω and that A is righ t complete. Let R ∈ Alg ( A ≥ 0 ) be a connectiv e E 1 -algebra. W e say that a left R -mo dule M is almost p erfect if there exists an in teger k suc h that M ∈ LMod R ( A ) ≥ k and is almost compact as an ob ject of LMo d R ( A ) ≥ k . W e let LMo d R ( A ) aperf ⊂ LMo d R ( A ) denote the full sub category spanned by the almost p erfect left R -mo dules. Remark 4.6. Under the assumption that A ≥ 0 ∈ P r L ω and A is righ t complete, if a left R -mo dule M is almost p erfect in LMo d R ( A ) ≥ k , then it is almost p erfect in LMo d R ( A ) ≥ k − 1 to o. Prop osition 4.7. Supp ose that A ≥ 0 ∈ P r L ω and that A is right c omplete. L et R ∈ Alg( A ≥ 0 ) . Then: (1) The ful l sub c ate gory LMo d R ( A ) ap erf ⊂ LMo d R ( A ) is close d under tr anslations and ﬁnite c olimits, and is ther efor e a stable sub c ate gory of LMo d R ( A ) . (2) The ful l sub c ate gory LMo d R ( A ) ap erf ⊂ LMo d R ( A ) is close d under the formation of r etr acts. (3) Every p erfe ct left R -mo dule is almost p erfe ct. (4) The ful l sub c ate gory LMo d R ( A ) ap erf ≥ 0 ⊂ LMod R ( A ) is close d under the formation of ge ometric r e alizations of simplicial obje cts. Pr o of. Pro of. Assertions (1) and (2) are obvious, and (3) follo ws from Remark 4.4 . T o prov e (4), it suﬃces to show that the collection of compact ob jects of LMo d R ( A ) [0 ,n ] is closed under geometric realizations, whic h follo ws from [ HA , Lemma 1.3.3.10]. Prop osition 4.8. Supp ose that A ≥ 0 is pr oje ctively gener ate d. L et R ∈ Alg ( A ≥ 0 ) and M ∈ LMod R ( A ) ap erf ≥ 0 b e a left R -mo dule which is c onne ctive and almost p erfe ct. Then M c an b e obtaine d as the ge ometric r e alization of a simplicial left R -mo dule P • such that e ach P n is a c omp act pr oje ctive left R -mo dule in LMo d R ( A ) ≥ 0 . Pr o of. W e mimic the pro of in [ HA , Prop. 7.2.4.11] and carefully replace “free” by “pro jectiv e”. In view of ∞ -categorical Dold-Kan corresp ondence, it will suﬃce to show that M can b e obtained as the colimit of a sequence D (0) f 1 − → D (1) f 2 − → D (2) → . . . where each coﬁb ( f n )[ − n ] is a compact pro jectiv e left R -mo dule; here we agree by conv en tion that f 0 denotes the zero map 0 → D (0) . The construction go es b y induction. Supp ose that the diagram D (0) → . . . → D ( n ) g − → M has already b een constructed, and that N = ﬁb ( g ) is n -connectiv e. Part (1) of Prop osition 4.7 implies that N is almost p erfect, so that the b ottom π n N is a compact ob ject in the category of left π 0 R -mo dules. It follows that there exists a map β : Q [ n ] → N , where Q is a compact pro jectiv e left R -mo dule b ecause LMo d R ( A ) ≥ 0 is pro jectiv ely generated. And β induces a surjection π 0 Q → π n N . W e no w deﬁne D ( n + 1) to b e the coﬁb e r of the comp osite map Q [ n ] β − → N → D ( n ) , and construct a diagram D (0) → . . . → D ( n ) → D ( n + 1) g ′ − → M 41 Using the octahed ral axiom of triangulated category , we obtain a ﬁb er sequence Q [ n ] → ﬁb( g ) → ﬁb( g ′ ) and the asso ciated long exact sequence in A ♡ pro ves that ﬁb ( g ′ ) is ( n + 1) -connective. In particular, w e conclude that for a ﬁxed m ≥ 0 , the maps π m D ( n ) → π m M are isomorphisms for n ≫ 0 , so that the natural map lim − → D ( n ) → M is an equiv alence of left R -mo dules by the left completeness, as desired. Prop osition 4.9. Supp ose that the ttt - ∞ -c ate gory ( A ⊗ , A ≥ 0 ) is pr oje ctively rigid. L et R ∈ Alg ( A ≥ 0 ) and let M b e a c onne ctive left R -mo dule. Then the fol lowing ar e e quivalent: (1) M is a c omp act pr oje ctive left R -mo dule. (2) M is a p erfe ct and ﬂat left R -mo dule. (3) M is an almost p erfe ct and ﬂat left R -mo dule. (4) M is a ﬂat left R -mo dule and π 0 M is ﬁnitely pr esente d over π 0 R . Pr o of. The directions (1) ⇒ (2) , (2) ⇒ (3) and (3) ⇒ (4) are ob vious. F or (4) ⇒ (1) , b y Corollary 1.32 , w e conclude that π 0 M is compact 1-pro jectiv e ov er π 0 R . Then by Prop osition 3.19 , we get that M is a compact pro jectiv e left R -mo dule. Deﬁnition 4.10. Let R ∈ Alg ( A ≥ 0 ) . W e will sa y a left R -mo dule M has T or-amplitude ≤ n if, for ev ery discrete right R -mo dule N , π i ( N ⊗ R M ) v anish for i > n . W e will say M is of ﬁnite T or-amplitude if it has T or-amplitude ≤ n for some integer n . Remark 4.11. Suppose that A is Grothendiec k. In view of Prop osition 2.15 , a connective left R -mo dule M has T or-amplitude ≤ 0 if and only if M is ﬂat. Prop osition 4.12. Supp ose that A is hyp er c omplete. Then a c onne ctive left R -mo dule M has T or- amplitude ≤ − 1 if and only if M = 0 . Pr o of. Assume M is connective and has T or-amplitude ≤ − 1 ; w e wish to show M ≃ 0 . Since R is connectiv e, its 0 -th truncation π 0 R = τ ≤ 0 R is a discrete right R -mo dule. By the deﬁnition of T or-amplitude ≤ − 1 , we hav e π i ( τ ≤ 0 R ⊗ R M ) = 0 for all i > − 1 ( i.e. i ≥ 0) . Because the t -structure is compatible with the symmetric monoidal structure, the tensor of tw o connective ob jects is connective. Since π 0 R ∈ A ≥ 0 and M ∈ A ≥ 0 , their tensor pro duct is connective: π 0 R ⊗ R M ∈ A ≥ 0 . Hence π i ( π 0 R ⊗ R M ) = 0 for all i < 0 . This shows all homotopy groups of π 0 R ⊗ R M v anish, so by h yp ercompleteness π 0 R ⊗ R M ≃ 0 . Consider the standard truncation ﬁb er sequence in right R -mo dules τ ⩾ 1 R − → R − → π 0 R. 42 T ensoring on the right with M yields a ﬁb er sequence ( τ ⩾ 1 R ) ⊗ R M − → R ⊗ R M − → π 0 R ⊗ R M . By step 2 the right-hand term v anishes, and R ⊗ R M ≃ M , so the left-hand map is an equiv alence : M ≃ ( τ ⩾ 1 R ) ⊗ R M . No w use connectivity estimates for tensor pro ducts: since M ∈ A ≥ 0 and τ ⩾ 1 R ∈ A ≥ 1 , their tensor lies in A ≥ 1+0 = A ≥ 1 , so M ∈ A ≥ 1 . Iterating the same argument gives M ≃ ( τ ⩾ 1 R ) ⊗ R M ∈ A ≥ 2 , and so on. By induction M ∈ A ≥ n for ev ery n ≥ 0 , hence π i ( M ) = 0 for all i ∈ Z . The hypercompleteness assumption on A implies M ≃ 0 . Prop osition 4.13. Supp ose that the ttt - ∞ -c ate gory ( A ⊗ , A ≥ 0 ) is pr oje ctively rigid. L et R ∈ Alg ( A ≥ 0 ) . Then: (1) If M is a left R -mo dule of T or-amplitude ≤ n , then M [ k ] has T or-amplitude ≤ n + k . (2) L et M ′ → M → M ′′ b e a ﬁb er se quenc e of left R -mo dules. If M ′ and M ′′ have T or-amplitude ≤ n , then so do es M. (3) L et M b e a left R -mo dule of T or-amplitude ≤ n . Then any r etr act of M has T or-amplitude ≤ n . (4) L et M b e an almost p erfe ct left mo dule over R . Then M is p erfe ct if and only if M has ﬁnite T or-amplitude. (5) L et M b e a left mo dule over R having T or-amplitude ≤ n . Then for every N ∈ RMo d R ( A ) ≤ 0 , π i ( N ⊗ R M ) vanishes for e ach i > n . Pr o of. W e mimic the pro of in [ HA , Prop. 7.2.4.23] but carefully replace “free” b y “pro jective”. The ﬁrst three assertions follow immediately from the exactness of the functor N 7→ N ⊗ R M . It follows that the collection left R -mo dules of ﬁnite T or-amplitude is stable under retracts and ﬁnite colimits and desusp ensions, and contains all compact pro jectiv e left R -mo dules. This prov es the “only if ” direction of (4) b y Prop osition 3.3 . F or the conv erse, let us supp ose that M is almost perfect and of ﬁnite T or-amplitude. W e wish to show that M is p erfect. W e ﬁrst apply (1) to reduce to the case where M is connectiv e. The pro of no w go es by induction on the T or-amplitude n of M . If n = 0 , then M is ﬂat and w e may conclude b y applying Prop osition 4.9 . W e ma y therefore assume n > 0 . Since M is almost p erfect, there exists a compact pro jective left R -mo dule P and a ﬁb er sequence M ′ → P f − → M where f induces an epimorphism on π 0 . T o prov e that M is p erfect, it will suﬃce to sho w that P and M ′ are p erfect. It is clear that P is p erfect, and it follows from Prop osition 4.7 that M ′ is almost p erfect. Moreo ver, since π 0 f is surjective, M ′ is connectiv e. W e will sho w that M ′ is of T or-amplitude ≤ n − 1 ; the inductiv e h yp othesis will then imply that M is p erfect, and the pro of will b e complete. 43 Let N b e a discrete right R -mo dule. W e wish to pro ve that π k ( N ⊗ R M ′ ) ≃ 0 for k ≥ n . Since the functor N ⊗ R • is exact, we obtain for each k an exact sequence π k +1 ( N ⊗ R M ) → π k ( N ⊗ R M ′ ) → π k ( N ⊗ R P ) The left en try v anishes in virtue of our assumption that M has T or-amplitude ≤ n . W e now complete the pro of of (4) b y observing that π k ( N ⊗ R P ) v anishes b ecause N is discrete and P is ﬂat and k ≥ n > 0 . W e now prov e (5). Assume that M has T or-amplitude ≤ m . Let N ∈ RMod R ( A ) ≤ 0 ; w e wish to prov e that π i ( N ⊗ R M ) ≃ 0 for i > n . Since N ≃ colim τ ≥− m N , it will suﬃce to prov e the v anishing after replacing N b y τ ≥− m N for every integer m . W e may therefore assume that N ∈ RMo d R ( A ) [ − m, 0] for some m ≥ 0 . W e pro ceed by induction on m . When m = 0 , the desired result follows immediately from our assumption on M . If m > 0 , we hav e a ﬁb er sequence τ ≥ 1 − m N → N → ( π − m N ) [ − m ] hence an exact sequence π i (( τ ≥ 1 − m N ) ⊗ R M ) → π i ( N ⊗ R M ) → π i + m ( π − m N ⊗ R M ) If i > n , then the ﬁrst group v anishes by the inductiv e hypothesis, and the third b y virtue of our assumption that M has T or-amplitude ≤ n . Prop osition 4.14. Supp ose that the ttt - ∞ -c ate gory ( A ⊗ , A ≥ 0 ) is pr oje ctively rigid. L et R ∈ Alg ( A ≥ 0 ) , and let C b e the smal lest stable sub c ate gory 8 of LMo d R ( A ) which c ontains al l c omp act pr oje ctive mo dules. Then C = LMo d p erf R ( A ) . Pr o of. The inclusion C ⊂ LMo d perf R ( A ) is ob vious. T o prov e the conv erse, we must show that every ob ject M ∈ LMo d perf R ( A ) b elongs to C . In v oking Prop osition 4.2 , we may reduce to the case where M is connectiv e. W e then work by induction on the (necessarily ﬁnite) T or-amplitude of M . If M is of T or-amplitude ≤ 0 , then M is ﬂat and the desired result follows from Prop osition 4.9 . In the general case, w e choose a compact pro jective R -mo dule P and a map f : P → M whic h induces a surjection π 0 P → π 0 M . W e may conclude that that ﬁb er K of f is a connective p erfect mo dule of smaller T or-amplitude than that of M , so that K ∈ C b y the inductive hypothesis. Since P ∈ C and C is stable under the formation of coﬁb ers, we conclude that M ∈ C as desired. 4.2 Finite presentation and almost of ﬁnite presentation W e now transition from the ﬁniteness of mo dules to the ﬁniteness of algebras. W e pro ve that ﬁnitely presen ted algebras can b e iteratively assembled via ﬁnite pushouts of symmetric algebras ev aluated on compact pro jectiv es. Deﬁnition 4.15. Let f : A → B b e a map in CAlg( A ≥ 0 ) . (1) W e say f : A → B is lo cally of ﬁnite presentation (or ﬁnitely presen ted) if B is a compact ob ject in CAlg( A ≥ 0 ) A/ ≃ CAlg(Mo d A ( A ≥ 0 )) . 8 W e don’t need to assume that C is idemp otent-complete. See [ HA , Remark 7.2.4.24] for the case of sp ectra. 44 (2) W e say f : A → B is almost of ﬁnite presentation if for each n ≥ 0 , τ ≤ n B is a compact ob ject in CAlg( A [0 ,n ] ) τ ≤ n A/ ≃ CAlg(Mo d A ( A ≥ 0 )) ≤ n . Remark 4.16. Suppose that A ≥ 0 ∈ P r L ω (then so is CAlg ( A ≥ 0 ) ). Given a commutativ e diagram in CAlg( A ≥ 0 ) A B C where B is of lo cally of ﬁnite presentation o ver A . Then C is lo cally of ﬁnite presentation ov er B if and only if C is lo cally of ﬁnite presen tation ov er A . This follows immediately from [ HTT , Prop osition 5.4.5.15]. Prop osition 4.17. Supp ose that A ≥ 0 is pr oje ctively gener ate d. L et f : A → B ∈ CAlg ( A ≥ 0 ) b e a map of ﬁnite pr esentation. Then ther e exists c omp act pr oje ctive A -mo dules M , N and a diagr am Sym ∗ A ( N ) A Sym ∗ A ( M ) B α ϕ such that the map B ′ → B induc es an isomorphism on π 0 , wher e B ′ is the pushout of ab ove diagr am in CAlg ( A ≥ 0 ) and α is the natur al augmentation. (Note that φ is not ne c essarily induc e d by a map of mo dules N → M ). Pr o of. Firstly , by Corollary 3.10 there exists a set of compact pro jectiv e A -mo dules { P α | α ∈ I } and a map P = ⊕ α P α → π 0 B of A -mo dules which induces an epimorphism on π 0 . Then there exists a lifting of A -mo dule map B P π 0 B b y Prop osition 3.9 . This lifting induces an A -algebra map Sym ∗ A ( P ) → B , whic h induces an epimorphism on π 0 (as ob jects in A ♡ ) b y our construction. Since f : A → B is of ﬁnite presen tation and A is Grothendiec k, π 0 B is a compact ob ject in CAlg ( Mo d π 0 A ( A ♡ )) . That implies there exists ﬁnite collection { P i } suc h that the comp osition Sym ∗ A ( ⊕ i P i ) → Sym ∗ A ( P ) → B induces an epimorphism on π 0 , by taking the ﬁltration of images of π 0 Sym ∗ A ( ⊕ j ∈ J P j ) → π 0 B where J ⊂ I is a ﬁnite subset. W e take M = ⊕ i P i and N ′ = ﬁb ( Sym ∗ A ( M ) → B ) , then N ′ is a connective A -mo dule by our construction. By similar argument as previous, there exists a set of compact pro jectiv e A -mo dules { Q α | α ∈ I 2 } and a map Q = ⊕ α Q α → N ′ of A -mo dules which induces an epimorphism on π 0 . Then there exists ﬁnite collection { Q i } suc h that the induced map ( ⊕ i Q i ) ⊗ A Sym ∗ A ( M ) → N ′ of Sym ∗ A ( M ) -mo dules induces an epimorphism on π 0 , by taking the ﬁltered diagram of π 0 Sym ∗ A ( M ) / Im π 0 ( ⊕ i Q i ⊗ A Sym ∗ A ( M )) → π 0 B where J 2 ⊂ I 2 is a ﬁnite subset. T ak e N = ⊕ i Q i , w e are done. Remark 4.18. A natural question arises: given a pro jectiv e ob ject P ∈ A ≥ 0 , is π 0 Sym ∗ ( P ) necessarily 1-pro jectiv e in the heart A ♡ ? How ev er, the answer is negativ e. Consider the pro jectiv ely rigid ttt - ∞ - 45 category A ≥ 0 = P Σ (Gr(Ab) 1 -cpro j ) arising in the con text of Dirac geometry as introduced by [ HP23 ]. Let P = Z [1] ∈ Syn ( Z ) cpro j ≥ 0 , whic h is a compact pro jectiv e ob ject. How ev er, its symmetric algebra satisﬁes π 0 Sym ∗ ( Z [1]) ∼ = Z [ x ] / (2 x 2 ) , which is not a 1-pro jective ob ject in the heart Gr(Ab) . 