Geometric Points in Tensor Triangular Geometry

GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI Abstract. In this paper, we study geometric p oints in tensor triangular ge- ometry . In doing so, we construct a coun ter-example to Balmer’s Nerves of Steel conjecture using free constructions in higher Zariski geometry . W e then go on to introduce and discuss constructible sp ectra in the con text of tensor triangular geometry . F or tensor triangulated categories satisfying a mild en- hancement condition, we use these spectra to construct geometric incarnations of (homological or triangular) primes via maps to “p ointlik e” tensor triangu- lated categories. Figure 1. On the p oints, W assily Kandinsky , 1928 1 Date : Marc h 27, 2026. 1 https://www.wikiart.org/en/wassily- kandinsky/on- the- points- 1928 (visited on 03/13/2026) 1 2 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI Contents 1. In tro duction 2 P art I. A Coun terexample to the Nerves of Steel Conjecture 8 2. Preliminaries on T ensor T riangular Geometry 8 3. F ree Constructions 15 4. The Aﬃne Line A 1 23 5. The Comparison F unctor 26 P art I I. Constructible Sp ectra in Higher Zariski Geometry 29 6. The E ∞ -Constructible Sp ectrum 29 7. Constructible Sp ectra in the Rational Case 33 8. The E n -Constructible Sp ectrum 45 9. Geometric Poin ts for Homological Sp ectra 52 App endix A. The Semi-Simplicit y of Rep( GL t ) 59 References 63 1. Introduction 1.A. Con text and main results. Hilb ert’s Nullstellensatz led to the idea that the prime ideals of a commutativ e ring R should corresp ond to the p oints of the aﬃne scheme Sp ec( R ), thus forming the origin of mo dern algebraic geometry . One categorical level up, tensor triangular geometry aims to dev elop an analogous geo- metric theory b y replacing commutativ e rings by tensor triangulated categories (henceforth, tt-categories), see [ Bal05 ; Aok+25 ]. The role of the Zariski sp ectrum for a tt-category C is pla yed by the Balmer spectrum Sp c( C ), which is built from prime tt-ideals in C together with a suitable top ology akin to the Zariski topology . This motiv ates the question, ﬁrst raised in [ Bal10b ]: So, what ar e tt-ﬁelds? Sp eciﬁcally , this calls for a deﬁnition of a theory of ﬁelds in tensor triangular geometry which allows us to realize the p oin ts of the sp ectrum of any tensor tri- angulated category C geometrically , i.e., through residue functors C → F . The minimal desiderata therefore are to giv e a deﬁnition of tt-ﬁelds such that: ( † ) If F is a tt-ﬁeld, then the sp ectrum Sp c( F ) = ∗ is a single p oint. ( ‡ ) F or ev ery prime P ∈ C , there exists a tt-functor C → F P whose image on sp ectra is precisely { P } . These are the t wo conditions singled out b y Balmer in [ Bal10b , Section 4.3], but as noted there, w e emphasize that b oth of them require sharp ening: F or instance, the ﬁrst one do es not rule out nil-extensions of ﬁelds, while the second one should b e strengthened to a uniqueness statemen t up to a suitable equiv alence relation. Unfortunately , suc h a general theory of tt-ﬁelds remains elusive and v arious attempts at partial solutions ha ve b een considered. Guided by classical examples from comm utative algebra, homotopy theory , and mo dular represen tation theory , t wo closely related deﬁnitions of tt-ﬁeld were prop osed in [ Bal10b ] and [ BKS19 ]. While both of them satisfy Desideratum ( † ), it is muc h harder to verify ( ‡ ) for these. In particular, we construct an example Prop osition 2.31 that sho ws that the t wo prop osed deﬁnitions of tt-ﬁelds given in [ Bal10b ] and [ BKS19 ] do not coincide, and for which ( ‡ ) fails, at least if w e are using the deﬁnition prop osed in [ BKS19 ]. GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 3 Changing persp ective and restricting attention to rigid tt-categories, Balmer, Krause, and Stev enson [ BKS19 ] relaxed the condition that the residue ﬁelds be tt-categories to instead study homological functors from C to ab elian residue ﬁelds. In a series of papers [ Bal20b ; Bal20a ; BC21 ], this idea has been dev elop ed in to a theory of homological sp ectrum Sp c h ( C ), homological primes, and homological supp ort, paralleling the triangular theory from [ Bal05 ]. In particular, Balmer shows that there is a surjective map φ : Sp c h ( C ) ↠ Spc( C ) , and that every homological prime B ∈ Sp c h ( C ) is detected by an ab elian residue ﬁeld. In addition, he pro ves an abstract nilp otence theorem for C based on Sp c h C . Since Sp c( C ) parametrized thick tensor ideals of C , from an abstract point of view the map φ expresses a formal relationship b et ween a nilpotence theorem and a thic k ideal theorem for C . Inspired by a broad v ariety of examples, the next conjecture reconciles the tw o p oints of view: Conjecture (Balmer’s Nerv es of Steel Conjecture) . F or any rigid tt -c ate gory C the c omp arison map φ : Sp c h ( C ) ↠ Spc( C ) is bije ctive. Besides it signiﬁcance in the aforementioned program to construct residue ﬁelds in tt-geometry , this conjecture would also hav e other imp ortan t applications, as re- view ed in Section 2 . The conjecture is known in all examples in whic h the sp ectrum has b een computed (see e.g., [ Bal20b , Section 5]) and enjoys a num b er of strong p ermanence (suc h as descent) prop erties [ BHS23 ; Bar+26 ]. Despite this evidence, our ﬁrst main result ( Theorem 5.7 ) disprov es the Nerves of Steel conjecture: Theorem A. The Nerves of Ste el Conje ctur e is false. Mor e pr e cisely, the c om- p arison map φ fails to b e inje ctive for the fr e e rigid c ommutative 2-ring A 1 , + on a p ointe d obje ct. The key ingredien ts in the proof of this result are a generalized version of the 1-dimensional cob ordism h yp othesis and Deligne’s semisimplicity theorem for his categories Rep( GL t ). In light of the dispro of of the Nerves of Steel conjecture, our second main ob jec- tiv e in this pap er is a diﬀerent approac h to the constructing of residue objects in tt- geometry . Based on the notion of Nullstellensatzian object in tro duced in [ BSY22 ], w e construct a hierarch y of c onstructible sp e ctr a Sp ec cons E n ( C ) for 1 ≤ n ≤ ∞ along with suitable comparison maps to the homological sp ectrum of C , pro vided the tt-category C admits a suitable enhancemen t as a rigid E m -2-ring for 1 ≤ n < m . As a ﬁrst pro of of concept, for rational rigid comm utative 2-rings C , we show that Nullstellensatzian categories supply a satisfactory theory of residue ﬁelds. Gener- alizing prior work of Mathew [ Mat17 ] in the No etherian case, Corollary 7.11 states: Theorem B. If R is a r ational E ∞ -ring, then Perf ( R ) has enough tt-ﬁelds, and p oints in the homolo gic al sp e ctrum of Perf ( R ) ar e al l witnesse d by maps fr om R into r ational 2-p erio dic ﬁelds. As one application, w e deduce that for an y module ov er a rational E ∞ -ring R , its naiv e homological supp ort agrees with the gen uine one, answering a question from [ Bar+26 ] in this case. The p ersp ective employ ed in the proof of Theorem B will also lead to a veriﬁcation of the rational (in fact: ﬁnite heigh t) monogenic Nerv es of Steel conjecture in work in progress by Burklund and separately Chedala v ada. 4 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI Returning to the general case, the theory of E n -constructible sp ectra w e dev elop along with an elab oration on Burklund’s m ultiplicative structure theorem ([ Bur22 ]) leads to the next result. It says that, as long as the given tt-category has a suitably structured enhancemen t, ev ery point in its homological sp ectrum is “geometric”, that is, detected b y a tt-category whose homological sp ectrum is a single p oint: Theorem C. L et C b e a rigid tt-c ate gory which admits an enhanc ement and let m ∈ Sp c h ( C ) b e a homolo gic al prime. Then ther e exists a tt-functor C → K to a rigid tt-c ate gory K such that (a) Sp c h ( K ) = ∗ a single p oint, (b) and such that the map Sp c h ( K ) → Spc h ( C ) has image exactly { m } . As a consequence, w e deduce that the residue tt-functors constructed in Theo- rem C satisfy the desiderata ( † ) and ( ‡ ). How ever, w e caution the reader that, as w e will explain, the constructions that go into the pro of of this theorem are not y et suﬃcien t to supply a fully satisfying theory of ﬁelds in tt-geometry . 1.B. Metho dology. F r e e c onstructions and the c ounter example to the Nerves of Ste el c onje ctur e. The starting p oin t for our dispro of of the Nerves of Steel conjecture is the exact nilp o- tenc e c ondition , due to Balmer [ Bal20a , Theorem A.1] and the second-named author [ Hys26 ]. This condition holding for all lo calizations of a given rigid tt-category is equiv alen t to the Nerv es of Steel conjecture for said category but, since it do es not mak e reference to the homological sp ectrum, it also makes sense outside the rigid con text. The main theorem of [ Hys26 ] shows that it fails for the free (non-rigid) comm utative 2-ring on a p ointed ob ject. Our strategy is to adapt this approach to the rigid context, whic h p oses substantial additional diﬃculties. There is a free rigid commutativ e 2-ring A 1 , + on a p ointed ob ject, as w ell as a free rigid commutativ e 2-ring A 1 on an ob ject. The category A 1 pla ys the role of the aﬃne line in higher Zariski geometry [ Aok+25 ], in that it represen ts the global sections functor. T o make our liv es easier, w e will implicitly assume that A 1 , + (resp. A 1 ) is rational, working with the free rigid comm utative 2-ring on a p oin ted object (resp. on an object) o ver the deriv ed category of the rationals D b ( Q ). In con trast to the non-rigid case, A 1 and A 1 , + are not lo cal and ha ve very large Balmer sp ectra, see Corollary 3.10 and Corollary 4.10 for a description of many p oin ts of A 1 . In particular, if we wish to study the exact-nilp otence condition, we m ust conten t ourselves with attempting to study some of their lo calizations. This applies, in particular, to a certain generic p oin t η of A 1 and likewise for A 1 , + . No w taking the free pointed ob ject to the p oin ted object 1 0 − → X induces a tt-functor can : A 1 , + → A 1 . In order to analyze this functor as w ell as the cate- gories it relates, we will make use of the 1-dimensional cob ordism hypothesis due to Lurie ([ Lur08 ], see also [ Har12 ]) and its (forthcoming) generalization b y Barkan– Steinebrunner [ BS ]. It provides identiﬁcations A 1 = F un((Cob) op , D ( Q )) ω and A 1 , + = F un((Cob + ) op , D ( Q )) ω , resulting in a ‘generators and relations descriptions’ of these categories. In par- ticular, in Section 4 w e are able to relate A 1 to Deligne’s category Rep( GL t ) (see [ Del07 ] and recalled in Section A ) and use his semisimplicity theorem. The k ey structural features w e establish are summarized in the follo wing result, collecting Theorem 4.6 and Proposition 5.1 : GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 5 Theorem D. The functor can extends to a tt-functor can η : A 1 , + η → A 1 η on generic p oints. Mor e over, we have: (a) The functor can η : A 1 , + η → A 1 η is c onservative. (b) The tt-c ate gory A 1 η is a semisimple tt-ﬁeld. As explained at the end of Section 5 , this theorem suﬃces to conclude that the exact-nilp otence condition fails for A 1 , + η , for the (image of the) univ ersal ﬁb er sequence Y → 1 → X , thereby pro ving Theorem A . E ∞ -c onstructible sp e ctr a and the r ational c ase. The dra wbac ks of the known notions of residue ﬁelds in tt-geometry—as exhibited ab ov e—lead us to a nov el approach, whic h is form ulated in the setting of higher Zariski geometry [ Aok+25 ]. This re- quires that the tt-category under consideration admits a suitable enhancemen t, i.e., arises as the homotopy category of a commutativ e 2-ring 2 . Since all tt-categories in nature are of this form and b ecause the salient op erations preserve the enhance- men t, we view this is as a v ery mild assumption. In [ BSY22 ] and motiv ated by applications to chromatic homotopy theory , Burk- lund, Sc hlank and Y uan introduced a general notion of Nullstellensatzian object whic h captures the essential features of algebraically closed ﬁeld abstractly . Ap- plying this concept to the category of commutativ e 2-rings leads to our deﬁnition of the c onstructible sp e ctrum Sp ec cons ( C ) of C , see Deﬁnition 6.5 . This sp ectrum is a compact T 1 -space whose p oin ts are given by Nullstellensatzian comm utative 2-rings. In Corollary 6.9 we construct a natural comparison map ψ : Sp ec cons ( C ) → Spc( C ) . The signiﬁcance of this map is then explained in Prop osition 6.10 : A p oint in Sp c( C ) is in the image of ψ if and only if it is can b e detected by a map of commutativ e 2- rings C → D with Spc( D ) = ∗ . In other w ords, the constructible sp ectrum provides an abstract setting whic h captures Desiderata ( † ) and ( ‡ ). In general, how ev er, the comparison map ψ is not surjectiv e, so not every prime ideal in the Balmer spec- trum can b e realized geometrically by a map of commutativ e 2-rings. W e remark that the failure of realizability holds b oth in the ﬁnite p ositive height situation ( Corollary 6.11 ) as w ell as at heigh t ∞ ( Example 6.12 ), that is in characteristic p . This is in sharp contrast to what happ ens in characteristic 0. W orking with a rational comm utative 2-ring C for the remainder of this section, w e ﬁrst observ e that Nullstellensatzian C -algebras correspond under decategoriﬁcation to Nullstel- lensatzian 1 C -algebra: Indeed, Prop osition 6.7 establishes a homeomorphism Sp ec cons ( C ) ≃ Spec cons CAlg(Ind( C )) ( 1 C ) . This aﬀords the construction of a comparison map ψ h : Sp ec cons ( C ) → Spc h ( C ) for rational C , which w e show in Theorem 7.1 to b e a bijection: Theorem E. L et C b e a r ational rigid c ommutative 2-ring. Then ther e is a natur al isomorphism of sets ψ h : Sp ec cons ( C ) ≃ Spc h ( C ) b etwe en the c onstructible sp e ctrum and the homolo gic al sp e ctrum of C . With this result in hand, it is then not to o diﬃcult to deduce our applications to rational monogenic commutativ e 2-rings in Theorem B . At the same time, this 2 by which we mean an (essen tially small) stably symmetric monoidal idempotent-complete ∞ -category 6 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI construction giv es a new top ology on the homological sp ectrum of a rational rigid comm utative 2-rings which is alwa ys compact T 1 . The question as to when it is compact Hausdorﬀ can be chec ked by understanding Nullstellensatzian algebras in a given category , and is closely related to when the homological sp ectrum is a sheaﬁﬁcation of the Balmer sp ectrum for a certain “canonical top ology” (see Theo- rem 7.23 ). It is p ossible that the top ology on the homological sp ectrum induced by Theorem E is alwa ys compact Hausdorﬀ, a question which is equiv alen t to a uni- form b ound on the exact-nilp otence condition for Nul lstel lensatzian rational rigid comm utative 2-rings, and which reduces to separating tw o sp eciﬁed p oin ts in the constructible spectrum of our counter-example to the nerves of steel conjecture (see Prop osition 7.25 for a precise statemen t). E n -c onstructible sp e ctrum and ge ometric p oints. As we hav e seen, outside the ra- tional setting, p oints of the Balmer sp ectrum of a tt-category can in general not b e realized geometrically through maps of comm utative 2-rings. Surprisingly , a minor mo diﬁcation of the constructible sp ectrum resolves this issue entirely . Generalizing the previous setup, supp ose for the remainder of this section that C is a rigid E m -2-ring, i.e., an (essentially small) rigid stably E m -monoidal idemp otent- complete ∞ -category for some m ≥ 3. The condition on m guarantees that the homotop y category of C is a rigid tt-category . F or any 1 ≤ n < m , we then deﬁne the E n -constructible sp ectrum of C as Sp ec cons E n ( 1 C ). This comes with a comparison map ψ h n : Sp ec cons E n ( 1 C ) → Spc h ( C ) induced b y sending a Nullstellensatzian E n -algebra to its homological support. Our main theorem ( Theorem 8.2 ) ab out the E n -constructible sp ectrum sho ws that ψ h n is bijective: Theorem F. L et Ind( C ) b e the Ind c ate gory of a rigid E m -2-ring C . Then for al l 1 ≤ n < m , ther e is a natur al bije ction ψ h n : Sp ec cons E n ( 1 C ) ≃ Spc h ( C ) . In fact, Theorem 8.17 provides an example that shows that the condition n < m in the statemen t ab o ve is optimal. The key ingredient in the pro of of Theorem F is the construction, for each homological prime m , of E n -algebra v ariants E n m of the weak rings E m in tro duced in [ Bal20a , Construction 2.11]. The latter play a distinguished role in the study of the homological spectrum, and likewise their structured analogues are essen tial in the analysis of Sp ec cons E n ( 1 C ). The construction of E n m relies essentially on Burklund’s w ork [ Bur22 ] on multiplicativ e structures on quotien t ob jects in higher algebra. In order to deduce Theorem C from Theorem F we need one additional step. While we cannot directly conclude that the homological sp ectrum of E n m is a p oin t, w e will pro v e in Section 9 that Spc h (P erf C ( E k m )) has the desired prop ert y for k suﬃ- cien tly large. This establishes a suﬃcient supplies of residue rigid E n -2-rings, which up on passage to homotop y categories giv es Theorem C . Finally , we remark that the o verall structure of Perf C ( E k m ) is not yet w ell-understo o d, so the extent to which these residue tt-categories should b e considered ﬁelds is still under inv estigation. 