5 Descendable algebras and idemp oten t algebras The goal of this section it to inv estigate descendable algebras and idemp otent algebras in the ttt - ∞ - categorical setting 5.1 F aithful algebras Descen t theory fundamentally requires a rigorous notion of faithful morphisms to ensure that relativ e tensor pro ducts do not obscure or annihilate non-trivial mo dules. In this subsection, w e deﬁne faithful and b oundedly faithful maps, proving a generalized Nak a y ama lemma for hypercomplete ttt - ∞ -categories: an y nilp otent thick ening is automatically b oundedly faithful. Deﬁnition 5.1. Let f : R → S ∈ Alg ( A ) . (1) W e say f is left faithful if the base change functor f ! = S ⊗ R ( − ) : LMo d R ( A ) → LMo d S ( A ) is conserv ative. (2) W e say f is b oundedly left faithful if f ∈ Alg ( A ≥ 0 ) and the tensor pro duct functor f ! = S ⊗ R ( − ) : LMo d R ( A ) − → LMo d S ( A ) − is conserv ative when restricting on b ounded b elow mo dules. Con v en tion 5.2. When R → S is a map of E 1 -algebras, w e simply refer to left faithful as faithful . Deﬁnition 5.3. Let α : ˜ A → A b e a map in Alg ( A ≥ 0 ) . W e say α is a nilp oten t thick ening if I = Ker ( π 0 α ) is a nilpotent ideal of π 0 ˜ A (i.e. I n = 0 for some n ≥ 1 ). Prop osition 5.4 (Nak a y ama lemma) . Assume that A is hyp er c omplete. L et α : ˜ A → A b e a nilp otent thickening in Alg ( A ≥ 0 ) . Then α is b oth b ounde d ly left and right faithful. Pr o of. W e only prov e the left b ounded faithfulness, as the argument for the right one is similar. Let I denote the Ker ( π 0 α ) . Supp ose that M ∈ LMod ˜ A ( A ) − is a b ounded b elo w mo dule suc h that A ⊗ ˜ A M = 0 . W e wish to show M = 0 . Now supp ose that M  = 0 , without loss of generalization, we can assume that M is connective and π 0 M  = 0 . Then π 0 A ⊗ π 0 ˜ A π 0 M ≃ τ ≤ 0 ( A ⊗ ˜ A M ) = 0 where ⊗ denotes the tensor pro duct in the heart. Therefore I · π 0 M = π 0 M . How ever I is nilp otent so π 0 M = 0 , which leads to a con tradiction. Remark 5.5. In general, a nilp oten t thick ening is not faithful, and hence not descendable. A basic coun ter-example is the truncation map S → H Z of E ∞ -ring sp ectra, which is a nilp otent thick ening but not faithful. Indeed, given a prime p and a natural num ber n , consider the sp ectrum K ( n ) of the corresp onding Morav a K-theory . Then we ha v e H ∗ ( K ( n )) = 0 , while π ∗ ( K ( n ))  = 0 (see [ Rud98 , Chap. IX.7.27]). That means the base c hange functor Sp ≃ Mo d S (Sp) → Mo d H Z (Sp) ≃ D ( Z ) is not conserv ative. 46 5.2 Descendable algebras Building conceptually on the condition of faithfulness, w e explore descendable algebras, which structurally generate the entire mo dule category as a thick ideal. By rigorously analyzing the augmented Čech nerve, w e demonstrate that a faithfully ﬂat map whose π 0 -truncation is an ℵ n -compact mo dule automatically is descendable, generalizing a result in [ Mat16 ]. Deﬁnition 5.6. Let C ⊗ ∈ CAlg ( P r L st ) and I ⊂ C b e a full sub category . W e s a y I is a thic k ideal of C if I ⊂ C is a stable sub category , closed under retractions and the following condition holds: F or an y x ∈ C and y ∈ I we hav e x ⊗ y ∈ I . Deﬁnition 5.7. Let C ⊗ ∈ CAlg ( P r L st ) and let f : A → B ∈ CAlg ( C ) . W e sa y f is descendable if the smallest thic k ideal of Mo d A ( C ) suc h that con tains B is Mo d A ( C ) itself. Prop osition 5.8 (See [ Mat16 ] 3.19) . L et C ⊗ ∈ CAlg ( P r L st ) and let f : A → B ∈ CAlg ( C ) . If f is desc endable, then f is faithful. Deﬁnition 5.9. Let C ⊗ ∈ CAlg ( P r L st ) and let f : A → B ∈ CAlg ( C ) . W e deﬁne the augmented cosimplical ob ject B • +1 : ∆ + → CAlg( C ) A/ b e the Cec h nerv e in (CAlg( C ) A/ ) op . Prop osition 5.10 (See [ Mat16 ] 3.20) . L et C ⊗ ∈ CAlg ( P r L st ) and let f : A → B ∈ CAlg ( C ) . Then the fol lowing c onditions ar e e quivalent: (1) A → B is desc endable. (2) B • +1 is a ∆ -limit diagr am in Pro(Mo d A ( C )) . Pr o of. W e ﬁrst prov e that (1) ⇒ (2) . Let B denote th e full sub category of Mo d A ( C ) spanned by those ob jects M for which the canonical map θ M : M ⊗ A A → M ⊗ A B • +1 is an equiv alence in Pro ( Mo d A ( C )) . By Corollary B.13 , we see that Pro ( Mo d A ( C )) inherits a symmetric monoidal structure that is compatible with limits, and hence B is a thick ideal of Mo d A ( C ) . It will suﬃce to show that B ∈ B . This is clear, since B ⊗ A B • +1 can b e iden tiﬁed with the split cosimplicial ob ject B • +1 . No w supp ose that (2) is satisﬁed. Let D denote the smallest thick ideal of Mo d A ( A ) which contains B . Then B • is a cosimplicial ob ject of D and each term T ot n ( B / A ) in the tow er T ot • ( B / A ) is in D to o. Assumption (2) implies that A ≃ lim ← − n T ot n ( B / A ) in Pro ( Mo d A ( A )) . How ev er, A is co compact in Pro ( Mo d A ( A )) , so A is equiv alen t to a retract of T ot n ( B / A ) for some in teger n , so that A ∈ D . That implies D = Mo d A ( A ) . Remark 5.11. F rom Prop osition 5.10 , w e see that the descendable condition is stronger than that B • +1 is a ∆ -limit merely in Mo d A ( A ) . Lemma 5.12. L et C b e a stable ∞ -c ate gory and let K b e a ﬁnite simplicial set. Given an arbitr ary diagr am F : K → C . The limit lim ← − K F c an b e expr esse d (c anonic al ly) as a ﬁnite c olimit of shifte d ﬁnite c opr o ducts of the obje cts F ( k ) . Similarly, the c olimit colim − − − → K F c an b e expr esse d (c anonic al ly) as a ﬁnite limit of shifte d ﬁnite pr o ducts of the obje cts F ( k ) . Pr o of. W e only prov e the ﬁrst statement, b ecause the pro of of the second one is totally parallel. W e compute the limit of F via its Bousﬁeld-Kan cosimplicial replacement (see [ Ram ]). Consider the 47 cosimplicial ob ject C • ∈ F un (∆ , C ) whose n -th degree term is giv en b y the pro duct ov er all n -simplices of K : C n = Y σ ∈ N ( K ) n F ( σ ( n )) , where N ( K ) n denotes the set of n -simplices of the nerve of K , and σ ( n ) ∈ K is the ﬁnal vertex of the simplex σ . Since K is a ﬁnite diagram, the set of non-degenerate simplices in N ( K ) is ﬁnite, meaning there exists a maximum dimension d ⩾ 0 such that all simplices in N ( K ) n for n > d are degenerate. F urthermore, b ecause C is a stable ∞ -category , ﬁnite pro ducts and ﬁnite copro ducts canonically coincide. Th us, we can rewrite eac h term as a ﬁnite direct sum: C n ≃ M σ ∈ N ( K ) n F ( σ ( n )) . By the general theory of ∞ -categorical limits, the limit of the diagram F is canonically equiv alen t to the limit (the totalization) of the cosimplicial ob ject C • : lim ← − K F ≃ T ot( C • ) = lim ← − ∆ C • . Because the normalized chain complex associated to the cosimplicial ob ject C • v anishes in degrees strictly greater than d , C • is a b ounded cosimplicial ob ject. In a stable ∞ -category , the totalization of a b ounded cosimplicial ob ject reduces to a ﬁnite limit, which by stabilit y is canonically equiv alen t to a ﬁnite colimit of its shifted terms. Speciﬁcally , it can b e computed as the geometric realization of the asso ciated shifted complex via iterated coﬁb ers: lim ← − K F ≃ colim  C d [ − d ] → C d − 1 [ − ( d − 1)] → · · · → C 1 [ − 1] → C 0  , where the connecting maps are constructed from the alternating sums of the asso ciated coface maps. Consequen tly , the limit lim ← − K F is obtained purely as a ﬁnite colim it of the ob jects C n [ − n ] , each of whic h is a shifted ﬁnite direct sum of the original ob jects ev aluated by F in K . Prop osition 5.13. L et C ⊗ ∈ CAlg ( P r L st ) and let f : A → B ∈ CAlg ( C ) . Then the fol lowing ar e e quivalent: (1) A → B is desc endable. (2) L et C ⊂ Mo d A ( C ) b e the smal lest thick sub c ate gory which c ontains the essential image of the for getful functor Mo d B ( C ) → Mo d A ( A ) . Then C = Mo d A ( C ) . (3) L et C ⊂ Mo d A ( C ) b e the smal lest thick ide al which c ontains the essential image of the for getful functor Mo d B ( C ) → Mo d A ( C ) . Then C = Mo d A ( C ) . (4) A , r e gar de d as an A -mo dule, c an b e obtaine d as a r etr act of a ﬁnite c olimit of a diagr am of A -mo dules, e ach of which c an b e lifte d to a B -mo dule. (5) A , r e gar de d as an A -mo dule, c an b e obtaine d as a r etr act of a ﬁnite limit of a diagr am of A -mo dules, e ach of which c an b e lifte d to a B -mo dule. Pr o of. The implications (2) ⇒ (3) and (4) ⇒ (3) are clear. By Lemma 5.12 , we obtain (4) ⇔ (5) . The equiv alence (1) ⇔ (2) follo ws from the same argument as in [ SAG , V ariant D.3.2.3]. It therefore suﬃces to pro ve (1) ⇒ (5) and (3) ⇒ (2) . 48 (1) ⇒ (5) Supp ose that f : A → B is descendable. By the pro of of Prop osition 5.10 , A is equiv alen t to a retract of T ot n ( B / A ) for some integer n ≥ 0 . Since the ﬁnite totalization T ot n ( B / A ) can b e built as a ﬁnite limit of the individual cosimplicial terms B k +1 = B ⊗ A · · · ⊗ A B ( 0 ⩽ k ⩽ n ). Since each B k +1 naturally admits the structure of a B -mo dule, A is indeed a retract of a ﬁnite limit of ob jects that can b e lifted to B -mo dules. (3) ⇒ (2) Let B ⊂ Mo d A ( C ) b e the smallest thick sub category which contains the essential image of the forgetful functor Mo d B ( C ) → Mo d A ( C ) . W e sho w that B has actually b een a thic k ideal. Let B ′ ⊂ B denote the full sub category spanned by those ob jects M suc h that N ⊗ A M ∈ B for any A -mo dule N . Since B ′ is a thick sub category to o, it suﬃces to sho w the case when M admits a B -mo dule structure B M . In this case N ⊗ A M ≃ A (( N ⊗ A B ) ⊗ B M ) admits a B -mo dule structure, hence M ∈ B ′ . Consequently , B = B ′ is a thic k ideal. Lemma 5.14 (See [ Mat16 ] 3.21, 3.24) . L et C ⊗ ∈ CAlg( P r L st ) . (1) L et f : R → S b e a desc endable morphism in CAlg ( C ) and R → A b e another map in CAlg ( C ) . Then the map A → A ⊗ R S given by the fol lowing pushout diagr am is desc endable. R S A A ⊗ R S (2) L et A → B → C b e maps in CAlg( C ) . Then the fol lowing hold: (a) If A → B and B → C ar e desc endable, so is A → C . (b) If A → C is desc endable, so do es A → B . Lemma 5.15 (See [ SAG ] D.3.3.6) . L et n b e a nonne gative inte ger, let J b e a ﬁlter e d p artial ly or der e d set of c ar dinality ≤ ℵ n , and let { X j } j ∈ J b e a diagr am of sp ac es indexe d by J op . If e ach of the sp ac es X j is m -c onne ctive for some inte ger m , then the inverse limit lim ← − j ∈ J X j is ( m − n ) -c onne ctive. Lemma 5.16. Supp ose that the ttt - ∞ -c at e gory ( A ⊗ , A ≥ 0 ) is pr oje ctively rigid. L et A ∈ Alg ( A ≥ 0 ) , let M b e a ﬂat left A -mo dule, and let N b e a c onne ctive left A -mo dule. Assume that π 0 M is an ℵ n -c omp act obje ct of the c ate gory of discr ete π 0 A -mo dules for some n ≥ 0 . Then Ext m A ( M , N ) ≃ 0 for m > n . Pr o of. The following argument is parallel with [ SAG , Lem. D.3.3.7]. Let us iden tify N with the limit of its P ostniko v tow er · · · → τ ≤ 2 N → τ ≤ 1 N → τ ≤ 0 N , so that w e ha ve a Milnor exact sequence 0 → lim 1  Ext m − 1 A ( M , τ ≤ k N )  → Ext m A ( M , N ) → lim ← − k { Ext m A ( M , τ ≤ k N ) } → 0 . It will therefore suﬃce to sho w that the ab elian groups lim ← − 1  Ext m − 1 A ( M , τ ≤ k N )  and lim ← − k { Ext m A ( M , τ ≤ k N ) } are trivial for m > n . T o prov e this, we will show that the maps Ext m − 1 A ( M , τ ≤ k N ) → Ext m − 1 A ( M , τ ≤ k − 1 N ) 49 are surjective for k ≥ 1 , and that the groups Ext m ( M , τ ≤ k N ) v anish for all k ≥ 0 . Using the exact sequences Ext m − 1 A ( M , τ ≤ k N ) → Ext m − 1 A ( M , τ ≤ k − 1 N ) → Ext m + k A ( M , π k N ) → Ext m A ( M , τ ≤ k N ) → Ext m A ( M , τ ≤ k − 1 N ) → Ext m +1+ k A ( M , π k N ) w e are reduced to proving that the groups Ext m + k A ( M , π k N ) v anish for all k ≥ 0 . Replacing m b y m + k and N b y π k N , we can further reduce to the case where N is discrete. In this case, w e hav e a canonical isomorphism Ext m A ( M , N ) ≃ Ext m π 0 A ( π 0 A ⊗ A M , N ) . W e may therefore replace A b y π 0 A (and M b y π 0 A ⊗ A M ) and thereby reduce to the case where A is discrete. Since M is ﬂat ov er A , it follows that M is also discrete. Since M is ﬂat ov er A , it can b e written as the colimit of a diagram { M α } α ∈ P indexed by a ﬁltered partially ordered set P , where each M α is a compact pro jectiv e left A -mo dule ( Theorem 3.20 ). In the case n = 0 , it follows that M is compact pro jectiv e and the conclusion is deduced by Prop osition 3.7 . In the case n ≥ 1 , for eac h ℵ n -small ﬁltered subset P ′ ⊂ P , let M P ′ denote the colimit lim − → α ∈ P ′ M α . Then b y [ Ker , 061J ], when n ≥ 1 , M can b e written as a ﬁltered colimit of the diagram { M P ′ } , where P ′ ranges o ver all ℵ n -small ﬁltered subsets of P . Since M is ℵ n -compact, the iden tity map id M : lim − → P ′ M P ′ → M factors through some M P ′ , so that M is a retract of M P ′ . W e ma y therefore replace M by M P ′ and P b y P ′ , and thereb y reduce to the case where P is ℵ n -small. W e ha ve a canonical isomorphism Ext