1.C. Structure of the document. There are tw o parts, consisting in the disproof of the Nerves of Steel conjecture and our analysis of the constructible sp ectrum, resp ectiv ely . GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 7 In the ﬁrst part, we b egin with some preliminaries on tt-geometry collected here for the conv enience of the reader. Most of this material in Section 2 is purely exp ositional, with the exception of the discussion of tt-ﬁelds. Section 3 deals with free constructions in higher Zariski geometry and the generalized 1-dimensional cob ordism h yp othesis, whic h is then applied in Section 4 to analyse the aﬃne line and its p oin ted v ariant. This part culminates in Section 5 , where the pro of of Theorem A is assem bled. The second part is concerned with the constructible spectrum and its applica- tions to the construction of geometric p oin ts in higher Zariski geometry . After a brief recollection on the theory of Nullstellensatzian ob jects from [ BSY22 ], Section 6 in tro duces the E ∞ -constructible sp ectrum of rigid commutativ e 2-rings, establishes its basic prop erties, and discusses its relev ance in tt-geometry . The end of this section also contains our coun terexamples to the most optimistic guesses ab out its utility , motiv ating the in-depth study of the rational case in Section 7 and the pro of of Theorem B . Finally , Section 8 and Section 9 in tro duce the E n -constructible sp ectrum of rigid 2-rings and, using a mo diﬁcation of Burklund’s approach to mul- tiplicativ e structures, verify Theorem C . This pap er concludes with an outline of a sp ecial case of Deligne’s pro of of the semi-simplicit y of Rep( GL t ) from [ Del07 ] in Section A . 1.D. Con ven tions and notation. Throughout the main b o dy of this w ork, we will w ork in the framework of higher Zariski geometry as developed in [ Aok+25 ], for- m ulated in the language of ∞ -categories ([ Lur09 ; Lur17 ]) and enhancing tensor tri- angular geometry as dev elop ed b y Balmer [ Bal05 ]. Our notations and terminology will conform to these references unless otherwise noted. In particular, tt-c ate gory refers to the classical 1-categorical notion of a tensor triangulated category , while w e use the term E m -2-ring with 0 ≤ m ≤ ∞ for an essentially small stably E m - monoidal ∞ -category , usually assumed to b e idemp otent-complete. When m ≥ 3, the homotop y category K = ho( C ) of an E m -2-ring C is a tt-category; in this case, C is said to b e an E m -enhanc ement of K . F or the remainder of this pap er, tt-category will implicitly mean rigid tt-category , and the term 2-ring without any mo diﬁers will implicitly mean comm utative 2-ring. A ckno wledgemen ts. W e thank P aul Balmer, Anish Chedala v ada, Akhil Mathew, Thomas Nikolaus, Phil P¨ utzst ¨ uc k, T omer Schlank, Robin Sroka, and Jan Steine- brunner for many helpful discussions. TB and LH were supp orted b y the Europ ean Researc h Council (ERC) under Horizon Europ e (grant No. 101042990) and are grateful to the Max Planc k Institute for Mathematics in Bonn for its hospital- it y and ﬁnancial supp ort. MR is funded by the Deutsc he F orsch ungsgemeinsc haft (DF G, German Research F oundation) – Pro ject-ID 427320536 – SFB 1442, as well as under Germany’s Excellence Strategy EX C 2044/2 –390685587, Mathematics M ¨ unster: Dynamics–Geometry–Structure. 8 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI P art I. A Counterexample to the Nerv es of Steel Conjecture 2. Preliminaries on Tensor Triangular Geometr y 2.A. Basic notions of tensor-triangular geometry . Motiv ated b y computa- tions due to Devinatz–Hopkins–Smith [ DHS88 ; HS98 ], Hopkins–Neeman [ Hop87 ; Nee92 ], and Thomason [ Tho97 ], Balmer [ Bal05 ] made the follo wing deﬁnition, marking the b eginning of the ﬁeld now known as tensor triangular geometry . Deﬁnition 2.1 ([ Bal05 ]) . Let C b e an essentially small tt-category . Then the Balmer sp e ctrum of C , Sp c( C ) is the set of prime ⊗ -ideals in C , topologized by deﬁning distinguished closed subsets Supp( x ) : = {P ∈ Sp c( C ) : x / ∈ P } , for ob jects x ∈ C . This deﬁnition parallels the construction of the Zariski spectrum in algebraic geometry , and thus leads to a geometric p ersp ective on tt-categories; for a recent accoun t, see [ A ok+25 ]. The Balmer sp ectrum comes equipp ed with natural maps to other spaces, such as the homogeneous sp ectrum of graded prime ideals in the graded endomorphism ring of the unit. Prop osition 2.2 ([ Bal10a , Theorem 5.3]) . Ther e exists a natur al c ontinuous map ρ : Sp c( C ) → Spec h ( π ∗ (End C ( 1 ))) fr om the Balmer sp e ctrum of a tt-c ate gory C to the sp e ctrum of homo gene ous prime ide als in the gr ade d-c ommutative ring π ∗ (End C ( 1 )) . The map ρ is or der-r eversing, in the sense that P ⊆ Q if and only if ρ ( P ) ⊇ ρ ( Q ) . Equipp ed with a go o d notion of geometry for tt-categories, the familiar words and concepts of algebraic geometry , or at least some of them, can b e adapted to the world of tensor triangular geometry . A particularly imp ortant notion for the presen t pap er is what it means to b e local, that is, ha ve the prop ert y that the sp ectrum has a unique closed p oin t. Pondering the deﬁnitions ([ Bal10a ]), closed p oin ts of the space Sp c( C ) corresp ond to minimal prime ideals in C , and one learns that Spc( C ) has a unique closed point if and only if the ideal (0) is prime. Collecting this into a deﬁnition, and rephrasing what it means for (0) to b e prime, we recall: Deﬁnition 2.3. A tt-category C is said to be lo c al is for all ob jects X, Y ∈ C , X ⊗ Y ≃ 0 implies that X ≃ 0 or Y ≃ 0. Another imp ortant play er in this paper is the notion of ⊗ -nilp otent maps. Deﬁnition 2.4. A map f : x → y in a tt-category is said to b e ⊗ -nilp otent if there exists some n ≥ 0 with f ⊗ n ≃ 0. Lemma 2.5. L et C b e a tt-c ate gory. The class of ⊗ -nilp otent morphisms in C is close d under: • T ensoring with any obje ct; • R etr acts; • Dualization. F urthermor e, if F : C → D is a tt-functor, it sends ⊗ -nilp otent morphisms to ⊗ - nilp otent morphisms. GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 9 Pr o of. The ﬁrst item follo ws from the fact that ( f ⊗ z ) ⊗ n ≃ f ⊗ n ⊗ z ⊗ n . The second item follows from the fact that if f 0 is a retract of f 1 , then f ⊗ n 0 is a retract of f ⊗ n 1 , and retracts of the 0 morphism are 0. The third item follo ws from the fact that dualization comm utes with tensor pro ducts and the dual of the 0 morphism is the 0 morphism. The last fact is also clear from the equiv alence F ( f ) ⊗ n ≃ F ( f ⊗ n ). □ Deﬁnition 2.6. An object x ∈ C in a tt-category C is said to b e ⊗ -faithful if x ⊗ − is conserv ative on morphisms, that is: if f : y → z is a morphism in C with x ⊗ f ≃ 0, then f ≃ 0. W e will make use of the follo wing elemen tary observ ation: Lemma 2.7. A n obje ct x in a tt-c ate gory C is ⊗ -faithful if and only if the c o- evaluation morphism co ev x : 1 → x ⊗ x ∨ is split inje ctive. In p articular, if every non-zer o obje ct of C is ⊗ -faithful, then the Balmer sp e ctrum of C is a single p oint. Pr o of. If 1 splits oﬀ of x ⊗ x ∨ , then x ⊗ x ∨ ⊗ − is conserv ativ e on morphisms, and the same must then b e true of the functor x ⊗ − . Conv ersely , if x ⊗ − is ⊗ -faithful, then since x ⊗ coev x is split injective, the map ﬁb(co ev x ) → 1 tensors with x to zero, and is th us itself zero, whic h implies that co ev x is split injective. The ﬁnal claim follows from the observ ation that every ⊗ -faithful ob ject gener- ates the unit ideal, which then m ust be true of every nonzero ob ject. □ 2.B. The homological sp ectrum. The ﬁrst Balmer sp ectrum computation, as with man y more to follo w, was carried out b y ﬁrst pro ving an abstract nilp otence theorem, then using this to pro duce a classiﬁcation of the thick ⊗ -ideals. F ollowing the idea that there should b e a general in terpla y betw een tensor triangular geometry and abstract nilpotence theorems and building on earlier w ork with Krause and Stev enson [ BKS19 ], Balmer in [ Bal20b , Remark 3.4] found a new w ay to attach a top ological space to a tt-category . W e recall the construction now. Deﬁnition 2.8. Let C b e a tt-category . The category mod( C ) is the full sub cate- gory of additive preshea ves on C v alued in ab elian groups F un ⊕ ( C op , Ab) generated by cokernels of maps よ ( x ) → よ ( y ) for x, y ∈ C . The category mo d( C ) is an ab elian category , which inherits a symmetric monoidal structure through Day con volution in suc h a wa y that the Y oneda em b edding よ : C → mo d( C ) is symmetric monoidal. In the absence of a general theory of “residue ﬁelds” for tt- categories, we turn our attention to nice “ab elian residue ﬁelds” instead. Explicitly , Balmer deﬁnes a homological residue ﬁeld of a tt-category C as a quotien t of mo d( C ) b y a maximal Serre ⊗ -ideal. This leads us to: Deﬁnition 2.9 ([ Bal20b , Remark 3.4]) . The homolo gic al sp e ctrum of a tt-category C is deﬁned as a set by Sp c h ( C ) : = { maximal Serre ⊗ -ideals in mo d( C ) } . 10 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI This becomes a topological space by deﬁning closed subsets to b e generated by those of the form Supp h ( x ) : = { m ∈ Spc h ( C ) : よ ( x ) / ∈ m } . A t any given p oint m in the homological sp ectrum, the homolo gic al r esidue ﬁeld at m is deﬁned to b e the composite functor C → mo d( C ) / m to the quotient by the maximal Serre ⊗ -ideal m . 2.C. W eak rings. Deﬁnition 2.10. Let C be a tt-category , and consider its big category Ind( C ). A we ak ring in Ind( C ) is a p ointed ob ject f : 1 → R which b ecomes split after tensoring with R , that it, suc h that R ⊗ f : R → R ⊗ R is split injective. Remark 2.11. Each p oin t of the homological sp ectrum m giv es rise to a unique w eak ring ob ject E m in the big category Ind( C ) corresponding to it, suc h that the kernel of tensoring with E m is exactly the k ernel of the comp osite Ind( C ) → Ind (mo d( C )) → Ind (mod( C ) / m ) [ BKS19 , Prop osition 3.9]. In [ BC21 ], Balmer– Cameron demonstrate techniques to compute a great n umber of these w eak rings in practice. The following prop erty of the weak rings attac hed to homological primes will b e useful later. Prop osition 2.12 ([ Bal20b , Proposition 5.3]) . L et C b e a tt-c ate gory, and c onsider homolo gic al primes m 1 , m 2 ∈ Sp c h ( C ) with asso ciate d we ak rings E m 1 , E m 2 . Then E m 1 ⊗ E m 2 ≃ 0 if and only if m 1  = m 2 . There are tw o natural notions of homological support attached to big ob jects in a given tt-category . Deﬁnition 2.13. Let C b e a tt-category , whic h has asso ciated big category “Ind( C )” (sa y , arising as the homotopy category the Ind-category of a rigid E n -2-ring with homotop y category C ), and let t ∈ Ind( C ) a big ob ject. Then we deﬁne • The naive homolo gic al supp ort of t is Supp n ( t ) : = { m ∈ Spc h ( C ) : t ⊗ E m  = 0 } . • The genuine homolo gic al supp ort of t is Supp h ( t ) : = { m ∈ Spc h ( C ) : hom( t, E m )  = 0 } . W e record the follo wing prop erties of these supp orts, whic h will be useful later. Prop osition 2.14 ([ Bal20a , Theorem 4.5]) . The genuine homolo gic al supp ort sat- isﬁes the so-c al le d tensor pr o duct pr op erty. Namely, Supp h ( t 1 ⊗ t 2 ) = Supp h ( t 1 ) ∩ Supp h ( t 2 ) . Prop osition 2.15 ([ Bal20a , Theorem 4.7]) . If t is a we ak ring, then ther e is an e quality Supp n ( t ) = Supp h ( t ) . GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 11 W e will frequen tly iden tify Supp h ( t ) with Supp n ( t ) implicitly when working with w eak rings later on. Finally , w e record also the following detection prop erty of the homological supp ort for weak rings. Prop osition 2.16 ([ Bal20a , Theorem 4.7]) . If w ∈ Ind( C ) is a we ak ring with homolo gic al supp ort Supp h ( w ) = ∅ , then w = 0 . 2.D. Nilp otence detection. Balmer in vestigated the connection of this homolog- ical sp ectrum to nilp otence theorems, proving that in some sense, proving abstract nilp otence theorems is equiv alent to computing the homological sp ectrum of a cat- egory . Prop osition 2.17 ([ Bal20b , Corollary 4.7, Theorem 5.4]) . Given a map f : x → Y for x ∈ C and Y ∈ Ind( C ) , if f vanishes in every homolo gic al r esidue ﬁeld of C , then f is ⊗ -nilp otent. F urthermor e, given a c ol le ction I = { m } ⊆ Sp c h ( C ) of p oints in the homolo gic al sp e ctrum with the pr op erty that f vanishing in the r esidue ﬁeld at m for al l m ∈ I implies f is ⊗ -nilp otent, then I = Sp c h ( C ) is the entir e sp ac e. An important related notion is that of nil-conserv ativity; w e recall the deﬁnition: Deﬁnition 2.18. Let { f i : C → D i } i ∈ I b e a family of functors from a ﬁxed tt- category C to v arious tt-categories D i . W e sa y that the family is jointly nil- c onservative if the induced functors on big categories (abusiv ely also denoted b y f i ) detect weak rings, that is: if R is a weak ring in Ind( C ) suc h that f i ( R ) ≃ 0 for all i ∈ I , then R ≃ 0. The connection b et ween join tly nil-conserv ativ e families and the homological sp ectrum is summarized by the follo wing theorem. Theorem 2.19 ([ Bar+24 , Theorem 1.9]) . L et { f i : C → D i } i ∈ I b e a family of tt-functors. Then the family is jointly nil-c onservative if and only if the induc e d map a i ∈ I Sp c h ( D i ) → Spc h ( C ) is surje ctive. W e will later require a basechange property for jointly nil-conserv ativ e families in the highly structured case, whic h in turn will require the follo wing lemma. Lemma 2.20. L et C and D b e rigid 2-rings, and let f ∗ : C → D b e a 2-ring map which is ful ly faithful (so that the right adjoint f ∗ on Ind-c ate gories has f ∗ f ∗ ≃ id Ind( C ) . Then for any map of rigid 2-rings g ∗ : C → E , the b ase change E → E ⊗ C D is also ful ly faithful. Pr o of. By [ Ram24 , Prop osition 4.18] (whic h applies since C is rigid), the adjunction f ∗ ⊣ f ∗ is Ind( C )-linear and hence, b y 2-functoriality of E ⊗ Ind( C ) − , it induces an adjunction up on tensoring with E . Since fully faithfulness can b e expressed in 2-categorical terms (that the unit map is an equiv alence), the result follo ws. □ Prop osition 2.21. L et C b e a rigid 2-ring. Supp ose we ar e given a family of functors F i : C → D i , i ∈ I of rigid 2-rings which is jointly nil-c onservative. Then, for any 2-ring map G : C → E to a rigid 2-ring E , the family { E → E ⊗ C D i } i ∈ I is jointly nil-c onservative as wel l. 12 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI Pr o of. Let S ∈ Ind( E ) b e a weak ring, and assume that F i ( S ) ∈ Ind( E ) ⊗ Ind( C ) Ind( D i ) is 0 for each i . Again, b y [ Ram24 , Prop osition 4.18], the adjunction G ⊣ G ∗ b et ween Ind( C ) and Ind( E ) induces an adjunction G ⊗ Ind( D i ) ⊣ G ∗ ⊗ Ind( D i ) b etw een Ind( D i ) and Ind( E ) ⊗ Ind( C ) Ind( D i ), which we abusively still denote G, G ∗ . W e therefore ﬁnd that G ∗ F i ( S ) = 0. W e claim that the following square is verti- cally righ t adjointable, i.e., that the canonical map F i G ∗ → G ∗ F i is an equiv alence: Ind( C ) Ind( D i ) Ind( E ) Ind( E ) ⊗ Ind( C ) Ind( D i ) F i G G F i Once this is prov ed, it will follo w that F i G ∗ ( S ) = 0 for all i , and hence that G ∗ ( S ) = 0 b y nil-conserv ativity . But there is a map of weak rings GG ∗ ( S ) → S by design, so this implies that S = 0. T o pro ve this adjointabilit y , we again use 2-functoriality of − ⊗ Ind( C ) − : the arro w Ind( C ) → Ind( D i ) sends the adjunction G ⊣ G ∗ to an adjunction in the arro w category , and those are precisely adjointable squares (this follows from the more general [ Hau21 , Theorem 4.6]). □ 2.E. The nerves of steel conjecture. F or an y tt-category C , there exists a nat- ural contin uouos surjection φ : Sp c h ( C ) ↠ Spc( C ) b y [ Bal20b , Corollary 3.9], whic h connects us bac k to the setting of tensor triangular geometry . The manner in which the v ery ﬁrst computation of a Balmer spectrum w as carried out in [ HS98 ] was essen tially by computing the homological sp ectrum of Sp ω , and then pro ving that the map Sp c h ( Sp ω ) → Sp c( Sp ω ) is an isomorphism. In mo dern terms, Hopkins–Smith prov ed the ﬁrst case of Balmer’s nerv es of steel conjecture, which we can no w state. Deﬁnition 2.22. W e sa y that the nerves of ste el c ondition holds for a tt-category C (written NoS( C )) if the map φ : Sp c h ( C ) → Sp c( C ) is a bijection. The nerves of ste el c onje ctur e asserts that ev ery tt-category satisﬁes the nerv es of steel condition. The nerves of steel conjecture has b een a center of atten tion for tt-geometers in recen t years, and man y equiv alent form ulations ha ve been discov ered. F or instance: Remark 2.23. F or a tt-category C , the comparison map φ exhibits Sp c( C ) as the K olomogorov quotient of Sp c h ( C ), i.e., the universal T 0 -space under it; see [ BHS23 , Lemma 4.2]. This implies the equiv alence of the following statements: (a) φ : Sp c h ( C ) → Spc( C ) is a bijective; (b) φ : Sp c h ( C ) → Spc( C ) is a homeomorphism; (c) Sp c h ( C ) is T 0 (and hence sp ectral). Remark 2.24. There is an alternative description of the homological sp ectrum constructed in [ BW25 ] via a space constructed from the Ziegler spectrum, relating homological primes to so-called “deﬁnable” ⊗ -subcategories. GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 13 Remark 2.25. W ork of [ Bar+26 ] sho ws that if the nerves of steel conjecture holds, then the triangular and homological notions of stratiﬁcation coincide for all rigidly- compactly generated tt-categories. In particular, stratiﬁcation would satisfy descen t along weakly descendable families of geometric tt-functors. 2.F. The exact-nilp otence condition. The nerves of steel conjecture was refor- m ulated in [ Bal20a , Theorem A.1], and again in [ Hys26 , Theorem 1.6] in terms of the so-called exact-nilp otence condition, which w e no w recall. Deﬁnition 2.26 ([ Hys26 , Deﬁnition 1.5]) . Let C b e a lo cal tt-category . W e say that the exact-nilp otenc e c ondition (r esp. exact-nilp otenc e c ondition to or der n ) (ENC( n )) holds for C if for all ﬁb er sequences of the form y g − → 1 f − → x, there exists a nonzero ob ject z ∈ C with either z ⊗ g or z ⊗ f b eing ⊗ -nilp otent (resp. either z ⊗ g ⊗ n ≃ 0 or z ⊗ f ⊗ n ≃ 0). As mentioned in Section 1.B , the starting p oint of this pap er is the following theorem. Theorem 2.27 ([ Bal20a , Theorem A.1]; [ Hys26 , Theorem 1.6]) . The nerves of ste el c onje ctur e holds if and only if the exact-nilp otenc e c ondition holds for every lo c al tt-c ate gory C . 