m A ( M , N ) ≃ π 0 Map LMod A ( M , Σ m N ) ≃ π 0 lim ← − α ∈ P Map LMod A ( M α , Σ m N ) T o show that this group v anishes, it will suﬃce (by virtue of Lemma 5.15 ) to show that the mapping spaces Map LMod A ( M α , Σ m N ) are n -connectiv e for each α ∈ P . This is clear, since M α is a compact pro jectiv e left A -mo dule and Σ m N is n -connective. Theorem 5.17. Supp ose that the ttt - ∞ -c ate gory ( A ⊗ , A ≥ 0 ) is pr oje ctively rigid. L et φ : A → B ∈ CAlg ( A ) b e a faithful ly ﬂat map such that π 0 B is a ℵ n -c omp act π 0 A -mo dule for some n ≥ 0 . Then f is desc endable. Pr o of. The following argument is parallel with the pro of of [ SAG , Prop. D.3.3.1]. Since B is ﬂat ov er A , b y Prop osition 2.22 we can iden tify B with the image of its connective cov er τ ≥ 0 B under the base change functor Mo d τ ≥ 0 A ( A ) → Mo d A ( A ) . By virtue of Lemma 5.14 , to pro ve that A → B is descendable, it will suﬃce to show that τ ≥ 0 A → τ ≥ 0 B is descendable. W e may therefore replace φ b y the induced map τ ≥ 0 A → τ ≥ 0 B and thereby reduce to the case where A is connective. Let C denote the smallest stable sub category of Mo d A ( A ) which contains all ob jects of the form M ⊗ A B and is closed under retracts. It will suﬃce to sho w that A b elongs to C . Let K b e the ﬁb er of the map φ : A → B , and let ρ : K → A b e the canonical map. F or eac h integer m ≥ 0 , let ρ ( m ) : K ⊗ m → A ⊗ m ≃ A b e the m th tensor p ow er of ρ , formed in the monoidal ∞ -category LMo d A . Then ρ ( m + 1) is giv en b y the comp osition K ⊗ m +1 id K ⊗ m − − − − → K ⊗ m ρ ( m ) − − − → A , so we hav e a ﬁb er sequence K ⊗ m ⊗ A B → coﬁb( ρ ( m + 1)) → coﬁb( ρ ( m )) It follows b y induction on m that each coﬁb ( ρ ( m )) b elongs to C . Consequently , to prov e that A ∈ C , it will suﬃce to show that A is a retract of coﬁb ( ρ ( m )) for some m ≥ 0 . This condition holds if the homotop y class of ρ ( m ) v anishes (when regarded as an element of Ext 0 A ( K ⊗ m , A ) ≃ Ext m A ((Σ K ) ⊗ m , A ) . The Prop osition 2.25 (4) implies that Σ K ≃ coﬁb ( φ ) is a ﬂat A -mo dule. Also w e ha ve that π 0 coﬁb ( K ) 50 is ℵ n -compact π 0 A -mo dule. It follows that (Σ K ) ⊗ m has the same prop erties for each m > 0 , so that Ext m A ((Σ K ) ⊗ m , A ) v anishes for m > n by virtue of Lemma 5.16 . 5.3 Almost algebra theory W e presen t a ma jor application of our geometric framework b y generalizing the higher almost ring theory in tro duced b y Heb estreit and Scholze [ HS24 ]. In this subsection, we completely classify epimorphic idemp oten t commutativ e algebras via idemp otent ideals. W e now introduce the main result of this subsection. Theorem 5.18. Assume that A is b oth right c omplete and left c omplete. L et R ∈ CAlg ( A ≥ 0 ) . Consider the ful l su b c ate gory LQ R of CAlg ( A ≥ 0 ) R/ sp anne d by the maps ϕ : R → S for which (1) the multiplic ation S ⊗ R S → S is an e quivalenc e, i.e. ϕ is idemp otent, (2) π 0 ( ϕ ) : π 0 R → π 0 S is epimorphic in A ♡ . Then the functor LQ R − →  I ⊂ π 0 R | I 2 = I  , ϕ 7− → Ker( π 0 ϕ ) is an e quivalenc e of c ate gories, wher e we again r e gar d the tar get as a p oset via the inclusion or dering. The inverse image of some I ⊂ π 0 ( R ) c an b e describ e d mor e dir e ctly as R/I ∞ , wher e I ∞ = lim ← − n ∈ N op J ⊗ R n I with J I → R the ﬁbr e of the c anonic al map R → H ( π 0 ( R ) /I ) . F urthermor e, this inverse system stabilises on π i for n > i + 1 . Mor e over, the image of the ful ly faithful r estriction functor Mo d R/I ∞ ( A ) → Mo d R ( A ) c onsists exactly of those mo dules whose homotopy is kil le d by I . Pr o of. W e only do slight mo diﬁcations of original pro of in [ HS24 ] to ﬁt into our framework. Firstly w e observ e that LQ R is indeed equiv alent to a p oset [see HA , Prop osition 4.8.2.9]. Let us also immediately v erify that Ker ( π 0 ϕ ) is indeed idemp otent for ϕ : R → S ∈ LQ R . T ensoring the ﬁbre sequence F → R → S with F giv es F ⊗ R F − → F − → F ⊗ R S and the righ t hand term v anishes since one has a ﬁbre sequence F ⊗ R S − → R ⊗ R S − → S ⊗ R S whose righ t hand map (after identifying R ⊗ R S ≃ S ) is a section of the multiplication S ⊗ R S → S and th us an equiv alence. But F is connective and the map π 0 ( F ) → Ker ( π 0 ϕ ) surjective by the long exact sequence of ϕ , whence a chase in the diagram shows that the multiplication Ker ( π 0 ϕ ) ⊗ π 0 R Ker ( π 0 ϕ ) → Ker ( π 0 ϕ ) is surjectiv e as desired. Next, w e v erify that the inv erse system J ⊗ R n I stabilises degreewise. In fact we show slightly more, namely that the coﬁbre R/J I ⊗ R J ⊗ R n I of the canonical map J ⊗ R n +1 I → J ⊗ R n I is n -connectiv e. Since 51 R/J I = H( π 0 ( R ) /I ) is annihilated by I , we immediately deduce that the homotopy groups of this coﬁbre are annihilated b y I (from b oth sides). No w, for n = 0 , the connectivity claim is clear, and if w e inductiv ely assume that R/J I ⊗ R J ⊗ R n I is n -connectiv e, then R/J I ⊗ R J ⊗ R n +1 I =  R/J I ⊗ R J ⊗ R n I  ⊗ R J I is clearly also n -connectiv e and its n th homotopy group is π n ( R/J I ⊗ R J ⊗ n R I ) ⊗ π 0 R I . Since the left hand term is annihilated by I , we compute π n  R/J I ⊗ R J ⊗ R n I  ⊗ π 0 R I = π n  R/J I ⊗ R J ⊗ R n I  ⊗ π 0 ( R ) /I π 0 ( R ) /I ⊗ π 0 R I = π n  R/J I ⊗ R J ⊗ R n I  ⊗ π 0 ( R ) /I I /I 2 = 0 whic h complete the induction. No w we claim lim ← − n ∈ N op J ⊗ R n I ! ⊗ R J I − → lim ← − n ∈ N op J ⊗ R n I is an equiv alence. Since the limit stabilises degreewise and J I is connectiv e, w e can mo ve the limit out of the tensor pro duct b y Prop osition 1.20 (the coﬁbre of the interc hange map is a limit of terms with gro wing connectivity), and then the statemen t follows from ﬁnality . By the same argumen t, for any n ≥ 1 the canonical map I ∞ ⊗ R J ⊗ R n I → I ∞ is equiv alen t to o and hence I ∞ ⊗ R I ∞ → I ∞ is an equiv alence—that is to say R → R/I ∞ is idemp otent and hence produ ces an element in CAlg ( A ≥ 0 ) idem R/ . As the next step, we show that the tautological map M = R ⊗ R M → R/I ∞ ⊗ R M is an equiv alence if and only if the homotop y of M is annihilated by I , or in other words that I ∞ ⊗ R M ≃ 0 . F or the “only if ” direction, it suﬃces to observe that π 0 ( R/I ∞ ) = π 0 R/I . F or “if ” direction, we start with the simplest case M = R/J I , where the claim was prov ed abov e. F or an arbitrary R -mo dule M concen trated in degree 0 and killed by the action of I , it naturally inherits a H( π 0 R/I ) -mo dule structure and hence w e get a retraction of R -mo dules I ∞ ⊗ R M → I ∞ ⊗ R M ⊗ R H ( π 0 R/I ) → I ∞ ⊗ R M . Ho wev er the middle one is zero by commutativit y of ⊗ R (note H( π 0 R/I ) = R/J I ), so I ∞ ⊗ R M = 0 . F or b ounded b elow M , we hav e I ∞ ⊗ R M ≃ I ∞ ⊗ R lim ← − k ∈ N op τ ≤ k M ! ≃ lim ← − k ∈ N ◦ p I ∞ ⊗ R τ ≤ k M ≃ 0 b y commuting the limit out using the same argument as ab ov e. Finally , for arbitrary M whose homotop y groups are killed by I , we ﬁnd I ∞ ⊗ R M ≃ I ∞ ⊗ R (colim k ∈ N τ ≥− k M ) ≃ colim k ∈ N I ∞ ⊗ R τ ≥− k M ≃ 0 . So combined with the idemp otent prop erty of R → R/I ∞ , we learn that the image of the fully faithful restriction functor Mo d R/I ∞ ( A ) → Mo d R ( A ) consists exactly of those mo dules whose homotopy is killed 52 b y I , as desired. Finally , we are ready to verify that the construction I 7→ ( R → R/I ∞ ) induces an inv erse to taking k ernels. The comp osition starting with an ideal is clearly the identit y . So we are left to show that for every ϕ : R → S in LQ R with I = Ker ( π 0 ϕ ) , the canonical map ψ : R/I ∞ → S , arising from the homotop y of S b eing annihilated b y I , is an equiv alence. Per construction it induces an equiv alence on π 0 . By Prop osition 5.4 , the functor ψ ! = S ⊗ R/I ∞ − : Mo d R/I ∞ ( A ) → Mo d S ( A ) is th us conserv ativ e when restricted to b ounded b elow mo dules. But the map S ≃ ψ ! ( R/I ∞ ) ψ ! ( φ ) − − − → ψ ! ( S ) = S ⊗ R/I ∞ S ≃ S ⊗ R S is induced b y the unit and thus an equiv alence since ϕ is idemp otent. W e can also pro ve the non-commutativ e v ersion of Theorem 5.18 , which in v olv es some diﬀerent tec hniques ab out asso ciative non-unital algebras. Lemma 5.19. L et C ⊗ ∈ Alg ( Cat ∅ ∞ ) b e a monoidal ∞ -c ate gory c omp atible with the empty c olimit. Then the for getful functor G : Alg nu ( C ) → C cr e ates the empty c olimit. Pr o of. W e can compute the relative left Kan extension of ∅ ∈ C ⊗ along the inclusion of the trivial op erad in to the op erad of non-unital algebras ∗ ≃ { [1] }  → ∆ op s, ≥ 1 , as follo ws: ∗ C ⊗ ∆ op s, ≥ 1 ∆ op ∅ A Because C ⊗ is compatible with the empt y colimit, this relative left Kan extension ev aluates to the initial ob ject, giving A ( ⟨ n ⟩ ) = ∅ for eac h n ≥ 1 . By the prop erty of the relative Kan extension, A is the initial ob ject of F un inert / ∆ op (∆ op s, ≥ 1 , C ⊗ ) . Since A lies in Alg nu ( C ) ⊂ F un inert / ∆ op (∆ op s, ≥ 1 , C ⊗ ) , we see that A is initial in Alg nu ( C ) , whose underlying ob ject is ∅ ∈ C . Consequently G creates the empty colimit b ecause G is conserv ative. Lemma 5.20. L et C ⊗ ∈ Alg ( Cat ∅ ∞ ) b e a monoidal ∞ -c ate gory c omp atible with the empty c olimit. Supp ose that C is p ointe d and admits ﬁnite limits. L et S ∈ Alg( C ) b e an E 1 -algebr a. Then the kernel I = ker( 1 → S ) i − → 1 is a non-unital E 1 -algebr a satisfying that µ I ≃ 1 ⊗ i ≃ i ⊗ 1 . Pr o of. Note that Alg nu ( C ) creates any limit that exists in C (see [ HA , §3.2.2]). By Lemma 5.19 , 0 is the initial ob ject in Alg nu ( C ) and hence a zero ob ject in it. Also, I admits a natural non-unital algebra 53 structure and c an b e identiﬁed with the ﬁb er of 1 → S . Considering the following commutativ e diagram: I ⊗ I 0 ⊗ 0 1 ⊗ 1 S ⊗ S I ⊗ 1 0 ⊗ S 1 ⊗ 1 S ⊗ S I 0 1 S µ I µ 1 µ S The back face represents the limits deﬁning the zero ob ject ov er S , while the top and fron t faces inv olv e the limit deﬁnition of I . Because the target of I ⊗ I o ver S is zero, the universal property of the limit induces the dashed map µ I : I ⊗ I → I . F urthermore, the commutativit y of the left and b ottom prisms forces this multiplication to factor through the left and righ t unit maps. Therefore, w e naturally obtain the equiv alences µ I ≃ 1 ⊗ i ≃ i ⊗ 1 , making I a non-unital E 1 -algebra with this trivialized m ultiplication. Lemma 5.21. L et φ : R → S ∈ Alg ( A ) . Then the b ase change functor LMo d R ( A ) → LMo d S ( A ) is a lo c alization if and only if the multiplic ation S ⊗ R S → S is an e quivalenc e (as R - R bimo dules). Pr o of. Let F : LMo d R ( A ) → LMo d S ( A ) denote the base change functor given by M 7→ S ⊗ R M , and let G : LMo d S ( A ) → LMo d R ( A ) b e its righ t adjoint, the restriction of scalars functor. The left adjoint F is a lo calization if and only if its right adjoint G is fully faithful, whic h is equiv alen t to the condition that the counit of the adjunction,  M : F ( G ( M )) → M , is an equiv alence for all S -mo dules M . (= ⇒ ) If F is a lo calization,  M is an equiv alence for all M . Ev aluating the counit at the free mo dule M = S , we obtain the multiplication map: S ⊗ R S ≃ F ( G ( S )) ϵ S − → S whic h must therefore b e an equiv alence. ( ⇐ =) Conv ersely , supp ose the multiplication map S ⊗ R S → S is an equiv alence. F or any S -mo dule M , w e can express M canonically as the relativ e tensor pro duct S ⊗ S M . The counit map applied to M corresp onds to the action map S ⊗ R M → M . W e can rewrite the domain as: S ⊗ R M ≃ S ⊗ R ( S ⊗ S M ) ≃ ( S ⊗ R S ) ⊗ S M Since S ⊗ R S ≃ − → S b y assumption, this reduces canonically to S ⊗ S M ≃ M . Thus, the counit is an equiv alence for all S -mo dules M , meaning G is fully faithful and F is a lo calization. Theorem 5.22. Assume that A is b oth right c omplete and left c omplete. L et R ∈ Alg ( A ≥ 0 ) . Consider the ful l su b c ate gory LQ R of Alg ( A ≥ 0 ) R/ sp anne d by E 1 -maps ϕ : R → S for which 54 (1) the multiplic ation S ⊗ R S → S is an e quivalenc e, i.e. ϕ is idemp otent, (2) π 0 ( ϕ ) : π 0 R → π 0 S is epimorphic in A ♡ . Then the functor LQ R − →  two-side d ide als I ⊂ π 0 R | I 2 = I  , ϕ 7− → Ker( π 0 ϕ ) is an e quivalenc e of c ate gories, wher e we again r e gar d the tar get as a p oset via the inclusion or dering. The inverse image of some I ⊂ π 0 ( R ) c an b e describ e d mor e dir e ctly as R/I ∞ , wher e I ∞ = lim ← − n ∈ N op J ⊗ R n I with J I → R the ﬁbr e of the c anonic al map R → H ( π 0 ( R ) /I ) . F urthermor e, this inverse system stabilises on π i for n > i + 1 . The image of the ful ly faithful r estriction functor LMo d R/I ∞ ( A ) → LMo d R ( A ) c onsists exactly of those mo dules whose homotopy is kil le d by I , as desir e d. Pr o of. The non-commutativ e case is a bit more tric ky . Firstly we observe that LQ R is indeed equiv alen t to a p oset [see HA , Prop osition 4.8.2.9]. Let us also immediately verify that Ker ( π 0 ϕ ) is indeed idemp otent for ϕ : R → S ∈ LQ R . Relativ ely tensoring the coﬁb er sequence (of R - R -bimo dules) F → R → S with F gives F ⊗ R F − → F − → F ⊗ R S and the righ t hand term v anishes since one has a coﬁb er sequence (of R - R -bimo dules) F ⊗ R S − → R ⊗ R S − → S ⊗ R S whose righ t hand map (after identifying R ⊗ R S ≃ S ) is a section of the multiplication S ⊗ R S → S and th us an equiv alence. But F is connective and the map π 0 ( F ) → Ker ( π 0 ϕ ) surjective by the long exact sequence from ϕ , whence a chase in the diagram shows that the multiplication Ker ( π 0 ϕ ) ⊗ π 0 R Ker ( π 0 ϕ ) → Ker ( π 0 ϕ ) is surjectiv e as desired. 