2.G. T ensor triangular ﬁelds. W e recall the deﬁnition of tt-ﬁelds prop osed b y Balmer, Krause, and Stevenson in [ BKS19 ] and discuss some of its basic prop er- ties. In particular, we show that it diﬀers from the earlier one given by Balmer in [ Bal10b ]. Deﬁnition 2.28 ([ BKS19 , Deﬁnition 1.1]) . A rigidly-compactly generated tt-category F is a tt-ﬁeld if it satisﬁes the following tw o conditions: (a) ev ery nonzero compact ob ject x ∈ F c is ⊗ -faithful, i.e., x ⊗ f = 0 implies f = 0 for morphisms f in F ; (b) ev ery ob ject in F is a copro duct of compacts. Occasionally , w e will also refer to the essen tially small tt-category F c as a tt-ﬁeld. Simple examples of tt-ﬁelds are giv en b y: Lemma 2.29. A semi-simple tt-c ate gory K with simple unit is a tt-ﬁeld, in that for any nonzer o obje ct z ∈ K , co ev : 1 → z ⊗ z ∨ is split inje ctive. Pr o of. In a semi-simple category , any nonzero map from an ob ject with simple endomorphism ring is split-injectiv e. □ Remark 2.30. An imp ortant consequence of the deﬁnition of tt-ﬁeld F is that F c is a Krull–Schmidt category , see for example [ BKS19 , Theorem 5.7]. In [ Bal10b , Section 4.3], Balmer prop osed a v ariant, only requiring condition (a) in Deﬁnition 2.28 . The next prop osition provides an example demonstrating that the tw o deﬁnitions do not coincide, thereby resolving a question which w as raised in [ BKS19 , Remark 1.2]. Prop osition 2.31. L et ( κ n ) n ∈ N b e an N -indexe d c ol le ction of ﬁelds and let U b e a non-princip al ultr aﬁlter on N . Consider the ultr apr o duct Q U P erf ( κ n ) taken in the c ate gory of 2-rings. 14 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI (a) Every nonzer o c omp act obje ct of Q U P erf ( κ n ) is ⊗ -faithful, i.e., the ultr apr o d- uct satisﬁes c ondition (a) of Deﬁnition 2.28 . (b) Ther e do es not exist any tt-functor Q U P erf ( κ n ) → F to a tt-ﬁeld F . In p artic- ular, Q U P erf ( κ n ) do es not satisfy c ondition (b) of Deﬁnition 2.28 Pr o of. Ev ery non-zero compact object in Q U P erf ( κ n ) is ⊗ -faithful since the co- ev aluation map of ev ery non-zero ob ject splits; this can b e c heck ed co ordinate-wise. In order to sho w that Q U P erf ( κ n ) do es not admit an y maps into a tt-ﬁeld, consider the image X of the ob ject ( L n i =1 κ n ) n ∈ N in the ultrapro duct. Since U is non-principal, for any m ∈ N there exists some V m ∈ U which do es not meet the in terv al [0 , m − 1]. In Q n ∈ V m P erf ( κ n ), the ob ject L m i =1 1 splits oﬀ ( L n i =1 κ n ) n ∈ V m , hence the same is true for X in the ultrapro duct. Its image under any tt-functor cannot satisfy the Krull–Sc hmidt property , so we conclude by Remark 2.30 . □ As an immediate consequence, we obtain: Corollary 2.32. The c ol le ction of tt-ﬁelds is not close d under ultr apr o ducts. Remark 2.33. In ligh t of the ab ov e, one might b e tempted to try to deﬁne a tt-ﬁeld as a rigidly-compactly generated tt-category where ev ery non-zero compact ob ject is ⊗ -faithful, follo wing [ Bal10b ]. This notion would b e closed under ultrapro ducts but, as we will show in Remark 3.29 b elow, there are categories with this property whic h act m uch more akin to a “nil-extension” of a ﬁeld. F urthermore, recent work of Riedel (in particular [ Rie25 , Remark 4.24]) can b e used to sho w that, Desideratum ( ‡ ) from the in tro duction also fails for the deﬁnition prop osed in [ Bal10b ] at least if one required the map to admit an enhancement as an E 1 -2-ring map. W e note that this do es not completely rule out the p ossibility of ( ‡ ) holding for this deﬁnition at the purely tensor triangulated level, just of there b eing some enhanced version of it. This concludes our review of the required bac kground material on tensor trian- gular geometry . GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 15 3. Free Constructions The goal of this section is to deﬁne and analyze the free categories we will need. Recall from Section 1.B that w e will need to analyze the free rational rigid 2-ring on a dualizable ob ject and the free rational rigid 2-ring on a p ointed dualizable ob ject. A ﬁrst step for this is to analyze the non-rational, non-stable v ersions of these categories. In a ﬁrst sub-section, we discuss the free symmetric monoidal category on a dualizable object as w ell as its Q -linearization, and in a second subsection w e compare it to the free symmetric monoidal category on a p ointed dualizable ob ject, resp ectiv ely to its Q -linearization. 3.A. The free symmetric monoidal category on a dualizable ob ject. The free symmetric monoidal category on a dualizable ob ject has the universal dualizable ob ject, X , its tensor p ow ers X ⊗ i , and similarly for its dual, X ∨ , as w ell as their tensors. Besides the symmetric groups Σ i × Σ j acting on X ⊗ i ⊗ X ∨ , ⊗ j , there are extra morphisms coming from the ev aluation pairing X ⊗ X ∨ → 1 and the ev aluation co-pairing 1 → X ∨ ⊗ X . ∅ + − ∅ + − co-ev aluation on X = + ev aluation pairing for X = + The graphical calculus that arises from the triangle identities suggested early on a connection to manifolds, as formalized by Baez and Dolan’s famous Cob ordism Hyp othesis [ BD95 ], which describes in general the free symmetric monoidal ( ∞ , n )- category on a dualizable object in terms of cobordism categories. The follo wing is an informal description of the answer in dimension 1. W e will single out below what we need more precisely: Deﬁnition 3.1. The 1-dimensional oriented cob ordism category Cob 1 d,or is the ( ∞ , 1)-category whose ob jects are orien ted closed 0-dimensional manifolds, i.e., ﬁ- nite sets with a sign ± on each element, and whose morphisms from M to N are oriented 1-dimensional compact manifolds with b oundary W with an orien ted iden tiﬁcation ∂ W ∼ = M ` N . Here, M is the manifold M with the reverse orientation. 16 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI + + iden tity of + = X . A pro of in general (for all n ) has b een sketc hed by Lurie in [ Lur08 ], though there do es not seem to b e a consensus about its completeness. On the other hand, in dimension 1, a complete pro of w as giv en b y Harpaz, so we can state: Theorem 3.2 ([ Har12 ],[ Lur08 ]) . The fr e e rigid symmetric monoidal ∞ -c ate gory on an obje ct is the c ate gory Cob 1 d,or of 1-dimensional oriente d c ob or disms b etwe en oriente d 0-manifolds. Since no higher dimensional, or unoriented cobordisms will app ear in this pap er, w e simplify notation: Notation 3.3. W e let Cob : = Cob 1 d,or . F or X ∈ Cob the universal dualizable ob ject (i.e., the singleton, as a p ositively orien ted 0-dimensional manifold), we let X i,j denote X ⊗ i ⊗ X ∨ , ⊗ j . The precise information w e need ab out Cob can be gathered in the follo wing: Prop osition 3.4. L et i, j, r, s b e natur al numb ers. (a) The symmetric monoidal dimension T of X in Cob is given by the cir cle as a c ob or dism fr om the empty manifold to itself, and Ω(Hom Cob ( ∅ , ∅ ) , T ) ≃ S 1 . F ur- thermor e, the induc e d map B S 1 → Hom Cob ( ∅ , ∅ ) witnesses the tar get as fr e e c om- mutative monoid over the sour c e; (b) The mapping sp ac e Hom Cob ( X i,j , X r,s ) is empty unless i − j = r − s ; (c) The mapping sp ac e Hom Cob ( X i,j , X r,s ) is fr e e on a discr ete set as a mo dule over Hom Cob ( 1 , 1 ) . In p articular, al l mapping sp ac es in Cob ar e ﬁnite disjoint unions of sp ac es of the form ` n (( B S 1 ) × n ) h Σ n . Pr o of. (a) This follows from the deﬁnition of Cob, together with the homotopy equiv alence of top ological groups S 1 → Diﬀ + ( S 1 ). (b) One can give a topological pro of, or a proof based on the univ ersal prop erty of Cob. T op olo gic al pr o of: Fix an orien ted compact 1-dimensional manifold with b ound- ary M , viewed as a cob ordism from ∂ 1 M to ∂ 2 M with asso ciated decomp osition of the boundary of M as ∂ M = ∂ 1 M ` ∂ 2 M , and write ∂ 2 M = ` r 1 X ` ` s 1 X ∨ , and ∂ 1 M = ` i 1 X ` ` j 1 X ∨ . The orien ted coun t of the size of the b oundary of M is zero by the classiﬁcation of compact 1-manifolds. F or our given M , this oriented count is nothing but r − s + j − i = ( r − i ) − ( s − j ), which is required to b e zero. But ( r − i ) − ( s − j ) is zero if and only if r − i = s − j . Universal pr op erty pr o of: Consider the wide sub category C of Cob spanned by morphisms X i,j → X r,s with i − j = r − s . It is clear that these morphisms are closed under comp osition, and under tensor pro ducts, and all symmetry/asso ciator GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 17 equiv alences, so that C → Cob is a symmetric monoidal functor. F urthermore, C con tains the ev aluation and co ev aluation, X ⊗ X ∨ → 1 and 1 → X ∨ ⊗ X , so X ∈ C is dualizable. It follows that the iden tity of Cob factors through C , which prov es the result. (c) Every 1-dimensional cob ordism is (uniquely) a disjoin t union of a simply- connected part and a disjoint union of circles. Consider then the full subspace Hom Cob ( X i,j , X r,s ) simple of Hom Cob ( X i,j , X r,s ) spanned b y the simply-connected cob ordisms. It is a space of interv als with ﬁxed endpoints, and is therefore discrete. The ab o ve observ ation shows that, as a Hom Cob ( 1 , 1 )-mo dule, Hom Cob ( X i,j , X r,s ) is free on Hom Cob ( X i,j , X r,s ) simple . □ Next, we note that “linearization” is an easy process: Prop osition 3.5. The fr e e rigid stably symmetric monoidal ∞ -c ate gory over D b ( Q ) on an obje ct is given by the c ate gory F un((Cob) op , D ( Q )) ω , with the Day c onvolution symmetric monoidal structur e. In p articular, it is gener ate d under ﬁnite c olimits, desusp ensions and r etr acts by images of the Y one da emb e dding, whose mapping sp e ctr a ar e Q -homolo gy sp e ctr a of the mapping sp ac es in Cob . More generally , following [ Lur17 ], the next result is explicitly sp elled out (in the case of sp ectra, but works equally well ov er an arbitrary stable base suc h as D ( Q )) in [ Aok+25 , Construction 2.7]: Prop osition 3.6. L et C b e a smal l symmetric monoidal c ate gory. The fr e e r ational stably symmetric monoidal c ate gory with a symmetric monoidal functor fr om C is F un( C op , D ( Q )) ω e quipp e d with the Day c onvolution structur e. It is gener ate d under ﬁnite c olimits, desusp ensions and r etr acts by images of the Y one da emb e dding, whose mapping sp e ctr a ar e Q -homolo gy sp e ctr a of the mapping sp ac es in C . Notation 3.7. W e deﬁne the (r ational) aﬃne line as A 1 : = F un((Cob) op , D ( Q )) ω , and let y : Cob → A 1 denote the Q -linearized Y oneda embedding. Remark 3.8. T o b e careful, one should add a Q -subscript to this notation, but no non-rational 2-ring will app ear in Part I of this paper, and we will not use this notation in Part I I, sav e for when w e are talking only ab out rational categories, so there is no risk for confusion. Lemma 3.9. The gr ade d endomorphism ring of the unit in A 1 is given by π ∗ ( 1 ) = Q [ t ][ t 1 , t 2 , t 3 , . . . ] with | t | = 0 and | t i | = 2 i for i > 0 . Pr o of. By Prop osition 3.5 and the ( Q -linear) Y oneda embedding, the graded endo- morphism ring of the unit is given by π ∗ ( 1 ) = Q (End Cob ( 1 )) . By Prop osition 3.4 , this mapping space is giv en b y F ree E ∞ ( B S 1 ) . 18 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI The Q -homology of this space is in turn given by the free E ∞ - Q -algebra on Q [ B S 1 ] ≃ Q [ x ] with | x | = 2, so we get that End A 1 ( 1 ) ≃ Q [ Q [ x ]] , whic h is a free algebra on generators in degree 2 n for eac h n ≥ 0. □ In the next section we will describ e more generally the graded mapping spectra b et ween v arious generators, see Lemma 4.1 . Corollary 3.10. The Balmer sp e ctrum Sp c( A 1 ) surje cts onto the sp e ctrum of ho- mo gene ous prime ide als in the gr ade d p olynomial ring Q [ t 0 , t 1 , t 2 , . . . ] . Pr o of. Com bine Lemma 3.9 with the fact that the Balmer spectrum of a tt-category surjects onto the graded sp ectrum of the unit, at least when the latter is coheren t, see [ Bal10a , Theorem 7.3]. □ The ﬁb er of the comparison map o ver the generic p oint of Sp ec( π ∗ ( 1 )) app ears as the Balmer spectrum of the following 2-ring: Notation 3.11. W e let A 1 η b e the quotient of A 1 b y the thic k tensor ideal generated b y coﬁb( a : Σ i 1 → 1 ) for all nonzero homogeneous elements a ∈ π i ( 1 ). Remark 3.12. Equiv alen tly , A 1 η can b e deﬁned as as the basechange of A 1 along the map of comm utative ring sp ectra End A 1 ( 1 ) → End A 1 ( 1 )[ a − 1 , a ∈ π k ( 1 ) \ { 0 } , k ∈ N ]. Here, the subscript η on A 1 η is supp osed to make one think of a generic p oint. 3.B. The free symmetric monoidal category on a p ointed dualizable ob- ject. The second free construction we will need to analyze is the free symmetric monoidal category on a p ointed dualizable ob ject, i.e., on a dualizable ob ject X equipp ed with a map 1 → X . W e will denote this category by Cob + . This category clearly receives a morphism i : Cob → Cob + pic king out the object X , and it is not hard to prov e that this morphism is essentially surjective, so one can view the objects of Cob + as oriented 0-dimensional manifolds as well. Heuristically , we ma y thus think of Cob + as a cob ordism category , where t wo new types of cob ordisms are allow ed: the half op en interv als ( −∞ , 0] and [ 0 , ∞ ) as cobordisms from ∅ to + and − to ∅ resp ectiv ely , though this is harder to pro ve rigorously . A precise pro of will appear in forthcoming work of Barkan and Steine- brunner [ BS ] as a sp ecial case of their generalized 1D cobordism h yp othesis. W e will, how ever, not need the full strength of this precise identiﬁcation and only cer- tain prop erties of Cob + that can actually establish more easily using their previous w ork [ BS25 ]. The precise facts w e will need about this category are the following: Prop osition 3.13. The fr e e symmetric monoidal c ate gory Cob + on a p ointe d du- alizable obje ct X and the induc e d functor i : Cob → Cob + have the fol lowing pr op- erties: (a) i is essential ly surje ctive; (b) i induc es e quivalenc es of mapping sp ac es Hom Cob ( X i,j , X r,s ) → Hom Cob + ( X i,j , X r,s ) whenever r − s ≤ i − j . In p articular, they ar e b oth empty when r − s < i − j . In p articular, i is an e quivalenc e on endomorphisms of the unit. GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 19 W e dela y the pro of to the end of the section, and instead indicate how we will use Cob + . F rom Proposition 3.6 , w e also hav e the following sp ecial case: Prop osition 3.14. The fr e e rigid stably symmetric monoidal ∞ -c ate gory over D b ( Q ) on a p ointe d obje ct is given by the c ate gory F un((Cob + ) op , D ( Q )) ω , with the Day c onvolution symmetric monoidal structur e. In p articular, it is gener ate d under ﬁnite c olimits, desusp ensions and r etr acts by images of the Y one da emb e dding, whose mapping sp e ctr a ar e Q -homolo gy sp e ctr a of the mapping sp ac es in Cob + . Notation 3.15. W e let A 1 , + : = F un((Cob + ) op , D ( Q )) ω and let y : Cob + → A 1 , + denote the Q -linearized Y oneda embedding. W e also abuse notation and let i : A 1 → A 1 , + denote the Q -linearization of the functor i : Cob → Cob + . Construction 3.16. Since A 1 , + is free on a p ointed ob ject, there is a 2-ring map ι : A 1 , + → A 1 , + whic h tak es our free p ointed object f : 1 → X to the dual of its ﬁb er g ∨ : 1 → y ∨ for some choice of ﬁber g of f (the c hoice of whic h is unique up to a contractible space of choices). W e ha ve: Prop osition 3.17. The functor ι is an auto-e quivalenc e. Mor e over, ι is an invo- lution in the sense that ι ◦ ι ≃ id A 1 , + . Pr o of. Since ι is exact and symmetric monoidal, w e ha ve that ι ( g ) ≃ ﬁb( ι ( f )) = ﬁb( g ∨ ) ≃ f ∨ , so that ι ( ι ( f )) ≃ ι ( g ∨ ) ≃ ι ( g ) ∨ ≃ f . Since A 1 , + is free on the p ointed dualizable ob ject f : 1 → X , the equiv alence ι ( ι ( f )) ≃ f extends to an equiv alence ι ◦ ι ≃ id A 1 , + . □ This functor will b e used crucially in the sequel to show that a lo cal category obtained from A 1 , + (sp eciﬁcally , A 1 , + η : = A 1 , + ⊗ A 1 A 1 η ) do es not satisfy the exact- nilp otence condition from Deﬁnition 2.26 . Remark 3.18. T aking into account the forthcoming work of Barkan–Steinebrunner [ BS ], Cob + has a geometric description. It seems rather nontrivial to deﬁne the functor ι while referring only to this geometric description, while its description in terms of universal prop erties is eviden t. W e conclude this section with the proof of Prop osition 3.13 . Recall that for us, Cob + is deﬁned as the free symmetric monoidal category on a p ointed ob ject, and not as a geometric ob ject. The ﬁrst thing we will need is the description of the free symmetric monoidal category on a pointed ob ject, with no dualizabilit y condition: 20 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI Lemma 3.19 ([ Hys26 , Lemma 2.1]) . The fr e e sy mmetric monoidal c ate gory on an obje ct X is Fin ≃ , and the fr e e symmetric monoidal c ate gory on an obje ct X e quipp e d with a map 1 → X is Fin inj , and the induc e d map sending X to X is the c anonic al inclusion Fin ≃ → Fin inj . Corollary 3.20. The c anonic al maps Fin ≃ → Cob and Fin inj → Cob + c oming fr om L emma 3.19 ﬁt in a pushout squar e of symmetric monoidal c ate gories: Fin ≃ Cob Fin inj Cob + Pr o of. This is immediate by comparing univ ersal prop erties. □ The idea of the pro of of Prop osition 3.13 is then to start by computing the pushout abov e in comm utative monoids in simplicial spaces, and analyze how far the result is from b eing a Segal space. In turn, the latter analysis will use Barkan and Steinebrunner’s formula for Segaliﬁcation, cf. [ BS25 ]. T o analyze the pushout in simplicial spaces, we ﬁrst need the following elemen- tary bit of com binatorics: Lemma 3.21. F or every [ n ] ∈ ∆ , the map (Fin ≃ ) × ((Fin inj ) [1 ,n ] ) ≃ ≃ (Fin ≃ ) [ n ] × ((Fin inj ) [1 ,n ] ) ≃ → ((Fin inj ) [ n ] ) ≃ deﬁne d by ( X , X 1  → X 2  → ...  → X n ) → ( X  → X ` X 1  → X ` X 2  → ...  → X ` X n ) is an e quivalenc e of Fin ≃ ≃ (Fin ≃ ) [ n ] -mo dules. Notation 3.22. Let N denote the Rezk nerve embedding categories into simplicial spaces via N C : [ n ] 7→ Hom ([ n ] , C ), and let L Seg denote the Segaliﬁcation functor from simplicial spaces to Segal spaces 3 . W e abuse notation b y using the same sym b ol for the induced functor on com- m utative monoids. Deﬁnition 3.23. Let P + : = N (Cob) a N (Fin ≃ ) N (Fin inj ) denote the pushout in commutativ e monoids in simplicial spaces. Remark 3.24. The canonical map L Seg ( P + ) → Cob + is a completion, and in particular mapping spaces in Cob + are the same as in L Seg ( P + ). Corollary 3.25. The map N (Cob) → P + is levelwise a monomorphism of sp ac es. Pr o of. The canonical map Cob → P + is levelwise given by (Cob [ n ] ) ≃ → (Fin inj ) [ n ] ⊗ (Fin ≃ ) [ n ] (Cob [ n ] ) ≃ Hence by Lemma 3.21 , it is equiv alen t to the inclusion (Cob [ n ] ) ≃ → (Fin inj ) [1 ,n ] × (Cob [ n ] ) ≃ at the comp onen t of ∅ → ... → ∅ . 