9 Next, w e verify that the inv erse system J ⊗ R n I (with the relativ e tensor pro duct ⊗ R ) stabilises degreewise. In fact we show sligh tly more, namely that the coﬁbre R/J I ⊗ R J ⊗ R n I of the canonical map J ⊗ R n +1 I → J ⊗ R n I is n -connectiv e. Now, for n = 0 , the connectivity claim is clear, and if we inductively assume that R /J I ⊗ R J ⊗ R n I is n -connectiv e, then R/J I ⊗ R J ⊗ R n +1 I =  R/J I ⊗ R J ⊗ R n I  ⊗ R J I is clearly also n -connectiv e and its n th homotopy group is π n ( R/J I ⊗ R J ⊗ n R I ) ⊗ π 0 R I . By Lemma 5.20 , 9 This paragraph is the main diﬀerence from the argumen t of Theorem 5.18 . 55 J I ∈ Alg nu ( R BMo d R ( A )) and we hav e the follo wing comm utative diagram J I ⊗ R J I R ⊗ R J I J I ⊗ R J I J I ⊗ R R J I i ⊗ 1 ∼ 1 ⊗ i ∼ hence R/J I ⊗ R J I ≃ J I ⊗ R R/J I in R BMo d R ( A ) . Therefore the left hand term R/J I ⊗ R J ⊗ R n I is annihilated b y I from b oth sides, we compute π n  R/J I ⊗ R J ⊗ R n I  ⊗ π 0 R I = π n  R/J I ⊗ R J ⊗ R n I  ⊗ π 0 ( R ) /I π 0 ( R ) /I ⊗ π 0 R I = π n  R/J I ⊗ R J ⊗ R n I  ⊗ π 0 ( R ) /I I /I 2 = 0 whic h complete the induction. No w we claim lim ← − n ∈ N op J ⊗ R n I ! ⊗ R J I − → lim ← − n ∈ N op J ⊗ R n I is an equiv alence. Since the limit stabilises degreewise and J I is connectiv e, w e can mo ve the limit out of the tensor pro duct b y Prop osition 1.20 (the coﬁbre of the interc hange map is a limit of terms with gro wing connectivity), and then the statemen t follows from ﬁnality . By the same argumen t, for any n ≥ 1 the canonical map I ∞ ⊗ R J ⊗ R n I → I ∞ is equiv alen t to o and hence I ∞ ⊗ R I ∞ → I ∞ is an equiv alence—that is to say R → R/I ∞ is idemp otent and hence produ ces an element in Alg ( A ≥ 0 ) idem R/ . As the next step, w e show that for a left R -mo dule M , the tautological map M = R ⊗ R M → R/I ∞ ⊗ R M is an equiv alence (in other w ords that I ∞ ⊗ R M ≃ 0 ) if and only if the homotopy of M is annihilated by I . F or the “only if ” direction, it suﬃces to observe that π 0 ( R/I ∞ ) = π 0 R/I . F or “if ” direction, we start with the simplest case M = R/J I , where the claim was prov ed ab ov e. F or an arbitrary left R -mo dule M concen trated in degree 0 and killed by the action of I , it naturally inherits a H( π 0 R/I ) -mo dule structure and hence w e ha ve equiv alences of left R -mo dules I ∞ ⊗ R M ≃ I ∞ ⊗ R H( π 0 R/I ) ⊗ H( π 0 R/I ) M . Ho wev er I ∞ ⊗ R H( π 0 R/I ) is zero by H( π 0 R/I ) = R/J I , so I ∞ ⊗ R M = 0 . F or b ounded b elow M , by left completeness we ha ve I ∞ ⊗ R M ≃ I ∞ ⊗ R lim ← − k ∈ N op τ ≤ k M ! ≃ lim ← − k ∈ N ◦ p I ∞ ⊗ R τ ≤ k M ≃ 0 b y commuting the limit out using the same argument as ab ov e. Finally , for arbitrary M whose homotop y groups are killed by I , we ﬁnd I ∞ ⊗ R M ≃ I ∞ ⊗ R (colim k ∈ N τ ≥− k M ) ≃ colim k ∈ N I ∞ ⊗ R τ ≥− k M ≃ 0 . So combined with the idemp otent prop erty of R → R/I ∞ , we learn that the image of the fully faithful restriction functor LMo d R/I ∞ ( A ) → LMo d R ( A ) consists exactly of those left mo dules whose homotop y 56 is killed b y I , as desired. Finally , we are ready to verify that the construction I 7→ ( R → R/I ∞ ) induces an inv erse to taking k ernels. The comp osition starting with an ideal is clearly the identit y . So we are left to show that for every ϕ : R → S in LQ R with I = Ker ( π 0 ϕ ) , the canonical map ψ : R/I ∞ → S , arising from the homotop y of S b eing annihilated by I , is an equiv alence. By construction it induces an equiv alence on π 0 . By Prop osition 5.4 , the functor ψ ! = S ⊗ R/I ∞ − : LMo d R/I ∞ ( A ) → LMo d S ( A ) is thus conserv ativ e when restricted to b ounded b elow mo dules. But the map S ≃ ψ ! ( R/I ∞ ) ψ ! ( φ ) − − − → ψ ! ( S ) = S ⊗ R/I ∞ S ≃ S ⊗ R S is induced b y the unit and thus an equiv alence since ϕ is idemp otent. Remark 5.23. The argumen t ab ov e also applies to E k -algebras. How ev er, since we do not treat E k -algebras in detail in this pap er, we choose to omit it. 6 Deformation theory and étale rigidit y T o eﬀectively study étale maps, we must ﬁrst p ort the machinery of deriv ations and cotangen t complexes in to our generalized categorical setting. F ortunately , most of them has b een developed in [ HA , §7.3-7.5]. 6.1 The cotangent complex formalism In this subsection, w e recall the absolute cotangent complex functor and extended deriv ations. W e establish that square-zero extensions in h yp ercomplete ttt - ∞ -categories b ehav e faithfully with resp ect to relative cotangen t exact sequences. Besides that, we show that any square-zero extension forms a descendable algebra. Deﬁnition 6.1 (See [ HA ] 7.3.2.14.) . Let C b e a presentable ∞ -category , and consider the asso ciated diagram T C F un(∆ 1 , C ) C G p q where q is given b y ev aluation at { 1 } ⊂ ∆ 1 . The functor G carries p -Cartesian morphisms to q -Cartesian morphisms, and for eac h ob ject A ∈ C the induced map G A : Sp ( C / A ) → C / A admits a left adjoin t Σ ∞ . Applying [ HA , Prop osition 7.3.2.6], we conclude that G admits a left adjoint relative to C , which we will denote b y F . The absolute cotangen t complex functor L : C → T C is deﬁned to b e the comp osition C → F un(∆ 1 , C ) F − → T C where the ﬁrst map is given by the diagonal em b edding. W e will denote the v alue of L on an ob ject A ∈ C b y L A ∈ Sp(C / A ) , and will refer to L A as the cotangen t complex of A . Deﬁnition 6.2 (See [ HA ] 7.4.1.1) . Let C b e a presentable ∞ -category , and let p : M T ( C ) → ∆ 1 × C denote a tangent corresp ondence to C (see [ HA , Deﬁnition 7.3.6.9]). A deriv ation in C is a map f : ∆ 1 → M T ( C ) suc h that p ◦ f coincides with the inclusion ∆ 1 × { A } ⊂ ∆ 1 × C , for some A ∈ C . In this case, w e will 57 iden tify f with a morphism η : A → M in M T ( C ) , where M ∈ T C × C { A } ≃ Sp ( C / A ) . W e will also sa y η : A → M is a deriv ation of A into M . W e let Der ( C ) denote the ﬁb er pro duct F un (∆ 1 , M T ( C )) × F un(∆ 1 , ∆ 1 × C ) C . W e will refer to Der ( C ) as the ∞ -category of deriv ations in C . Remark 6.3. W e primarily care the case C = CAlg ( A ) . In this case, an ob ject in Der ( C ) can be informally describ ed as a triple data ( A, M , η : A → M [1]) where A ∈ CAlg ( A ) , M ∈ Mo d A ( A ) and η is a deriv ation. Deﬁnition 6.4 (See [ HA ] 7.4.1.3) . Let C b e a presentable ∞ -category , and let p : M T ( C ) → ∆ 1 × C b e a tangen t corresp ondence for C . An extended deriv ation is a diagram σ ˜ A A 0 M [1] f η in M T ( C ) with the following prop erties: (1) The ob ject 0 ∈ T C is a zero ob ject of Sp  C / A  . Equiv alen tly , 0 is a p -initial vertex of M T ( C ) . (2) The diagram σ is a pullbac k square. (3) The ob jects e A and A b elong to C ⊂ M T ( C ) , while 0 and M b elong to T C ⊂ M T ( C ) . (4) Let f : ∆ 1 → C b e the map whic h classiﬁes the morphism f app earing in the diagram ab o ve, and let e : ∆ 1 × ∆ 1 → ∆ 1 b e the unique map such that e − 1 { 0 } = { 0 } × { 0 } . Then the diagram is comm utative. ∆ 1 × ∆ 1 M T ( C ) ∆ 1 × C ∆ 1 C σ e p f W e let g Der( C ) denote the full sub category of F un  ∆ 1 × ∆ 1 , M T ( C )  × F un(∆ 1 × ∆ 1 , ∆ 1 × C ) F un  ∆ 1 , C  spanned b y the extended deriv ations. Notation 6.5. Throughout Section 6 , we let Der = Der ( CAlg ( A )) denote the ∞ -category of deriv ations in CAlg( A ) . W e let A η = ﬁb( η ) denote the corresp onding square-zero extension of A . W e deﬁne a sub category Der + ⊂ Der as follo ws: (1) An ob ject ( η : A → M [1]) ∈ Der b elongs to Der + if and only if b oth A and M are connective. Equiv alently , η b elongs to Der + if b oth A and A η are connectiv e, and the map π 0 A η → π 0 A is an epimorphism in A ♡ . (2) Let f : ( η : A → M [1]) → ( η ′ : B → N [1]) b e a morphism in Der b et ween ob jects which b elong to Der + . Then f b elongs to Der + if and only if the in duced map B ⊗ A M → N is an equiv alence of B -mo dules. W e now recall some basic prop erties ab out deriv ations. 58 Prop osition 6.6 (See [ HA ] 7.3.3.6) . L et A B C f h g b e morphisms in CAlg ( A ) . Then ther e exists a c anonic al c oﬁb er se quenc e C ⊗ B L B / A → L C / A → L C /B in Mo d C ( A ) . Prop osition 6.7 (See [ HA ] 7.3.3.7) . L et A ′ A B ′ B b e a pushout diagr am in CAlg ( A ) . Then ther e exists a c anonic al e quivalenc e L B / A ≃ B ⊗ B ′ L B ′ / A ′ in Mo d B ( A ) . Lemma 6.8. Assume that A is hyp er c omplete. L et f : ( η : A → M [1]) → ( η ′ : B → N [1]) b e a morphism in Der + . If the induc e d map A η → B η ′ is an e quivalenc e in CAlg ( A ) , then f is an e quivalenc e. (See [ HA , Lem. 7.4.2.9] for the case of sp ectra.) Pr o of. The morphism f determines a map of ﬁb er sequences A η A M [1] B η ′ B N [1] f 0 f 1 Since the left vertical map is an equiv alence, we obtain an equiv alence α : coﬁb ( f 0 ) ≃ coﬁb ( f 1 ) . T o complete the pro of, it will suﬃce to show that coﬁb ( f 0 ) v anishes. Supp ose otherwise. Since coﬁb ( f 0 ) is connectiv e and A is hypercomplete, there exists some smallest integer n suc h that π n coﬁb ( f 0 )  = 0 . In particular, coﬁb( f 0 ) is n -connective. Since f induces an equiv alence B ⊗ A M → N , coﬁb ( f 1 ) can b e iden tiﬁed with coﬁb ( f 0 ) ⊗ A M [1] . Since M is connective, we deduce that coﬁb ( f 1 ) is ( n + 1) -connectiv e. Using the equiv alence α , we conclude that coﬁb( f 0 ) is ( n + 1) -connectiv e, whic h contradicts our assumption that π n coﬁb( f 0 )  = 0 . Deﬁnition 6.9. W e deﬁne a sub category F un + (∆ 1 , CAlg( A )) as follows: (1) An ob ject f : e A → A of F un (∆ 1 , CAlg ( A )) b elongs to F un + (∆ 1 , CAlg ( A )) if and only if b oth A and e A are connective, and f induces a surjection π 0 e A → π 0 A . (2) Let f , g ∈ F un + (∆ 1 , CAlg ( A )) , and let α : f → g b e a morphism in F un (∆ 1 , CAlg ( A )) . Then α b elongs to F un + (∆ 1 , CAlg ( A )) if and only if it classiﬁes a pushout square in the ∞ -category CAlg( A ) . 59 Prop osition 6.10. Assume that A is hyp er c omplete. L et Φ : g Der → F un (∆ 1 , CAlg ( A )) b e the functor given by ( η : A → M [1]) 7→ ( A η → A ) . Then Φ induc es a functor Φ + : g Der + → F un + (∆ 1 , CAlg ( A )) . Mor e over, the functor Φ + is a left ﬁbr ation. Pr o of. It is a parallel pro of of [ HA , Lem. 7.4.2.7]. Remark 6.11. The hypercomplete condition in Prop osition 6.10 can not b e remo ved. How ev er, the only part inv olving the h yp ercompleteness in the argumen t of [ HA , Lem. 7.4.2.7] is Lemma 6.8 . Corollary 6.12. Assume that A is hyp er c omplete. L et A ∈ CAlg ( A ≥ 0 ) , M a c onne ctive A -mo dule, and η : A → M [1] a derivation. Then the functor Φ induc es an e quivalenc e of ∞ -c ate gories Der + η / → CAlg cn A η given on obje cts by ( η ′ : B → N [1]) 7→ B η ′ . W e end with a surprising result that any square zero extension is descendable 10 . Prop osition 6.13. L et η : A → M ∈ Der and α : ˜ A → A b e the induc e d squar e-zer o extension in CAlg( A ) . Then α is desc endable. Pr o of. By the deﬁnition of a square-zero extension, the deriv ation η : A → M induces a ﬁb er sequence in Mo d ˜ A ( A ) : M − → ˜ A α − → A. T o apply Prop osition 5.13 , we must verify that b oth A and M (view ed as ˜ A -mo dules) admit A -mo dule structures: • The ob ject M is naturally an A -mo dule by the deﬁnition of the deriv ation η ∈ Der ( A, M ) . Its ˜ A -mo dule structure is simply obtained via restriction of scalars along α . • The ob ject A is trivially an A -mo dule. Th us, ˜ A is obtained as a ﬁnite limit of a diagram of ˜ A -mo dules, each of which admits the structure of a mo dule ov er A . T aking f = α , Prop osition 5.13 directly implies that the morphism α : ˜ A → A is descendable. Corollary 6.14. L et η : A → M ∈ Der and α : ˜ A → A b e the induc e d squar e-zer o extension in CAlg ( A ) . Then α is faithful. First pr o of: It follows by combining Prop osition 5.8 and Prop osition 6.13 . W e can also chec k it directly: Se c ond pr o of: W e consider the following pullback diagram in CAlg ( A ) , and hence also a pullback diagram in Mo d ˜ A ( A ) . ˜ A A A A ⊕ M α α δ δ 0 10 W e thank Germán Stefanic h for p ointing this out to the author. 60 No w given an ˜ A -mo dule X suc h that X ⊗ ˜ A A = 0 , w e wish to show that X = 0 too. Because the follo wing diagram is pullbac k in Mo d ˜ A ( A ) , X = X ⊗ ˜ A ˜ A X ⊗ ˜ A A = 0 0 = X ⊗ ˜ A A X ⊗ ˜ A ( A ⊕ M ) = 0 w e get X = 0 . 6.2 L-étale algebras With the cotangent complex established, we formalize the notion of L-étale (formally étale) morphisms as those explicitly p ossessing a v anishing relative cotangent complex. W e demonstrate that L-étale morphisms corresp ond precisely to co cartesian edges within the tangent corresp ondence, providing a robust, op erational geometric framework for deformation theory in arbitrary ttt - ∞ -categories. Deﬁnition 6.15. W e say a map of E ∞ -algebras A → B ∈ CAlg ( A ) is L-étale (or formally étale) 11 if the relativ e cotangen t complex L B / A v anishes. Lemma 6.16. L et A B C f h g b e morphisms in CAlg ( A ) . Then: (1) Supp ose that f is L-étale. Then g is L-étale if and only if h is L-étale. (2) If g is L-étale and faithful ly ﬂat. Then f is L-étale if and only if h is L-étale. Pr o of. It follows from the coﬁb er sequence C ⊗ B L B / A → L C / A → L C /B in Mo d C ( A ) . Prop osition 6.17. L et f : R → S ∈ CAlg ( A ) b e map such that S is an idemp otent c ommutative R -algebr a. Then f is L-étale. Pr o of. Since S ⊗ R S ≃ S , by Prop osition 6.7 we hav e L S/R ≃ S ⊗ S L S/R ≃ L S/S = 0 . Corollary 6.18. L et f : R → S ∈ CAlg ( A ) b e a ﬂat map b etwe en discr ete E ∞ -algebr as such that f is idemp otent in the he art A ♡ . Then f : R → S is L-étale. Lemma 6.19. Given a diagr am of ∞ -c ate gories C D E F p q 11 In deformation theory , the L-étale condition seems only interesting in the connectiv e case. 61 wher e p, q ar e c o c artesian ﬁbr ations and F pr eserves c o c artesian e dges. L et s ∈ E and θ : K ▷ → C s b e a diagr am. If for any morphism f : s → t ∈ E the e dge f ! ◦ θ : K ▷ → C t is an F s -c olimit, then θ is an F -c olimit. Pr o of. This is the relativ e version of [ HTT , Prop. 4.3.1.10]. Corollary 6.20. Given a diagr am of ∞ -c ate gories C D E F p q wher e p, q ar e c o c artesian ﬁbr ations and F pr eserves c o c artesian e dges. L et s ∈ E and θ : x → y b e a morphism in the ﬁb er C s . If for any morphism f : s → t ∈ E , the e dge f ! ( θ ) : f ! ( x ) → f ! ( y ) is F s -c o c artesian, then θ is an F -c o c artesian. Prop osition 6.21. L et f : A → B ∈ CAlg ( A ) ⊂ M T ( CAlg ( A )) b e a morphism of E ∞ -algebr as. Then f is F -c o c artesian if and only if f is L-étale, wher e F : M T ( CAlg ( A )) → ∆ 1 × CAlg ( A ) is the natur al pr oje ction. Pr o of. Applying Corollary 6.20 to the following diagram, we win. M T (CAlg( A )) ∆ 1 × CAlg( A ) ∆ 1 F p q Deﬁnition 6.22. W e deﬁne a (non-full) sub category Der L - et ⊂ Der as follo ws: (1) A deriv ation η : A → M [1] b elongs to Der L - et if and only if A and M are connective. (2) Let φ : ( η : A → M [1]) → ( η ′ : B → N [1]) b e a morphism b etw een deriv ations b elonging to Der L - et . Then φ b elongs to Der L - et if and only the map A → B is L-étale, and φ induces an equiv alence M ⊗ A B → N . W e deﬁne a sub category CAlg ( A ≥ 0 ) L - et ⊂ CAlg( A ) as follows: (1) An ob ject A ∈ CAlg( A ) b elongs to CAlg ( A ≥ 0 ) L - et if and only if A is connective. (2) A morphism f : A → B of connective E ∞ -algebras b elongs to CAlg ( A ≥ 0 ) L - et if and only if f is L-étale. Prop osition 6.23. L et f : Der → CAlg ( A ) denote the for getful functor ( η : A → M ) 7→ A . Then f induc es a left ﬁbr ation Der L - et → CAlg( A ≥ 0 ) L - et . Pr o of. Fix 0 ≤ i < n ; we must show that every lifting problem of the form Λ n i Der L - et ∆ n CAlg( A ≥ 0 ) L - et l 62 admits a solution l . Considering the following diagram, Λ n i Der L - et Der F un(∆ 1 , M T (CAlg( A ))) ∆ n CAlg( A ≥ 0 ) L - et CAlg( A ) F un(∆ 1 , ∆ 1 × CAlg( A )) ⌜ l l ′ l ′′ then there exists a lifting l ′′ b y Prop osition 6.21 , and hence there exists a lifting l ′ . W e observe that l ′ actually lies in Der L - et b y Lemma 6.16 , hence w e ﬁnd a solution l . 6.3 Étale rigidity W e arrive at the main deformation theorem of this section: Étale Rigidity . W e formally prov e that in a pro jectiv ely rigid ttt - ∞ -category , the ∞ -category of étale algebras ov er a connective E ∞ -algebra is en tirely equiv alen t to the ordinary category of discrete étale extensions o ver its π 0 -truncation. This demonstrates that étale morphisms are top ologically rigid and admit no higher homotopical deformations along the P ostnik ov tow er. Deﬁnition 6.24. W e sa y that a map f : A → B in CAlg ( A ) is étale if f is ﬂat and the map τ ≥ 0 f : τ ≥ 0 A → τ ≥ 0 B is L-étale and ﬁnitely presented. Remark 6.25. One may naturally ask whether a ﬁnitely presented L-étale morphism f : A → B ∈ CAlg ( A ≥ 0 ) of connective E ∞ -algebras is necessarily ﬂat, which holds in the case of sp ectra (see [ SA G , Lemma B.1.3.3]). In other words, can the ﬂatness condition in the deﬁnition of an étale morphism b e omitted? Ho wev er, the answer is negative, ev en in the pro jectiv ely rigid case. F or example, consider the smashing lo calization A = Sp C p → Sp C p / Sp B C p ≃ Sp , where C p denotes the cyclic group of order p for a prime p . The corresp onding idemp otent algebra S C p → S in CAlg(Sp cn C p ) is ﬁnitely presented and L-étale, but it is not ﬂat. The main result in this subsection is the following theorem (see [ HA , §7.5] in the case of sp ectra). Theorem 6.26 (Étale rigidity) . Assume that A is Gr othendie ck and left c omplete. L et A ∈ CAlg ( A ) . Then: (1) L et CAlg ( A ) ﬂ ,L - et A/ denote the ful l sub c ate gory of CAlg ( A ) A/ sp anne d by the ﬂat L-étale maps A → B . If A is c onne ctive 12 , then the functor π 0 induc es an e quivalenc e CAlg( A ) ﬂ ,L - et A/ ∼ − → CAlg( A ♡ ) ﬂ ,L - et π 0 A/ with the (1-)c ate gory of the discr ete ﬂat L-étale c ommutative π 0 A -algebr as. (2) Supp ose further that A ⊗ ≥ 0 is pr oje ctively rigid. L et CAlg ( A ) et A/ denote the ful l sub c ate gory of CAlg( A ) A/ sp anne d by the étale maps A → B . Then the functor π 0 induc es an e quivalenc e CAlg( A ) et A/ ∼ − → CAlg( A ♡ ) et π 0 A/ 12 The connectivity can b e remov ed if we replace the L -étale condition on A → B by the L -étale condition on the connective cov er τ ≥ 0 A → τ ≥ 0 B . 63 with the (1-)c ate gory of the discr ete étale c ommutative π 0 A -algebr as. Remark 6.27. The ﬂat condition in the heart CAlg ( A ♡ ) ab o ve should b e understo o d as the (derived) ﬂatness in the sense of Deﬁnition 2.18 , which is generally a stronger condition than the 1-ﬂatness (see Warning 2.26 ). One consequence is the following identiﬁcation b etw een ﬂat idemp otent algebras. Corollary 6.28. Supp ose that A is Gr othendie ck and left c omplete. L et A ∈ CAlg ( A ) . Then the functor π 0 induc es an e quivalenc e CAlg( A ) ﬂ , idem A/ ∼ − → CAlg( A ♡ ) ﬂ , idem π 0 A/ of c ate gories of ﬂat idemp otent E ∞ -algebr as. Pr o of. Note that ﬂat idemp otent E ∞ -algebras o v er A can be identiﬁed with ﬂat idemp otent E ∞ -algebras o ver τ ≥ 0 A , b ecause b y Prop osition 2.20 (3), there is a natural equiv alence of symmetric monoidal ∞ - categories Mo d τ ≥ 0 A ( A ) f l, ⊗ ∼ − → Mo d A ( A ) f l, ⊗ . It implies that their connective co v ers are automatically L-étale ov er τ ≥ 0 A . F urthermore, under the ﬂatness assumption, an idemp otent algebra in the heart CAlg ( A ♡ ) is idemp otent in CAlg ( A ) and is therefore also L-étale. By Theorem 6.26 (1), it suﬃces to show that for any ﬂat morphism A → B in CAlg ( A ) such that τ ≥ 0 A → τ ≥ 0 B is L-étale and π 0 B ⊗ π 0 A π 0 B ≃ π 0 B , we hav e B ⊗ A B ≃ B . This, how ev er, follows directly from Prop osition 2.25 (2). The pro of of Theorem 6.26 will o ccupy the remainder of this section. Prop osition 6.29. L et η : A → M ∈ Der + and α : ˜ A → A b e the induc e d squar e-zer o extension in CAlg( A ≥ 0 ) . Now given a pushout diagr am in CAlg( A ≥ 0 ) . ˜ A A ˜ B B f ′ 0 α f 0 Then: (1) f ′ 0 is L-étale if and only if f 0 is L-étale. (2) Assume that A is Gr othendie ck. Then f ′ 0 is ﬂat if and only if f 0 is ﬂat. (3) If f ′ 0 is L-étale, then f ′ 0 is lo c al ly of ﬁnite pr esentation if and only if f 0 is lo c al ly of ﬁnite pr esentation. Pr o of. (1) The “only if ” direction is obvious. The “if ” direction follows from the equiv alences L B / A ≃ B ⊗ ˜ B L ˜ B / ˜ A ≃ A ⊗ ˜ A L ˜ B / ˜ A and Corollary 6.14 . (2) The “only if ” direction is obvious. F or the con verse, supp ose that B is ﬂat ov er A : it suﬃces to show that for every discrete e A -mo dule N , the relativ e tensor pro duct e B ⊗ e A N is discrete by Prop osition 2.15 (3). T o prov e this, let I ⊂ π 0 e A b e the kernel of the surjective map π 0 e A → π 0 A , so that w e hav e a short exact sequence of modu les ov er π 0 e A : 0 → I N → N → N /I N → 0 It will therefore suﬃce to show that the tensor pro ducts e B ⊗ e A I N and e B ⊗ e A N /I N are discrete. Replacing N b y I N or N /I N , w e can reduce to the case where I N = 0 , so that N has the structure of an A -mo dule. 64 Then e B ⊗ e A N ≃ B ⊗ A N is discrete by virtue of the assumption that B is ﬂat ov er A . (3) The proof is parallel to that of [ DA GXII I , Lem. 2.5.4]. Prop osition 6.30. Assume that A is hyp er c omplete. L et A ∈ CAlg ( A ≥ 0 ) , M b e a c onne ctive A -mo dule, and η : A → M [1] b e a derivation. Then the squar e-zer o extension ˜ A → A induc es an e quivalenc e ( − ) ⊗ ˜ A A : CAlg( A ≥ 0 ) L - et ˜ A/ ∼ − → CAlg( A ≥ 0 ) L - et A/ b etwe en ∞ -c ate gories of c onne ctive L-étale c ommutative ˜ A -algebr as and A -algebr as. Pr o of. An y square-zero extension e A → A is asso ciated to some deriv ation ( η : A → M ) ∈ Der L - et . Let Φ : Der → F un (∆ 1 , CAlg ( A )) b e the functor deﬁned in Prop osition 6.10 . Let Φ 0 , Φ 1 : Der → CAlg ( A ) denote the comp osition of Φ with ev aluation at the vertices { 0 } , { 1 } ∈ ∆ 1 . The functors Φ 0 and Φ 1 induce maps CAlg( A ≥ 0 ) L - et ˜ A/ Φ ′ 0 ← − − Der L - et Φ ′ 1 − − → CAlg ( A ≥ 0 ) L - et A/ Moreo ver, the functor Φ exhibits Φ ′ 1 as equiv alen t to the comp osition of Φ ′ 0 with the relative tensor pro duct ⊗ e A A . Consequently , it will suﬃce to prov e the following: (1) The functor Φ ′ 0 is fully faithful, and its essential image consists precisely of the connective L-étale comm utative e A -algebras. (2) The functor Φ ′ 1 is fully faithful, and its essential image consists precisely of the connective L-étale comm utative A -algebras. The (1) follows from Corollary 6.12 and Prop osition 6.29 (1). And the (2) follo ws from Prop osition 6.23 . Deﬁnition 6.31 (See [ HA ] 7.4.1.18) . F or n ≥ 0 , w e say a morphism f : A → B in CAlg ( A ≥ 0 ) is n -connectiv e if ﬁb ( f ) b elongs to A ≥ n . And we say f is an n -small extension if the following further conditions are satisﬁed: (1) The ﬁb er ﬁb( f ) b elongs to A [ n, 2 n ] . (2) The m ultiplication map ﬁb( f ) ⊗ A ﬁb( f ) → ﬁb( f ) is nullhomotopic. W e let F un n − con  ∆ 1 , CAlg( A )  denote the full sub category of F un  ∆ 1 , CAlg( A )  spanned b y the n -connectiv e extensions, and F un n − sm  ∆ 1 , CAlg( A )  the full subcategory of F un n − con  ∆ 1 , CAlg( A )  spanned b y the n -small extensions. W e let F un n − sm  ∆ 1 , CAlg( A )  denote the full subcategory of F un  ∆ 1 , CAlg( A )  spanned by the n -small extensions. Deﬁnition 6.32. F or A ∈ CAlg ( A ) , w e let L A ∈ Sp  CAlg( A ) / A  ≃ Mo d A ( A ) denote its cotangent complex. Let Der denote the ∞ -category Der (CAlg( A )) of deriv ations in CAlg ( A ) , so that the ob jects of Der can b e iden tiﬁed with pairs ( A, η : L A → M [1]) where A is an E ∞ -algebra of A and η is a morphism in Mo d A ( A ) . W e let Der n − sm denote the full sub category of Der spanned by those pairs ( A, η : L A → M [1]) suc h that A is connective and the image of M b elongs to A [ n, 2 n ] . 65 Theorem 6.33 (See [ HA ] 7.4.1.26) . L et Φ : Der → F un  ∆ 1 , CAlg( A )  b e the functor given by ( η : A → B ) 7→ ( A η → A ) . Then the functor Φ ( k ) r estricts to an e quivalenc e of ∞ -c ate gories Φ n − sm : Der n − sm → F un n − sm  ∆ 1 , CAlg( A )  Corollary 6.34. (1) Every n -smal l extension in CAlg( A ≥ 0 ) is a squar e-zer o extension. (2) L et A ∈ CAlg ( A ≥ 0 ) . Then every map in the Postnikov tower . . . → τ ≤ 3 A → τ ≤ 2 A → τ ≤ 1 A → τ ≤ 0 A is a squar e-zer o extension. Theorem 6.35. Assume that A is left c omplete. L et f : A → B ∈ CAlg ( A ) b e a ﬂat map such that τ ≥ 0 f : τ ≥ 0 A → τ ≥ 0 B is L-étale. Given an arbitr ary C ∈ CAlg ( A ) . The c anonic al map Map CAlg( A ) A/ ( B , C ) → Map CAlg( A ) π 0 A/ ( π 0 B , π 0 C ) is a homotopy e quivalenc e. In p articular, Map CAlg( A ) A/ ( B , C ) is homotopy e quivalent to a discr ete sp ac e. Pr o of. The follo wing pro of is similar as [ DA GIV , Prop. 3.4.13]. Let A 0 , B 0 , and C 0 b e connectiv e cov ers of A, B , and C , resp ectively . W e hav e a pushout diagram A 0 A B 0 B f 0 f where f 0 is ﬂat L-étale. It follows that the induced maps Map CAlg( A ) A/ ( B , C ) → Map CAlg( A ) A 0 / ( B 0 , C ) ← Map CAlg( A ) A 0 / ( B 0 , C 0 ) are homotopy equiv alences. W e may therefore replace A, B and C b y their connective cov ers, and thereb y reduce to the case where A, B , and C are connective. W e hav e a commutativ e diagram Map CAlg( A ) A/ ( B , π 0 C ) Map CAlg( A ) A/ ( B , C ) Map CAlg( A ) π 0 A/ ( π 0 B , π 0 C ) ψ ϕ where the map ψ is a homotopy equiv alence. It will therefore suﬃce to show that φ is a homotopy equiv alence. Let us say a map g : D → D ′ of commutativ e A -algebras is go o d if the induced map φ g : Map CAlg( A ) A/ ( B , D ) → Map CAlg( A ) A/ ( B , D ′ ) is a homotopy equiv alence. Equiv alently , g is go o d if e B ( g ) is an equiv alence, where e B : CAlg ( A ) A/ → S is the functor corepresented by B . W e wish to show that the truncation map C → π 0 C is go o d. W e will employ the following chain of reasoning: (1) Let D b e a commutativ e A -algebra, let M b e a D -mo dule, and let g : D ⊕ M → D b e the pro jection. F or every map of comm utative A -algebras h : B → D , the homotopy ﬁb er of φ g o ver the p oint h 66 can b e identiﬁed with Map Mod B ( L B / A , M ) ≃ Map Mod D ( L B / A ⊗ B D , M ) . Since f is L-étale, the homotop y ﬁb ers of φ g are con tractible. It follows that φ g is a homotopy equiv alence, so that g is go o d. (2) The collection of go o d morphisms is stable under pullback. This follows immediately from the observ ation that e B preserv es limits. (3) An y square-zero extension is go od. This follows from ( a ) and (b). (4) Supp ose giv en a sequence of go o d morphisms . . . → D 2 → D 1 → D 0 Then the induced map lim ← − { D i } → D 0 is go o d. This follows again from the observ ation that e B preserv es limits. (5) F or every connective commutativ e A -algebra C , the truncation map C → π 0 C is go od. This follows b y applying (d) to the P ostnik ov tow er . . . → τ ≤ 2 C → τ ≤ 1 C → τ ≤ 0 C ≃ π 0 C whic h is a sequence of square-zero extensions. Prop osition 6.36. Assume that A is Gr othendie ck and hyp er c omplete. L et A ∈ CAlg ( A ≥ 0 ) ≤ n +1 b e ( n + 1) -trunc ate d c onne ctive. Then the trunc ation functor τ ≤ n : CAlg ( A ) A/ → CAlg ( A ) τ ≤ n A/ r estricts to: (1) A n e quivalenc e CAlg ( A ) ﬂ ,L - et A/ ∼ − → CAlg ( A ) ﬂ ,L - et τ ≤ n A/ fr om the ∞ -c ate gory of ﬂat L-étale c ommutative A -algebr as to the ∞ -c ate gory of ﬂat L-étale c ommutative τ ≤ n A -algebr as. (2) A n e quivalenc e CAlg ( A ) et A/ ∼ − → CAlg ( A ) et τ ≤ n A/ fr om the ∞ -c ate gory of étale c ommutative A -algebr as to the ∞ -c ate gory of étale c ommutative τ ≤ n A -algebr as. Pr o of. It follows by combining Prop osition 2.22 (2), Prop osition 6.29 (2) and Prop osition 6.30 . Prop osition 6.37 (See [ HA ] 7.4.3.17) . L et f : A → B b e a morphism in CAlg ( A ≥ 0 ) . Supp ose that n ≥ 0 and that f induc es an e quivalenc e τ ≤ n A → τ ≤ n B . Then τ ≤ n L B / A ≃ 0 . Corollary 6.38. L et f : A → B b e a map in CAlg ( A ≥ 0 ) . Assume that coﬁb ( f ) is n -c onne ctive, for n ≥ 0 . Then the r elative c otangent c omplex L B / A is n -c onne ctive. The c onverse holds pr ovide d that f induc es an isomorphism π 0 A → π 0 B . Prop osition 6.39. Assume that A is hyp er c omplete. L et f : A → B ∈ CAlg( A ≥ 0 ) . Then: (1) f : A → B is L-étale if τ ≤ n f : τ ≤ n A → τ ≤ n B is L-étale for any n ≥ 0 . (2) Assume further that A is Gr othendie ck. Then f : A → B is ﬂat if and only if τ ≤ n f : τ ≤ n A → τ ≤ n B is ﬂat for any n ≥ 0 . Pr o of. (1) F or any n ≥ 0 , we hav e the coﬁb er sequence τ ≤ n B ⊗ B L B / A → L τ ≤ n B / A → L τ ≤ n B /B . 