3 This is almost left adjoint to N , up to the distinction b etw een Segal spaces and complete Segal spaces. Since completion does not change the mapping spaces [ HS25 , Corollary 3.15], this distinction will be irrelevan t for us. GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 21 Since the comp onent of ∅ → ... → ∅ in (Fin inj ) [1 ,n ] is con tractible, w e ﬁnd that the canonical map N (Cob) → P + is lev elwise an inclusion of comp onen ts, as claimed. □ W e now need to discuss this monomorphism in more detail. F or this, w e need to recall the notion of necklace from [ BS25 ]: Deﬁnition 3.26 ([ BS25 , Deﬁnition 1.10]) . A ne cklac e is a bip ointed simplicial space equiv alen t to a w edge of the form ∆ n 1 ∨ ... ∨ ∆ n k , where all the wedges ∆ n ∨ ∆ m are taken at the p oin t n ∈ ∆ n , 0 ∈ ∆ m , bipointed at the image of 0 ∈ ∆ n 1 , n k ∈ ∆ n k . The relev ance of this notion is that if x, y ∈ X 0 are p oin ts in the 0 simplices of a simplicial space X , we can compute the mapping space in L Seg X from x to y as a colimit along necklaces N of mapping spaces of bip ointed simplicial spaces N → ( X, x, y ), cf. [ BS25 , Theorem A]. W e no w need to make the following observ ation: Lemma 3.27. L et i, j b e inte gers and c onsider X i,j ∈ Cob as a p oint in N (Cob) 0 , and henc e in ( P + ) 0 . L et N b e a ne cklac e in the sense of Deﬁnition 3.26 . A map f : N → P + of simplicial sp ac es factors thr ough 4 N (Cob) if and only if it sends the b asep oints of N to X i,j , X r,s for some inte gers i, j, r, s with r − s ≤ i − j . Pr o of. A necklace N is b y deﬁnition isomorphic to the iterated wedge ∆ n 1 ∨ ... ∨ ∆ n k for a sequence of natural num b ers n 1 , ..., n k , and a map N → P + is th us the data of a p oin t in ( P + ) n 1 × ( P + ) 0 ... × ( P + ) 0 ( P + ) n k . Un winding the equiv alence ( P + ) n ≃ ((Fin inj ) [1 ,n ] ) ≃ × (Cob [ n ] ) ≃ w e ﬁnd the ev aluation at 0 ∈ ∆ n amoun ts to projection on Cob [ n ] follo wed by ev aluation at 0, while ev aluation at n ∈ ∆ n amoun ts to ev aluating b oth terms at n , and using the ob vious sum operation Fin ≃ × Cob → Cob. So let ( a 1 , ..., a k ) b e a p oin t in ( P + ) n 1 × ( P + ) 0 ... × ( P + ) 0 ( P + ) n k where a q has endp oin ts ( X i q ,j q , X r q ,s q ) ∈ Cob ≃ × Cob ≃ . W e learn from the pullback condition that r q = i q +1 , s q = j q +1 and that X r q ,s q = X n,m ⊗ X t, 0 for some t ≥ 0 and some n, m so that there exists a map X i q ,j q → X n,m in Cob. In particular, m = s q , n ≤ r q . F rom the latter, w e learn that i q − j q = n − m ≤ r q − s q . F or them to be equal, t has to b e 0, which means that the last term of the sequence of injections in (Fin inj ) [1 ,n ] is empty , so all the terms need to b e empty . It follows that all the terms are in the image of Cob (see Corollary 3.25 ), as was claimed. □ W e obtain the following from [ BS25 , Theorem A], which is exactly item (b) in Prop osition 3.13 : Corollary 3.28. The map Cob = L Seg ( N (Cob)) → L Seg ( P + ) → Cob + is ful ly faithful on hom( X i,j , X r,s ) whenever r − s ≤ i − j . Pr o of. By [ BS25 , Theorem A], the mapping space in L Seg ( P + ) (and hence in Cob + ) from X i,j to X r,s can b e computed as a colimit o ver all nec klaces N of maps N → P + sending the basep oints to X i,j , X r,s resp ectiv ely . By the previous lemma, the conditions on ( i, j, r , s ) guaran tee that this is the same as the colimit of maps N → N (Cob) sending the basepoints to the ob jects with the same name, which is exactly Hom Cob ( X i,j , X r,s ), as was to b e sho wn. □ 4 Necessarily uniquely , by Corollary 3.25 . 22 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI Pr o of of Pr op osition 3.13 . The previous corollary is exactly item (b), so we are left with item (a). Consider the full sub category of Cob + spanned b y the image of Cob. This is a symmetric monoidal subcategory , and it contains the pointed ob ject X . Hence, by the universal prop erty of Cob + , the iden tity of Cob + factors through this sub category , whic h prov es that they are equal. □ 3.C. A brief digression, and an example. Using the free symmetric monoidal rigid rational additive 1-category on a p ointed ob ject, whic h we can describe from the ab ov e construction, w e can present the promised example of a tt-category with all non-zero compact objects b eing ⊗ -faithful, but which acts more akin to a nil- extension of a ﬁeld than an actual tt-ﬁeld itself. Remark 3.29. Consider the free idemp otent complete rigid symmetric monoidal Q - linear additiv e 1-category on a p oin ted ob ject, whic h we will denote b y Q [hCob + ] to connect it to Prop osition 3.13 , and lo calize this category at the generic p oin t of the unit (a p olynomial ring in a v ariable t ), to get the category we will call Q [hCob + ] η . The indecomp osable ob jects in this category are all summands of ob jects of the form X i,j (and are in bijection with the simple ob jects of Rep( GL t ; Q ( t ))). If we sa y that a summand of such an ob ject has grading i − j , this giv es a well-deﬁned grading on the indecomposable ob jects in this category , in suc h a w a y that the non-zero morphisms are grading non-decreasing. Let I 2 ⊆ Q [hCob + ] η denote the additiv e ⊗ -ideal of morphisms whic h increase grading b y at least 2, and consider the additive quotient C : = Q [hCob + ] η / I 2 . W e claim that ev ery non-zero compact ob ject in K b ( C ) is ⊗ -faithful. Since we can further quotient out the ideal I 1 of morphisms of grading at least 1 (in which case w e get back to Rep( GL t ; A ), whic h is semi-simple b y Theorem A.2 ), this category acts as a sort of “nil-extension of a ﬁeld. ” W e claim that ev ery non-zero compact object of K b ( C ) is ⊗ -faithful. Since this category is Krull-Schmidt, it suﬃces to show that the dimension of any indecom- p osable object is in vertible. Since morphisms in C of grading zero are split (which follo ws from Theorem A.2 ), and morphisms of grading ≥ 2 v anish, we can kill oﬀ con tractible chain complexes to see that every indecomp osable ob ject Z • ∈ K b ( C ) is represented by a c hain complex of the form . . . → Z i → Z i − 1 → . . . , with each Z i either zero or a sum of indecomp osable ob jects all with grading k − i for some ﬁxed k dep ending only on Z • . The form ula given at the end of [ Del96 ] (see also Corollary A.6 ) shows that the dimension of each Z i is a p olynomial, say f i ( t ), with p ositive leading co eﬃcient, such that the parity of the degree of f i ( t ) equals the parity of k − i . In particular, w e ha ve dim( Z • ) = X i ∈ Z ( − 1) i f i ( t )  = 0 , where the sum is non-zero since we can look at the f i ( t )s with highest degree, and w e see that the indices where such f i ( t )s app ear all ha ve the same parit y . Since the dimension of this ob ject is non-zero, and the endomorphism ring of the unit in this category is a ﬁeld, the dimension of every non-zero indecomp osable ob ject is in vertible, as claimed. GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 23 4. The Affine Line A 1 4.A. Semi-Simplicit y of A 1 η . The goal of this subsection is to prov e that the category A 1 η deﬁned in Section 3 is semi-simple, and is even a tt-ﬁeld in the sense of [ BKS19 , Deﬁnition 1.1]. T o wards this end, we ﬁrst hav e to study the mapping spaces of generators, beginning with the case of the aﬃne line. In the classical language of tensor categories, the graded mapping spectra b e- t ween v arious generators in A 1 corresp ond to the endomorphism rings of the gen- erators in Deligne’s category Rep( GL t ) [ Del07 ], recalled in Section A : Lemma 4.1. L et i, j b e natur al numb ers. Ther e is an isomorphism of gr ade d rings π ∗ End A 1 ( X i,j ) ∼ = π ∗ ( 1 ) ⊗ π 0 ( 1 ) End Rep( GL t ) ( X i,j ) Q . In p articular, the gr ade d endomorphism ring is c onc entr ate d in even de gr e es. Mor e gener al ly, for any family of indic es i k , j k , π ∗ End A 1 ( M k X i k ,j k ) ∼ = π ∗ ( 1 ) ⊗ π 0 ( 1 ) End Rep( GL t ) ( M k X i k ,j k ) and b oth sides ar e c onc entr ate d in even de gr e es. Pr o of. It follows from Prop osition 3.4 and Prop osition 3.5 that for any i 0 , j 0 , i 1 , j 1 , Hom A 1 ( X i 0 ,j 0 , X i 1 j 1 ) is free on a discrete set as a mo dule ov er End A 1 ( 1 ), and thus this is preserved by passing to π ∗ . Sp eciﬁcally , it is free on the set of (isomorphism classes of) simply-connected cobordisms from X i 0 ,j 0 to X i 1 ,j 1 . Thus its π ∗ is indeed basec hanged from its π 0 , and we simply hav e to identify π 0 . The result then follows from the deﬁnition of Deligne’s Rep( GL t ) as the free additiv ely symmetric monoidal 1-category on a dualizable ob ject, cf. Deﬁnition A.1 , or directly from Deligne’s original deﬁnition [ Del07 , Deﬁnition 10.2]. □ Remark 4.2. In representation theoretic language, these graded endomorphism rings are isomorphic to the “walled Brauer algebras B i,j ( t ) with parameter t on the ring π ∗ ( 1 ) ∼ = Q [ t, t 1 , t 2 , ... ], see [ Co x+08 , Section 2]. Indeed, it is clear that the set of these cob ordisms is in bijection with the generators of B i,j ( t ) from [ Co x+08 ], and each connected comp onent arising in the multiplication of tw o suc h generators corresp onds to a dimension of X , i.e., to a multiplication b y t , so the multiplication of these generators also corresp onds to the one on B i,j ( t ). Lemma 4.3. With notation as in Se ction 3 , for any i, j , the gr ade d endomorphism ring π ∗ End A 1 η ( X i,j ) is isomorphic to is isomorphic to the endomorphism ring of X i,j in Deligne’s Rep( GL t , π ∗ ( 1 A 1 η )) , and similarly for ﬁnite dir e ct sums of gener- ators. Equivalently, it is the wal le d Br auer algebr a B i,j ( t ) with p ar ameter t in the gr ade d ﬁeld π ∗ ( 1 ) ≃ Q ( t, t 1 , t 2 . . . ) . In p articular, the ring π 0 (End A 1 η ( X i,j )) is isomorphic to the wal le d Br auer algebr a B i,j ( t ) with p ar ameter t in the ﬁeld π 0 ( 1 ) . Pr o of. This follo ws from Lemma 4.1 since A 1 η is obtained from A 1 b y localizing at the prime ideal (0) in the unit of A 1 . □ Prop osition 4.4. The algebr a π 0 (End A 1 η ( X i,j )) is semi-simple for al l i, j . In fact, for any ﬁnite family of indic es ( i s , j s ) , s ∈ S , the algebr a π 0 (End A 1 η ( L s ∈ S X i s ,j s )) is semi-simple. 