67 Since τ ≤ n L τ ≤ n B /B = 0 b y Prop osition 6.37 , w e get that τ ≤ n − 1 ( τ ≤ n B ⊗ B L B / A ) ≃ τ ≤ n − 1 L τ ≤ n B / A . Now consider another coﬁb er sequence τ ≤ n B ⊗ τ ≤ n A L τ ≤ n A/ A → L τ ≤ n B / A → L τ ≤ n B /τ ≤ n A . L τ ≤ n B /τ ≤ n A ab o ve v anishes b y the assumption and τ ≤ n L τ ≤ n A/ A = 0 by Proposition 6.37 , so τ ≤ n L τ ≤ n B / A = τ ≤ n ( τ ≤ n B ⊗ τ ≤ n A L τ ≤ n A/ A ) = 0 . Then combining Lemma 3.6 (1) and equations ab o ve we get τ ≤ n − 1 ( L B / A ) ≃ τ ≤ n − 1 ( τ ≤ n B ⊗ B L B / A ) ≃ τ ≤ n − 1 L τ ≤ n B / A = 0 . By the h yp ercompleteness, w e get L B / A = 0 . (2) The “only if ” direction can b e deduced by Prop osition 2.22 . Now supp ose τ ≤ n f : τ ≤ n A → τ ≤ n B is ﬂat for any n ≥ 0 . Since B is connective, it suﬃces to show that given any dis crete M ∈ Mo d A ( A ) ♡ w e ha ve B ⊗ A M ∈ Mod B ( A ) ♡ is discrete to o. Now we hav e τ ≤ n ( B ⊗ A M ) ≃ τ ≤ n ( τ ≤ n B ⊗ A M ) b y Lemma 3.6 (1). Also we hav e τ ≤ n ( τ ≤ n B ⊗ A M ) ≃ τ ≤ n ( τ ≤ n B ⊗ τ ≤ n A τ ≤ n A ⊗ A M ) ≃ τ ≤ n B ⊗ τ ≤ n A τ ≤ n ( τ ≤ n A ⊗ A M ) where the second equality comes from the ﬂatness of τ ≤ n f . Note that τ ≤ n B ⊗ τ ≤ n A τ ≤ n ( τ ≤ n A ⊗ A M ) ≃ τ ≤ n B ⊗ τ ≤ n A τ ≤ n M . Com bining these w e get an equiv alence τ ≤ n ( B ⊗ A M ) ≃ τ ≤ n B ⊗ τ ≤ n A τ ≤ n M , then by the ﬂatness of τ ≤ n f again we conclude that for any n ≥ 0 , τ ≤ n ( B ⊗ A M ) is discrete. Hence B ⊗ A M is discrete by the h yp ercompleteness. W e mimic the pro of of [ HA , Theorem 7.4.3.18] with light mo diﬁcation to get the following statements. Prop osition 6.40. Supp ose that A is right c omplete and that A ≥ 0 is c omp actly gener ate d. L et A ∈ CAlg( A ≥ 0 ) , and let B b e a c onne ctive E ∞ -algebr a over A . Then: (1) If B is lo c al ly of ﬁnite pr esentation over A , then L B / A is p erfe ct as a B -mo dule. The c onverse holds pr ovide d that A ⊗ ≥ 0 is pr oje ctively rigid and that π 0 B is ﬁnitely pr esente d as a c ommutative π 0 A -algebr a in the sense of Deﬁnition 1.25 . (2) If B is almost of ﬁnite pr esentation over A , then L B / A is almost p erfe ct as a B -mo dule. The c onverse holds pr ovide d that A ≥ 0 is pr oje ctively gener ate d and that π 0 B is ﬁnitely pr esente d as a c ommutative π 0 A -algebr a in the sense of Deﬁnition 1.25 . Pr o of. W e ﬁrst pro ve the forward implications. It will b e conv enien t to phrase these results in a sligh tly more general form. Supp ose given a comm utative diagram σ : A B C f h g 68 in CAlg( A ≥ 0 ) , and let F ( σ ) = L B / A ⊗ B C . W e will show: (1 ′ ) If B is lo cally of ﬁnite presentation as an E ∞ -algebra o ver A , then F ( σ ) is p erfect as a C -mo dule. (2 ′ ) If B is almost of ﬁnite presentation as an E ∞ -algebra o v er A , then F ( σ ) is almost p erfect as a C -mo dule. W e will obtain the forward implications of (1) and (2) by applying these results in the case B = C . W e ﬁrst observe that the construction σ 7→ F ( σ ) deﬁnes a functor CAlg ( A ) A//C → Mo d C ( A ) . Note that the functor F can b e identiﬁed with the ﬁb er of the relative adjunction F un(∆ 1 , CAlg( A ) A/ ) T CAlg( A ) A/ Mo d(Mo d A ( A )) CAlg( A ) A/ ∼ on C ∈ CAlg ( A ) A/ , we deduce that this functor preserv es colimits. Since the collection of ﬁnitely presen ted C -mo dules is closed under ﬁnite colimits and retracts, it will suﬃce to prov e (1 ′ ) in the case where B = Sym ∗ A M for some connective p erfect A -mo dule M . Using Prop osition [ HA , Prop osition 7.4.3.14], w e deduce that F ( σ ) ≃ M ⊗ A C is a p erfect C -mo dule, as desired. W e now pro v e (2 ′ ) . By [ HTT , Corollary 5.5.7.4], for any n ≥ 2 there exists a ﬁnitely presented comm utative A -algebra B ′ ∈ CAlg ( A ≥ 0 ) A/ suc h that τ ≤ n B is a retraction of τ ≤ n B ′ as commutativ e A -algebras. Note that the retraction can b e lifted in CAlg ( A ≥ 0 ) A//τ ≤ n C b y [ Ker , 04KB ], as the following. τ ≤ n B ′ τ ≤ n B τ ≤ n C r i No w consider the diagram B ′ τ ≤ n B ′ τ ≤ n B τ ≤ n C A B C r W e claim that τ ≤ n − 2 ( L B / A ⊗ B C ) is a retraction of τ ≤ n − 2 ( L B ′ / A ⊗ B ′ C ) . How ev er, assertion (1 ′ ) implies that L B ′ / A ⊗ B ′ C will b e p erfect b ecause B ′ is lo cally of ﬁnitely presentation as a commutativ e A -algebra. Then L B ′ / A ⊗ B ′ C is p erfect as a retraction of a p erfect mo dule and L B / A ⊗ B C is almost p erfect. Now using Prop osition 6.37 , w e see that L τ ≤ n B /B and L τ ≤ n B ′ /B ′ are n -connectiv e, thus we ha ve the natural equiv alences τ ≤ n − 2 ( L B / A ⊗ B τ ≤ n B ) ∼ − → τ ≤ n − 2 L τ ≤ n B / A , τ ≤ n − 2 ( L B ′ / A ⊗ B ′ τ ≤ n B ′ ) ∼ − → τ ≤ n − 2 L τ ≤ n B ′ / A . So τ ≤ n − 2 ( L B / A ⊗ B C ) ∼ − → τ ≤ n − 2 ( L B / A ⊗ B τ ≤ n C ) ∼ − → τ ≤ n − 2 ( L τ ≤ n B / A ⊗ τ ≤ n B τ ≤ n C ) are equiv alences by Lemma 3.6 (1). By assumption we hav e that τ ≤ n − 2 ( L τ ≤ n B / A ⊗ τ ≤ n B τ ≤ n C ) is a 69 retraction of τ ≤ n − 2 ( L τ ≤ n B ′ / A ⊗ τ ≤ n B ′ τ ≤ n C ) . Again by Lemma 3.6 (1), w e get the equiv alences τ ≤ n − 2 ( L B ′ / A ⊗ B ′ C ) ∼ − → τ ≤ n − 2 ( L B ′ / A ⊗ B ′ τ ≤ n C ) ∼ − → τ ≤ n − 2 ( L τ ≤ n B ′ / A ⊗ τ ≤ n B ′ τ ≤ n C ) . Com bining these, w e in fact conclude that τ ≤ n − 2 ( L B / A ⊗ B C ) is a retraction of τ ≤ n − 2 ( L B ′ / A ⊗ B ′ C ) . W e now prov e the reverse implication of (2). Assume that L B / A is almost p erfect and that π 0 B is a ﬁnitely presen ted as a commutativ e π 0 A -algebra. T o prov e (2), it will suﬃce to construct a sequence of maps A → B ( − 1) → B (0) → B (1) → . . . → B suc h that each B ( n ) is lo cally of ﬁnite presentation as a commutativ e A -algebra, and eac h map f n : B ( n ) → B is ( n + 1) -connectiv e. W e b egin by constructing B ( − 1) with an even stronger prop erty: the map f − 1 induces an isomorphism π 0 B ( − 1) → π 0 B . By Prop osition 4.17 , there exists compact pro jectiv e A -mo dules M , N and a diagram Sym ∗ A ( N ) A Sym ∗ A ( M ) B α ϕ suc h that the map B ( − 1) → B induces an equiv alence on π 0 B , where w e take B ( − 1) as the pushout of ab o ve diagram. W e no w pro ceed in an inductiv e fashion. Assume that we hav e already constructed a connective comm utative A -algebra B ( n ) whic h is of ﬁnite presentation ov er A , and an ( n + 1) -connectiv e morphism f n : B ( n ) → B of commutativ e A -algebras. Moreov er, we assume that the induced map π 0 B ( n ) → π 0 B is an isomorphism (if n ≥ 0 this is automatic; for n = − 1 it follo ws from the sp eciﬁc construction giv en ab o ve). W e hav e a ﬁb er sequence of B -modules L B ( n ) / A ⊗ B ( n ) B → L B / A → L B /B ( n ) By assumption, L B / A is almost p erfect. Assertion (2 ′ ) implies that L B ( n ) / A ⊗ B ( n ) B is p erfect. Using Prop osition 4.7 , we deduce that the relative cotangen t complex L B /B ( n ) is almost p erfect. Moreo v er, Prop osition 6.37 ensures that L B /B ( n ) is ( n + 2) -connectiv e. It follows that π n +2 L B /B ( n ) is a compact mo dule o v er π 0 B . Using [ HA , Theorem 7.4.3.12] and the isomorphism π 0 B ( n ) → π 0 B , w e deduce that the canonical map π n +1 ﬁb( f n ) → π n +2 L B /B ( n ) is an isomorphism. Cho ose a compact pro jectiv e B ( n ) -mo dule M and a map M [ n + 1] → ﬁb ( f n ) such that the composition π 0 M ≃ π n +1 M [ n + 1] → π n +1 ﬁb( f ) ≃ π n +2 L B /B ( n ) is epimorphic. By construction, we ha ve a commutativ e diagram of B ( n ) -mo dules M [ n + 1] 0 B ( n ) B 70 A djoint to this, w e obtain a diagram in CAlg ( A ≥ 0 ) A/ . Sym ∗ B ( n ) M [ n + 1] B ( n ) B ( n ) B W e now deﬁne B ( n + 1) to b e the pushout B ( n ) ⊗ Sym ∗ B ( n ) M [ n +1] B ( n ) , and f n +1 : B ( n + 1) → B to b e the induced map. It is clear that B ( n + 1) is lo cally of ﬁnite presentation o ver B ( n ) , and therefore lo cally of ﬁnite presentation o ver A ( Remark 4.16 ). T o complete the pro of of (2), it w ill suﬃce to show that the ﬁb er of f n +1 is ( n + 2) -connectiv e. By construction, w e ha ve a comm utative diagram π 0 B ( n + 1) π 0 B ( n ) π 0 B e ′′ e ′ e where the map e ′ is epimorphic and e is isomorphic. It follows that e ′ and e ′′ are also isomorphic. In view of Corollary 6.38 , it will now suﬃce to show L B /B ( n +1) is ( n + 3) -connective. W e hav e a ﬁb er sequence of B -mo dules L B ( n +1) /B ( n ) ⊗ B ( n +1) B → L B /B ( n ) → L B /B ( n +1) Using [ HA , Proposition 7.4.3.14] and Prop osition 6.7 , w e conclude that L B ( n +1) /B ( n ) is canonically equiv alent to M [ n + 2] ⊗ B ( n ) B ( n + 1) . W e may therefore rewrite our ﬁb er sequence as M [ n + 2] ⊗ B ( n ) B → L B /B ( n ) → L B /B ( n +1) . The inductiv e hypothesis and Corollary 6.38 guarantee that L B /B ( n ) is ( n + 2) -connectiv e. The ( n + 3) - connectiv eness of L B /B ( n +1) is therefore equiv alent to the surjectivity of the map π 0 M ≃ π n +2  M [ n + 2] ⊗ B ( n ) B  → π n +2 L B /B ( n ) whic h is eviden t from our construction. This completes the pro of of (2). T o complete the conv erse of (1), we use the same strategy but mak e a more careful c hoice of M . Let us assume that L B / A is p erfect. It follows from the ab o ve construction that each cotangent complex L B /B ( n ) is likewise p erfect. Using Prop osition 4.13 , w e may assume L B /B ( − 1) is of T or-amplitude ≤ k + 2 for some k ≥ 0 . Moreov er, for each n ≥ 0 we hav e a ﬁb er sequence of B -modules L B /B ( n − 1) → L B /B ( n ) → P [ n + 2] ⊗ B ( n ) B , where P is compact pro jectiv e by our construction, and therefore of T or-amplitude ≤ 0 . Using Prop osi- tion 4.13 and induction on n , we deduce that the T or-amplitude of L B /B ( n ) is ≤ k + 2 for n ≤ k . In 71 particular, the B -mo dule M = L B /B ( k ) [ − k − 2] is connective and has T or-amplitude ≤ 0 . It follows from Remark 4.11 that M is a ﬂat B -mo dule. Inv oking Prop osition 4.9 13 , w e conclude that M is a compact pro jectiv e B -mo dule. Using Prop osition 3.8 , we can choose a compact pro jectiv e B ( k ) -mo dule M and an equiv alance M [ k + 2] ⊗ B ( k ) B ≃ L B /B ( k ) . Using this map in the construction outlined ab ov e, we guaran tee that the relative cotangent complex L B /B ( k +1) v anishes. It follo ws from Corollary 7.4.3.4 (whic h also works in our general setting) that the map f k +1 : B ( k + 1) → B is an equiv alence, so that B is lo cally of ﬁnite presentation as an E ∞ -algebra o ver A , as desired. Corollary 6.41. Supp ose that the ttt - ∞ -c ate gory ( A ⊗ , A ≥ 0 ) is pr oje ctively rigid. L et f : A → B ∈ CAlg( A ≥ 0 ) . Then f is étale if and only if τ ≤ n f : τ ≤ n A → τ ≤ n B is étale for every n ≥ 0 . Pr o of. It follows immediately by combining Prop osition 6.39 and Prop osition 6.40 . No w we can giv e the proof of our étale rigidity . Our pro of is parallel with the proof of [ DA GIV , Theorem 3.4.1]. Pr o of of The or em 6.26 : (1) First, using Prop osition 2.20 (3), we may reduce to the case where A is connectiv e. F or each 0 ≤ n ≤ ∞ , let C n denote the full sub category of F un (∆ 1 , CAlg ( A ≥ 0 )) spanned by those morphisms f : B → B ′ suc h that B and B ′ are connective and n -truncated, and let C f l,L - et n denote the full sub category of C n spanned by those morphisms which are also ﬂat and L-étale. Using the left completeness, w e deduce that C ∞ is the homotop y in verse limit of the to wer . . . → C 2 τ ≤ 1 − − → C 1 τ ≤ 0 − − → C 0 . Using Prop osition 6.39 , we deduce that C f l,L - et ∞ is the homotop y in verse limit of the restricted to w er . . . → C f l,L - et 2 → C f l,L - et 1 → C f l,L - et 0 Cho ose a P ostniko v tow er A → . . . → τ ≤ 2 A → τ ≤ 1 A → τ ≤ 0 A F or 0 ≤ n ≤ ∞ , let D n denote the ﬁb er pro duct C f l,L - et n × CAlg( A ≥ 0 ) { τ ≤ n A } , so that we can identify D n with the full sub category CAlg ( A ≥ 0 ) f l,L - et τ ≤ n A/ ⊂ CAlg ( A ≥ 0 ) τ ≤ n A/ spanned by the ﬂat L-étale morphisms f : τ ≤ n A → B . It follows from Prop osition 6.39 that D ∞ is the homotop y in verse limit of the to wer . . . → D 2 g 1 − → D 1 g 0 − → D 0 W e wish to prov e that the truncation functor induces an equiv alence D ∞ → D 0 . F or this, it will suﬃce to sho w that eac h of the functors g i is an equiv alence. Cons equen tly , it follows from Prop osition 6.36 . (2) The pro of is totally parallel with (1). W e only need to replace “ﬂat L-étale” by “étale” and replace “ Prop osition 6.39 ” b y “ Corollary 6.41 ”. 7 The ∞ -category of pro jectiv ely rigid ttt - ∞ -categories Ha ving exhaustiv ely explored the in ternal higher algebraic theory in a ttt - ∞ -category , we step back to analyze the moduli of such categories themselves. 13 This is a key fact which requires pro jective rigidity . 72 7.1 The universal example via 1-dimensional cob ordism In this subsection, w e prov e that the ∞ -category of pro jectively rigid ttt - ∞ -categories is compactly generated. Astonishingly , the universal (compact) generator is precisely the presheaf category on the 1-dimensional framed cob ordism category , connecting our abstract algebraic framework directly to top ological ﬁeld theories. Notation 7.1. Let V ∈ CAlg ( P r L ) . W e denote CAlg rig , at V to b e the full sub category of CAlg ( P r L V ) spanned b y rigid and atomically generated commutativ e V -algebras. W e refer the reader to [ Ram24a ; Ram24b ] for more d etails ab out the atomic generation and the rigidit y . Remark 7.2. By [ Ram24b , Lemma 4.50], a pro jectiv ely rigid symmetric monoidal Grothendieck prestable ∞ -category in the sense of Deﬁnition 3.12 can be identiﬁed with a Sp ≥ 0 -atomically generated rigid comm utative algebra W ∈ CAlg rig , at Sp ≥ 0 . Remark 7.3. By Corollary 1.14 , a right complete ttt - ∞ -category ( B ⊗ , B ≥ 0 ) can b e totally recov ered from the connective part B ⊗ ≥ 0 via the stabilization. So there is an equiv alence b etw een the ∞ -category of pro jectively rigid ttt - ∞ -categories and the ∞ -category of pro jectively rigid symmetric monoidal Grothendiec k prestable ∞ -categories CAlg alg ( P r t - rex st ) ∼ − → CAlg rig , at Sp ≥ 0 giv en by ( B ⊗ , B ≥ 0 ) 7→ B ⊗ ≥ 0 . And the inv erse is giv en b y C ⊗ 7→ (Sp( C ) ⊗ , Sp( C ) ≥ 0 ) , see Corollary 1.14 . W e can see the presentabilit y from the following result. Prop osition 7.4. F or any V ∈ CAlg( P r L ) , CAlg rig , at V is pr esentable. Pr o of. T o see that, ﬁrstly we ha v e that CAlg rig , at V = CAlg rig V × P r dbl V P r at V is accessible by com bining [ Ram24a , Corollary 3.15] and [ Ram24b , Corollary 5.13, 5.14]. Then the presentabilit y follows from that CAlg rig , at V admits small colimits, which is obtained by observing the inclusion CAlg rig , at V ⊂ CAlg V is closed under small colimits. Alternativ ely , we can give a more straightforw ard pro of of the presentabilit y of CAlg rig , at V in the case V ⊗ = Sp ⊗ ≥ 0 , and ev en further giv e a compact generator whic h is link ed to cob ordism h yp othesis. Before that, let us recall a lemma, which we learned from Germán Stefanich. Lemma 7.5. L et Cat × ∞ , ad denote the ∞ -c ate gory of smal l additive ∞ -c ate gories with ﬁnite pr o duct pr eserving functors. Then the c or e functor ( − ) ≃ : Cat × ∞ , ad → S is c onservative. Pr o of. F or any x, y ∈ C , w e hav e a natural map of retractions: Map C ( x, y ) Iso C ( x ⊕ y , x ⊕ y ) Map C ( x, y ) Map D ( F x, F y ) Iso D ( F x ⊕ F y , F x ⊕ F y ) Map D ( F x, F y )  1 ∗ 0 1  F p 12 F F  1 ∗ 0 1  p 12 Since F ≃ : C ≃ → D ≃ (regarded as a functor) is fully faithful, and since Iso C ( x ⊕ y , x ⊕ y ) can b e identiﬁed with the mapping anima Map C ≃ ( x ⊕ y , x ⊕ y ) , we conclude that F is fully faithful. 73 Remark 7.6. Note that the additiv e condition in the abov e lemma can not b e weak ened to the semi-additiv e, otherwise the endmorphism  1 f 0 1  is not necessarily an automorphism. Deﬁnition 7.7. W e say a symmetric monoidal ∞ -category is small rigid if it is small and every ob ject in it is dualizable. Prop osition 7.8. W e have a natur al e quivalenc e CAlg rig , at Sp ≥ 0 ∼ − → CAlg rig (Cat idem ∞ , ad ) given by C ⊗ 7→ ( C cpro j ) ⊗ , wher e CAlg rig ( Cat idem ∞ , ad ) denotes the ful l sub c ate gory of CAlg ( Cat idem ∞ , ad ) sp anne d by smal l rigid idemp otent-c omplete additively symmetric monoidal ∞ -c ate gories. Pr o of. It suﬃces to observe that any colimit-preserving symmetric monoidal functor b etw een pro jectiv ely rigid comm utative algebras B ⊗ ≥ 0 → C ⊗ ≥ 0 preserv es compact pro jectiv e ob jects. No w let us recall the (1-dimensional) cob ordism hypothesis, whic h w as originally formulated in [ BD95 ] and w as pro ved by Hopkins–Lurie in [ Lur09 ]. Theorem 7.9 ([ Lur09 ] Cob ordism hypothesis of dim=1) . L et Cob ⊗ 1 denote the oriente d 1-dimensional c ob or dism ( ∞ , 1) -c ate gory with the symmetric monoidal structur e given by disjoint union. Then Cob ⊗ 1 is smal l rigid and satisﬁes the fol lowing universal pr op erty: L et C b e a symmetric monoidal ( ∞ , 1) -c ate gory. Then the evaluation functor Z 7→ Z ( ∗ ) induc es an e quivalenc e of ∞ -c ate gories F un ⊗ ( Cob 1 , C ) → ( C d ) ≃ wher e F un ⊗ denotes the ∞ -c ate gory of symmetric monoidal functors. Remark 7.10. Note that the 1-dimensional orien ted and framed cob ordism ∞ -categories are equiv alent Cob 1 ≃ Bord fr 1 [see Lur09 , §4.2], but that do es not hold in higher dimensional cases. W e now prov e the main theorem of this subsection. Theorem 7.11 (Universal example) . The ∞ -c ate gory CAlg rig , at Sp ≥ 0 is c omp actly gener ate d by a single element F un( Cob op 1 , Sp ≥ 0 ) ⊗ ∈ CAlg rig , at Sp ≥ 0 wher e the symmetric monoidal structur e F un( Cob op 1 , Sp ≥ 0 ) ⊗ is given by Day c onvolution. Pr o of. By Prop osition 7.8 , we hav e an equiv alence CAlg rig , at Sp ≥ 0 ∼ − → CAlg rig (Cat idem ∞ , ad ) . W e ﬁrst observe that by Theorem 7.9 , the comp osite CAlg rig , at Sp ≥ 0 ∼ − → CAlg rig (Cat idem ∞ , ad )  → CAlg(Cat idem ∞ , ad ) ( − ) ≃ − − − → S is represen ted b y F un ( Cob op 1 , Sp ≥ 0 ) ⊗ ∈ CAlg rig , at Sp ≥ 0 . Since the second functor preserv es small col- imits by [ Ram24b , Proposition 4.53] and the third functor preserves ﬁltered colimits, we see that F un ( Cob op 1 , Sp ≥ 0 ) ⊗ is compact in CAlg rig , at Sp ≥ 0 . Moreo v er, the representable functor b y F un ( Cob op 1 , Sp ≥ 0 ) ⊗ is conserv ative by Lemma 7.5 . Consequently , it is a compact generator of CAlg rig , at Sp ≥ 0 . 74 7.2 Algebraic functors W e formalize the morphisms b etw een pro jectiv ely rigid ttt - ∞ -categories, termed algebraic functors. In this subsection, w e demonstrate that an algebraic functor preserve all the essential algebraic prop erties w e hav e deﬁned: ﬂatness, pro jectiv e mo dules, ﬁnite presentation, and étale maps. Deﬁnition 7.12. W e call a righ t t -exact colimit-preserving symmetric monoidal functor ( B ⊗ , B ≥ 0 ) → ( C ⊗ , C ≥ 0 ) b et ween pro jectiv ely rigid ttt - ∞ -categories by an algebraic functor. Prop osition 7.13. L et F : ( B ⊗ , B ≥ 0 ) → ( C ⊗ , C ≥ 0 ) b e an algebr aic functor b etwe en pr oje ctively rigid ttt - ∞ -c ate gories, and let R ∈ Alg( B ≥ 0 ) . Then the functor LMo d R ( B ≥ 0 ) → LMo d F ( R ) ( C ≥ 0 ) pr eserves (1) c omp act pr oje ctives; (2) c omp acts; (3) pr oje ctives; (4) ﬂats; (5) almost p erfe cts. Pr o of. (1)-(3) are obvious. (4) and (5) follo w b y Theorem 3.20 and Prop osition 4.8 . Prop osition 7.14. L et F : ( B ⊗ , B ≥ 0 ) → ( C ⊗ , C ≥ 0 ) b e an algebr aic functor b etwe en pr oje ctively rigid ttt - ∞ -c ate gories, and let R ∈ CAlg ( B ≥ 0 ) . Then the functor CAlg ( B ≥ 0 ) R/ → CAlg ( C ≥ 0 ) F ( R ) / pr eserves (1) ﬁnitely pr esente d algebr as; (2) almost ﬁnitely pr esente d algebr as; (3) ﬂat algebr as; (4) L-étale algebr as; (5) étale algebr as. Pr o of. Let G denote the right adjoint to F . F or (1), it follows from the following diagram B ≥ 0 C ≥ 0 CAlg( B ≥ 0 ) CAlg( C ≥ 0 ) Sym ∗ F ≥ 0 Sym ∗ CAlg( F ≥ 0 ) that CAlg ( F ≥ 0 ) sends free commutativ e E ∞ -algebras of compact ob jects in B ≥ 0 to free commutativ e E ∞ -algebras of c ompact ob jects in C ≥ 0 , whic h are compact generators of CAlg ( B ≥ 0 ) . F or (2), it suﬃces to observe that for any n ≥ 0 , the functor CAlg( B ≥ 0 ) ≤ n CAlg( F ≥ 0 ) ⊗ S ≤ n − − − − − − − − − − − → CAlg ( B ≥ 0 ) ≤ n preserv es compact ob jects, b ecause CAlg ( B ≥ 0 ) CAlg( F ≥ 0 ) − − − − − − − → CAlg ( C ≥ 0 ) do es. F or (3), it follows from Prop osition 7.13 (4). 75 F or (4), it suﬃces to show that for any A → B ∈ CAlg( B ) , we hav e F ( L B / A ) ≃ L F ( B ) /F ( A ) in Mo d F ( B ) ( C ) . W e ﬁrst observe that the equiv alence in [ HA , Theorem 7.3.4.18] is natural, i.e., we hav e the follo wing diagram: F un(∆ 1 , CAlg( C )) T CAlg( C ) Mo d( C ) F un(∆ 1 , CAlg( B )) T CAlg( B ) Mo d( B ) CAlg( C ) CAlg( B ) F un(∆ 1 , CAlg( G )) ∼ p 2 Mod( G ) ∼ p 1 By taking the left adjoints of the diagram, we obtain the following diagram: CAlg( B ) F un(∆ 1 , CAlg( B )) T CAlg( B ) Mo d( B ) CAlg( C ) F un(∆ 1 , CAlg( C )) T CAlg( C ) Mo d( C ) δ CAlg( F ) F un(∆ 1 , CAlg( F )) p L 1 T L CAlg( G ) ∼ Mod( F ) δ p L 2 ∼ whic h exactly means F ( L A ) ≃ L F ( A ) for an y A ∈ CAlg ( B ) . Since the follo wing functor preserv es co cartesian edges, Mo d( B ) Mod( C ) CAlg( B ) CAlg ( C ) Mod( F ) CAlg( F ) w e see that Mo d ( F ) preserves relative colimits by [ HTT , Prop osition 4.3.1.10]. Since L B / A is deﬁned to b e the relativ e pushout as the following diagram, L A L B 0 L B / A A B A B w e conclude that F ( L B / A ) ≃ L F ( B ) /F ( A ) . F or (5), it follows from (1), (3) and (4). 76 8 Questions and future directions In this ﬁnal section, we catalogue several op en questions regarding ﬂatness, étaleness, and the criteria for faithful ﬂatness, particularly in the non-connective and hypercomplete settings. W e then conclude b y outlining a broader vision for future research building up on the framework established in this pap er. Question 8.1. Assume that the ttt - ∞ -c ate gory ( A ⊗ , A ≥ 0 ) is pr oje ctively rigid. (1) Given a c ommutative diagr am in CAlg ( A ≥ 0 ) : A B C f h g This c onﬁgur ation r aises sever al desc ent-the or etic questions for étale morphisms: • If f and h ar e étale, is g ne c essarily étale? • Assuming g is faithful ly ﬂat and étale, is f étale if and only if h is étale? • If g is étale, do es ther e exist a ﬁnitely pr esente d c ommutative A -algebr a B 0 and an étale map B 0 → C 0 such that C ≃ C 0 ⊗ B 0 A ? (2) L et R ∈ Alg ( A ) b e non-c onne ctive. If M is a ﬂat left R -mo dule, do es the c onservativity of the tensor pr o duct functor ( − ) ⊗ R M : LMo d R ( A ) → A imply that M is faithful ly ﬂat? (While this is likely false, a deﬁnitive c ounter example r emains elusive.) (3) Under what c onditions is ﬂatness e quivalent to the pr op erty that the natur al map π n ( R ) ⊗ π 0 R π 0 M → π n M is an e quivalenc e for al l n ∈ Z ? (A known suﬃcient c ondition is that P ∗ is a pr oje ctive left P ∗ -mo dule for any c omp act pr oje ctive left R -mo dule P .) (4) Given a pr oje ctive obje ct P ∈ A ≥ 0 , is P ∗ 1 -pr oje ctive as an obje ct in Mo d R ∗ (Gr( A ♡ )) ? (5) L et f : A → B ∈ CAlg ( A ) b e a ﬂat morphism in the sense of Deﬁnition 2.18 . Is f L-étale if and only if its c onne ctive c over τ ≥ 0 f : τ ≥ 0 A → τ ≥ 0 B is L-étale? (This app e ars false, as suggeste d by the map of E ∞ -rings k u → K U .) Question 8.2. Assume that A is Gr othendie ck and hyp er c omplete. (1) Consider a pushout diagr am in CAlg( A ≥ 0 ) : ˜ A A ˜ B B f ′ 0 α f 0 If f ′ 0 is L-étale and ﬂat, is f ′ 0 almost of ﬁnite pr esentation if and only if f 0 is almost of ﬁnite pr esentation? (2) L et ˜ A → A b e a nilp otent thickening in CAlg ( A ≥ 0 ) . Do es the tensor pr o duct functor r estricte d to b ounde d-b elow mo dules, Mo d ˜ A ( A ) − → Mo d A ( A ) − , r eﬂe ct c omp act obje cts? 77 Question 8.3. L et A ⊗ b e a sy mmetric monoidal Gr othendie ck ab elian c ate gory and let f : A → B ∈ CAlg( A ) . (1) If Cok er ( f ) is a ﬂat A -mo dule, is f faithful ly ﬂat? (Note that this holds true if ther e exists a pr esentably symmetric monoidal pr estable enhanc ement of A ⊗ .) F uture directions: Deriv ed geometry internal to A With the foundational framework of higher algebra o v er an arbitrary pro jectiv ely rigid ttt - ∞ -category A no w ﬁrmly established, a natural and comp elling next step is to systematically dev elop derived algebraic geometry within this setting. Just as classical derived algebraic geometry is built up on E ∞ -rings in sp ectra, our scaﬀolding allows one to deﬁne and study derived schemes, stacks, and mo duli problems mo deled lo cally on E ∞ -algebras internal to A . This op ens the do or to imp orting deep geometric in tuitions and intersection theory into new, exotic environmen ts, ranging from equiv arian t and motivic domains to spec ialized analytic settings, ultimately broadening the scop e and applicabilit y of mo dern geometric metho ds. A Dualit y F or the reader’s con venience, we review the general theory of dualizable ob jects within monoidal ∞ -categories. A.1 Dualizable ob jects Con v en tion A.1. Throughout App endix A.1 , we ﬁx a symmetric monoidal ∞ -category C ⊗ → Comm ⊗ . Deﬁnition A.2. W e sa y an ob ject X ∈ C is dualizable if there exists an ob ject X ∨ and a pair of morphisms c : 1 → X ⊗ X ∨ e : X ∨ ⊗ X → 1 where 1 denotes the unit ob ject of C . These morphisms are required to satisfy the follo wing conditions: The comp osite maps X c ⊗ id − − − → X ⊗ X ∨ ⊗ X id ⊗ e − − − → X X ∨ id ⊗ c − − − → X ∨ ⊗ X ⊗ X ∨ e ⊗ id − − − → X ∨ are homotopic to the identit y on X and X ∨ , resp ectiv ely . Deﬁnition A.3. W e say an ob ject X ∈ C is cotensorable if the tensor pro duct functor ( − ) ⊗ X : C → C admits a righ t adjoin t. If so, we denote this right adjoint b y Map C ( X, − ) . Remark A.4. If C ⊗ is a presentably symmetric monoidal ∞ -category , then any ob ject in it is cotensorable. Prop osition A.5. L et X ∈ C b e an obje ct. Then X is dualizable if and only if X is c otensor able and for any Y ∈ C , the natur al map Map C ( X, 1 ) ⊗ Y → Map C ( X, Y ) is an e quivalenc e in C . Pr o of. Assume that X is dualizable. Then X is cotensorable since we hav e Map C ( X, − ) ≃ X ∨ ⊗ ( − ) . P articularly , Map C ( X, 1 ) ≃ X ∨ . Now let Y ∈ C . W e wish to show that the comp osite map φ : Map C ( − , Y ⊗ X ∨ ) → Map C ( − ⊗ X , Y ⊗ X ∨ ⊗ X ) e − → Map C ( − ⊗ X , Y ) 78 is a homotop y equiv alence. Let ψ denote the comp osition Map C ( − ⊗ X , Y ) → Map C ( − ⊗ X ⊗ X ∨ , Y ⊗ X ∨ ) c − → Map C ( − , Y ⊗ X ∨ ) . Using the compatibility of e and c , we deduce that φ and ψ are homotopy inv erses to one another. By the Y oneda lemma, φ can b e identiﬁed with the map Map C ( X, 1 ) ⊗ Y → Map C ( X, Y ) . Assume that X is cotensorable and that for any Y ∈ C , the natural map Map C ( X, 1 ) ⊗ Y → Map C ( X, Y ) is an equiv alence in C . Particularly , w e hav e an equiv alence Map C ( X, 1 ) ⊗ X ∼ − → Map C ( X, X ) . Let c : 1 → Map C ( X, 1 ) ⊗ X b e the in v erse image of the iden tity map id : Map C ( X, X ) → Map C ( X, X ) . Then it is straightforw ard to chec k that the counit e : Map C ( X, 1 ) ⊗ X → 1 and c : 1 → Map C ( X, 1 ) ⊗ X form a d ualit y datum. Prop osition A.6. L et C d ⊂ C b e the ful l sub c ate gory c onsisting of dualizable obje cts. Then the pr ofunctor C d × C d → S given by Map C ( 1 , − ⊗ − ) is a b alanc e d pr ofunctor (see [ Ker , 03MM ]) , which induc es a natur al e quivalenc e of ∞ -c ate gories ( − ) ∨ =: Map C ( − , 1 ) : ( C d ) op ∼ − → C d . F urthermor e, ( − ) ∨∨ ≃ Id C d is e quivalent to the identity functor. Pr o of. It suﬃces to observe that if c : 1 → X ⊗ Y is part of a duality datum for X , then it is also part of a d ualit y datum for Y . Remark A.7. In fact, this p erfect pairing can b e enhanced to a symmetric monoidal p erfect pairing and hence induces an equiv alence of symmetric monoidal ∞ -categories ( − ) ∨ =: Map C ( − , 1 ) : ( C op d ) ⊗ ∼ − → ( C d ) ⊗ , see [ ECI , Propos ition 3.2.4]. Prop osition A.8. The ful l sub c ate gory C d ⊂ C is close d under tensor pr o duct, henc e it forms a symmetric monoidal ful l sub c ate gory. Pr o of. Let X, Y ∈ C d . Choosing c = c X ⊗ c Y : 1 ≃ 1 ⊗ 1 → ( X ⊗ X ∨ ) ⊗ ( Y ⊗ Y ∨ ) ≃ ( X ⊗ Y ) ⊗ ( Y ∨ ⊗ X ∨ ) , w e see that c exhibits Y ∨ ⊗ X ∨ as a dual of X ⊗ Y . Deﬁnition A.9. Let C cot ⊂ C b e the full sub category consisting of cotensorable ob jects. W e deﬁne the functor Map C ( − , − ) : ( C cot ) op × C → F un ′ ( C op , S ) ≃ C giv en by ( X, Y ) 7→ Map C ( − ⊗ X , Y ) , where F un ′ ( C op , S ) ≃ C denotes the full sub category of representable functors. Lemma A.10. L et K b e a c ol le ction of simplicial sets. If C is K -c o c omplete and the monoidal structur e on it is c omp atible with K -c olimits for any K ∈ K (me aning − ⊗ − pr eserves K -c olimits sep ar ately), then for any K ∈ K , the ful l sub c ate gory C cot ⊂ C is close d under K -c olimits and for any diagr am X ( − ) : K → C cot , the natur al map lim ← − α ∈ K Map C ( X α , − ) ≃ Map C ( lim − → α ∈ K X α , − ) is an e quivalenc e in F un( C , C ) . 79 Pr o of. Consider the following diagram: ( C cot ) op C op F un( C , C ) op F un( C , C ) F un( C op × C , S ) X 7→ Map C ( X, − ) X 7→ ( − ) ⊗ X ϕ i L i R where i L is given by F 7→ Map C ( F ( − ) , − ) and i R is given by G 7→ Map C ( − , G ( − )) . An ob ject X ∈ C is cotensorable if and only if φ ( X ) lies in the image of i R . Now given a diagram K ∈ K , it suﬃces to observ e that: (1) ( C cot ) op = φ − 1 (Im( i R )) . (2) φ preserv es K op -limits and the inclusion i R is closed under K op -limits. Prop osition A.11. (1) If C ⊗ is p ointe d ly symmetric monoidal (me aning that C is p ointe d and the tensor pr o duct of the zer o obje ct with any obje ct is zer o), then the zer o obje ct ∗ is dualizable. (2) If C is idemp otent c omplete, then C d ⊂ C is close d under r etr actions. (3) If C ⊗ is semiadditively symmetric monoidal, then C d ⊂ C is close d under ﬁnite c opr o ducts and henc e forms a ful l semiadditive sub c ate gory. (4) If C ⊗ is stably symmetric monoidal, then C d ⊂ C is close d under ﬁnite c olimits and ﬁnite limits and henc e forms a ful l stable sub c ate gory. Pr o of. Applying Prop osition A.5 and Lemma A.10 to K = { ∅ } , { N( Idem ) } , { ﬁnite discrete diagrams } , { ﬁnite diagrams } resp ectiv ely , we prov ed (1), (2), (3) and the “closed under ﬁnite colimits” part of (4). F or the “closed under ﬁnite limits” part of (4), it suﬃces to show that C d ⊂ C is closed under desusp ension. This follo ws from Σ − 1 X = (Σ X ) ∨ for a dualizabl e ob ject X ∈ C d . A.2 Dualit y of Bimo dules Extending b eyond symmetric monoidal categories, we de tail the higher duality theory sp eciﬁcally adapted for bimo dules o ver E 1 -algebras. Con v en tion A.12. Throughout App endix A.2 , we ﬁx a monoidal ∞ -category C ⊗ → Ass ⊗ whic h admits geometric realizations of simplicial ob jects and suc h that the tensor pro duct ⊗ : C × C → C preserv es geometric realizations of simplicial ob jects. Deﬁnition A.13. Let X ∈ A BMo d B ( C ) and Y ∈ B BMo d A ( C ) . Let c : B → Y ⊗ A X b e a map in B BMo d B ( C ) . W e say c exhibits X as the right dual of Y , or c exhibits Y as the left dual of X , if there exists a map e : X ⊗ B Y → A in A BMo d A ( C ) suc h that X ≃ X ⊗ B B id ⊗ c − − − → X ⊗ B Y ⊗ A X e ⊗ id − − − → A ⊗ A X ≃ X Y ≃ B ⊗ B Y c ⊗ id − − − → Y ⊗ A X ⊗ B Y id ⊗ e − − − → Y ⊗ A A ≃ Y are homotopic to id X and id Y , resp ectiv ely . 80 Prop osition A.14 (See [ HA ] 4.6.2.18) . L et A ∈ Alg ( C ) , let X ∈ LMo d A ( C ) , let Y ∈ RMo d A ( C ) , and let c : 1 → Y ⊗ A X b e a map in C . Then the fol lowing ar e e quivalent: (1) The map c : 1 → Y ⊗ A X exhibits Y as a left dual of X . (2) F or e ach C ∈ C and e ach M ∈ RMo d A ( C ) , the c omp osite map Map RMod A ( C ) ( C ⊗ Y , M ) → Map C ( C ⊗ Y ⊗ A X, M ⊗ A X ) ◦ c − → Map C ( C, M ⊗ A X ) is a homotopy e quivalenc e. (3) F or e ach C ∈ C and e ach N ∈ LMo d A ( C ) , the c omp osite map Map LMod A ( C ) ( X ⊗ C, N ) → Map C ( Y ⊗ A X ⊗ C, Y ⊗ A N ) ◦ c − → Map C ( C, Y ⊗ A N ) is a homotopy e quivalenc e. Corollary A.15. L et A ∈ Alg ( C ) . L et LMo d A ( C ) ld ⊂ LMo d A ( C ) denote the ful l sub c ate gory of left dualizable left A -mo dules, and let RMo d A ( C ) rd ⊂ RMo d A ( C ) denote the ful l sub c ate gory of right dualizable right A -mo dules. Then the pr ofunctor RMo d A ( C ) rd × LMo d A ( C ) ld → S given by Map C ( 1 , − ⊗ A − ) is a b alanc e d pr ofunctor (see [ Ker , 03MM ]) , which induc es a natur al e quivalenc e of ∞ -c ate gories ∨ ( − ) : LMo d A ( C ) ld ∼ ⇄ (RMo d A ( C ) rd ) op : ( − ) ∨ . Pr o of. It suﬃces to observe that c : 1 → Y ⊗ A X exhibits Y as a left dual of X if and only if it exhibits X as a right dual of Y . Corollary A.16. Supp ose that C ⊗ is a c o c ompletely symmetric monoidal (p otential ly lar ge) ∞ -c ate gory, i.e., C admits smal l c olimits and the tensor pr o duct ⊗ : C × C → C pr eserves smal l c olimits sep ar ately. L et A ∈ Alg ( C ) , let X ∈ LMo d A ( C ) , let Y ∈ RMo d A ( C ) , and let c : 1 → Y ⊗ A X b e a map in C . If C is gener ate d by dualizable obje cts under smal l c olimits, then the fol lowing ar e e quivalent: (1) The map c : 1 → Y ⊗ A X in C exhibits Y as a left dual of X . (2) The functor Map C ( 1 , Y ⊗ A − ) : LMo d A ( C ) → b S is c or epr esente d by X with the element c : 1 → Y ⊗ A X . (3) The functor Map C ( 1 , − ⊗ A X ) : RMo d A ( C ) → b S is c or epr esente d by Y with the element c : 1 → Y ⊗ A X . (Note that we use the notation b S ab ove b e c ause C is not ne c essarily smal l her e.) B Ind(Pro)-completion of large ∞ -categories The ∞ -categorical theory frequently encounters set-theoretic size issues. In this app endix, we resolve the Ind- and Pro-completions of large ∞ -categories relative to large regular cardinals. 81 Con v en tion B.1. W e work relative to a chain of strongly inaccessible cardinals δ 0 < δ 1 < δ 2 . Then V δ i is a Grothendieck universe, and elements of V δ 0 , V δ 1 , V δ 2 are called small, large and very large, resp ectiv ely . Theorem B.2 (See [ HTT ] 5.3.6.10) . L et K ⊂ K ′ b e δ 1 -smal l c ol le ctions of simplicial sets. L et d Cat K ∞ denote the sub c ate gory sp anne d by those ∞ -c ate gories which admit K -indexe d c olimits and those functors which pr eserve K -indexe d c olimits, and let d Cat K ′ ∞ b e deﬁne d likewise. Then the inclusion d Cat K ′ ∞ ⊂ d Cat K ∞ admits a left adjoint given by C 7→ P K ′ K ( C ) . Prop osition B.3 (See [ HP24 ] A.2) . L et C b e a c o ac c essible ∞ -c ate gory (i.e. C op is ac c essible). F or a functor X : C op → S , the fol lowing c onditions ar e e quivalent: (1) The functor X : C op → S is ac c essible. (2) The functor X : C op → S is the left Kan extension of a functor Y : ( C κ ) op → S along the c anonic al inclusion i : ( C κ ) op → C op for some smal l r e gular c ar dinal κ , wher e C κ ⊂ C denotes the ful l sub c ate gory of κ -c o c omp act obje cts. (3) The fu nctor X : C op → S is a c olimit in F un ( C op , S ) of a smal l diagr am of r epr esentable functors. Corollary B.4 (See [ HP24 ] A.4) . L et C b e a c o ac c essible ∞ -c ate gory. Then the Y one da emb e dding C → F un ac ( C op , S ) exhibits F un ac ( C op , S ) ≃ P small ∅ ( C ) , wher e F un ac ( C op , S ) denotes the ful l sub c ate gory of ac c essible functors. Prop osition B.5 (See [ HP24 ] A.9) . L et C b e a c o ac c essible ∞ -c ate gory. The ∞ -c ate gory P ac ( C ) of ac c essible pr eshe aves of anima on C admits al l smal l limits and c olimits, and b oth ar e c alculate d p ointwise. Prop osition B.6. L et C b e a c opr esentable ∞ -c ate gory and κ b e a smal l r e gular c ar dinal. Then the Y one da emb e dding C → F un ac κ − lex ( C op , S ) exhibits F un ac κ − lex ( C op , S ) ≃ P small κ − ﬁl ∅ ( C ) = Ind κ ( C ) . Pr o of. First we observe that Ind κ ( C ) = F un ac κ − lex ( C op , S ) ⊂ F un κ − lex ( C op , b S ) = d Ind κ ( C ) is closed under small κ -ﬁltered colimits. W e claim that any ob ject in F un ac κ − lex ( C op , S ) can b e written as the retraction of a s mall κ -ﬁltered colimit of representable functors. Then the result immediately follows. No w let F ∈ F un ac κ − lex ( C op , S ) . Since F ∈ d Ind κ ( C ) , it can be written as a large κ -ﬁltered colimit of representable functors F ≃ lim − → i ∈ I h X i . F or each small κ -ﬁltered full subcategory I ′ ⊂ I , let F I ′ denote the colimit lim − → α ∈ I ′ h X α . Then by [ Ker , 0620 ], F can b e written as a large δ 0 -ﬁltered colimit of the diagram { F I ′ } , where I ′ ranges ov er all small κ -ﬁltered full sub categories of I . Ho w ev er, b y Prop osition B.3 , F is largely δ 0 -compact in d Ind κ ( C ) , so F is a retraction of some F I ′ . Prop osition B.7. L et C b e a c opr esentable ∞ -c ate gory and κ b e a smal l r e gular c ar dinal. Then Ind κ ( C ) ≃ P small κ − small ( C ) . Pr o of. By the construction in [ HTT , Corollary 5.3.6.10], it is equiv alen t to prov e that Ind κ ( C ) ⊂ d Ind κ ( C ) is the smallest full sub category which contains representables and closed under small colimits, i.e. to pro ve 82 Ind κ ( C ) = d Ind κ ( C ) δ 0 . Since the representables generate d Ind κ ( C ) under l arge colimits, it suﬃces to show that Ind κ ( C ) ⊂ d Ind κ ( C ) δ 0 and Ind κ ( C ) is idemp otent complete. Those are implied by Prop osition B.3 and Prop osition B.5 . Deﬁnition B.8. Let d Cat κ − lex ∞ denote the sub category spanned by those ∞ -categories which admit ﬁnite limits and those functors which preserve κ -small limits, where κ < δ 1 is a large regular cardinal. Prop osition B.9. L et κ < λ < δ 1 b e two lar ge r e gular c ar dinals. Then ther e exists an adjoint p air d Cat κ − lex ∞ Pro λ κ ⇄ d Cat λ − lex ∞ by the dual version of [ HTT , Corollary 5.3.6.10]. Remark B.10. W e hav e the identiﬁcation Pro λ κ ( C ) ≃ Ind λ κ ( C op ) op . Prop osition B.11. The ab ove adjunction c an b e pr omote d to a symmetric monoidal adjunction d Cat κ − lex , ⊗ ∞ Pro λ κ ⇄ d Cat λ − lex , ⊗ ∞ by the dual version of [ HA , Prop osition 4.8.1.3]. Remark B.12 (Dual of [ HA ] 4.8.1.9) . CAlg ( d Cat ∞ ) can b e identiﬁed with the v ery large ∞ -category of large symmetric monoidal ∞ -categories. Un winding the deﬁnitions, we see that CAlg ( d Cat κ − lex ∞ ) can b e iden tiﬁed with the sub category of CAlg ( d Cat ∞ ) spanned by the symmetric monoidal ∞ -categories which are compatible with κ -small limits (meaning the tensor pro duct − ⊗ − preserv es κ -small limits separately), and those symmetric monoidal functors whic h preserv e κ -small limits. 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Higher algebra in $t$-structured tensor triangulated $\infty$-categories

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