24 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI Pr o of. Using Lemma 4.3 , this follo ws from [ Del07 , Th´ eor` eme 10.5], a pro of of which is explained in Theorem A.2 . □ Lemma 4.5. L et C b e an additive c ate gory. • L et x ∈ C such that End C ( x ) is semi-simple. F or any summand y of x , End C ( y ) is also semi-simple. • If x, z ar e ortho gonal, and End C ( x ) , End C ( z ) ar e b oth semi-simple, then so is End C ( x ⊕ z ) . Pr o of. F or the ﬁrst item, we note that if e ∈ End C ( x ) is an idemp oten t corresp ond- ing to y , we hav e End C ( y ) ∼ = e End C ( x ) e , so it suﬃces to fo cus on the latter. No w, b y Artin–W edderburn, we may assume that End C ( x ) ∼ = M n ( D ) for some division algebra D , so that e is the projection onto some sub-v ector space V ⊂ D n . Then, eM n ( D ) e ∼ = End D ( V ) ∼ = M k ( D ) for some k is semi-simple. F or the second item, if x, z are orthogonal, then End C ( x ⊕ z ) ∼ = End C ( x ) × End C ( z ) and pro ducts of semi-simple rings are semi-simple. □ Theorem 4.6. The c ate gory A 1 η is semi-simple with simple unit. In p articular, this c ate gory is a tt-ﬁeld, and ther e is a unique p oint of Sp c( A 1 ) over the generic p oint of Sp ec( π ∗ ( 1 A 1 )) . Pr o of. By the Y oneda lemma and our explicit description of A 1 from Section 3 , A 1 , and hence A 1 η , is generated under ﬁnite colimits and retracts by the image of the Y oneda em b edding, that is, the ob jects X i,j . Consider the sub category C of ho( A 1 η ) spanned by sums of the form y ⊕ Σ z , where y , z are summands of ob jects of the form L s ∈ S X i s ,j s for some ﬁnite family ( i s , j s ) , s ∈ S of indices. Note that since all the generators ha v e even perio dic homotop y rings, y and Σ z are orthogonal in ho( A 1 η ) for any suc h y , z . By Prop osition 4.4 and Lemma 4.5 , every ob ject in C has a semi-simple endo- morphism ring. By 2-perio dicity , C is closed under shifts as well, and it is clearly closed under retracts. This implies directly that its preimage in A 1 η is closed under co/ﬁb ers, since any map b et ween objects of a semi-simple category is isomorphic to one of the form id ⊕ 0 : x ⊕ y → x ⊕ z . Thus this preimage is equal to A 1 η , which prov es the claim in ligh t of lemma 2.29 . □ 4.B. P oints of Spc( A 1 ) . W e note that all the arguments in this section directly generalize to A 1 K := A 1 ⊗ End( 1 A 1 ) K for any map of commutativ e ring sp ectra End( 1 A 1 ) → K , where K is a graded ﬁeld and on π 0 , Q [ t ] → K do es not kill ( t − n ) for any integer n ∈ Z . W e also observe the follo wing : Construction 4.7. Since End( 1 A 1 ) is free as a rational commutativ e ring sp ec- trum on classes in ev en degrees, it is also formal , i.e., equiv alen t to the commu- tativ e ring presented b y the cdga Q [ t, t i , ∈ N ≥ 1 ] , | t i | = 2 i with 0 diﬀerentials. It follo ws that for an y homogeneous prime p ⊂ π ∗ End( 1 A 1 ), one can realize the map π ∗ End( 1 A 1 ) → π ∗ End( 1 A 1 ) / p as π ∗ of a map of (formal) comm utative ring sp ectra, and a fortiori also π ∗ End( 1 A 1 ) → F rac( π ∗ End( 1 A 1 ) / p ) can b e realized as a map of formal commutativ e ring sp ectra (here, F rac is the graded fraction ﬁeld, obtained b y inv erting all nonzero homogeneous elements). GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 25 W e denote the outcome of this construction by End( 1 A 1 ) → K ( p ). Remark 4.8. In characteristic 0, all ﬁelds are formally smo oth and so by defor- mation theory any graded ﬁeld K as abov e is equiv alen t, as a commutativ e ring sp ectrum, to L or L [ u ± 1 ] for some ordinary ﬁeld L and | u | = 2 k , k ≥ 1; either wa y , it is formal. Thus, p ossibly up to reparametrization of u , an y map End( 1 A 1 ) → K to a graded ﬁeld factors weakly uniquely ov er K ( p ) for some homogeneous prime ideal p . Another construction of K ( p ) is given b y Mathew in [ Mat17 , Theorem 1.2], up to the (in this case, minor) issue that Q [ t, t i , i ∈ N ≥ 1 ] is not no etherian. He also discusses uniqueness in his situation. Remark 4.9. F or the ab ov e construction, one can also apply Theorem 7.1 , prov en later on. In terpreted for rational E ∞ -rings R , this says that the p oin ts in the homological sp ectrum of P erf ( R ) are in bijection with equiv alence classes of maps R → L , where L is an ev en 2-p erio dic rational E ∞ -ring with π 0 ( L ) an algebraically closed ﬁeld, under the equiv alence relation L 1 ∼ L 2 iﬀ L 1 ⊗ R L 2  = 0. Since we are w orking on a free algebra on even degree classes, one can chec k that these equiv alence classes of homological residue ﬁelds are determined by homogeneous prime ideals in the graded endomorphism ring of the unit, and a residue ﬁeld exists for each such prime. Corollary 4.10. L et S b e the set of primes in Q [ t ] of the form ( t − n ) . On the c omplement of the inverse image of S under the map Sp ec h ( π ∗ 1 A 1 ) → Sp ec( Q [ t ]) induc e d by Q [ t ] → Q [ t, t i : i ∈ N ≥ 1 ] , the map Sp c( A 1 ) → Sp ec h ( π ∗ 1 A 1 ) is a bije ction. Pr o of. By Prop osition 2.21 and the discussion preceding the construction, the ﬁb er o ver p ∈ Sp ec h ( π ∗ 1 1 A ) is the Balmer spectrum of a tt-ﬁeld, and hence is reduced to a p oint. □ 26 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI 5. The Comp arison Functor There is a functor A 1 , + → A 1 taking the free p ointed ob ject to the map 1 0 − → x , that is, to the zero map to the free object. This induces an equiv alence on endomorphism rings of the unit, and so w e ma y in vert all endomorphisms of the unit and get an induced functor of 2-rings p : A 1 , + η → A 1 η . Our goal in this section is to lev erage the functor p to get information about A 1 , + η from information about A 1 η . The key result is: Prop osition 5.1. The functor p : A 1 , + η → A 1 η is c onservative. As an immediate corollary , we hav e: Corollary 5.2. The 2-ring A 1 , + η is lo c al. Pr o of. It follo ws directly from Theorem 4.6 that A 1 η is lo cal, and it is clear that if a tt-category has a conserv ative symmetric monoidal functor to a lo cal tt-category , it is itself local. □ Th us, A 1 , + η is a p otential counterexample, and is in fact one of the univ ersal examples (the universal examples to test the exact nilp otence condition are exactly the A 1 , + / P , P ∈ Sp ec( A 1 , + )). Before the pro of, we need a few lemmas. Lemma 5.3. L et S b e a non-ne gatively gr ade d ring sp e ctrum, and M , N two gr ade d S -mo dules. Supp ose M is in gr adings ≥ n + 1 , and N in gr adings ≤ n . Then any map M → N of gr ade d S -mo dules is nul l. The same is true for bimo dules. Pr o of. The bimo dule statement has either the same pro of, or follows from the mo dule case b y using S ⊗ S op , so we only treat the module case. First, we note that without the S -module structure, this is obvious: an y map of graded sp ectra M → N is 0. Second, we note that since S is non-negativ ely graded, all tensors S ⊗ k ⊗ M are in gradings ≥ n + 1. Using the bar resolution for M and the case of graded sp ectra with no S -mo dule structure, w e obtain the result. □ Corollary 5.4. L et f : S → R b e a map of gr ade d ring sp e ctr a, wher e R, S ar e nonne gatively gr ade d, and supp ose furthermor e that S is in gr adings ≤ n for some n . If the ﬁb er I is in gr adings ≥ 1 , then for k ≥ n + 1 , the map I ⊗ S k → S is nul l as a map of gr ade d S -bimo dules, and henc e also when for getting the gr ading. Pr o of. Since S is nonnegatively graded, I ⊗ S k is in gradings ≥ k ≥ n + 1 and so the previous lemma applies directly . □ F or a map of ring sp ectra S → R with ﬁb er I , the map I ⊗ S k → S b eing n ull is a very strong nilp otence condition, and buys us the following: Lemma 5.5. L et f : S → R b e a map of ring sp e cta with ﬁb er I . If for n lar ge enough, the map I ⊗ S n → S is nul l as a map of S -bimo dules, then tensoring with R is c onservative on S -mo dules. Pr o of. Let M b e an S -mo dule. By assumption, for n large enough, the map I ⊗ S n ⊗ S M → M is null. Thus, it can only b e an equiv alence if M ≃ 0. But if R ⊗ S M ≃ 0, then I ⊗ S M ≃ M and so I ⊗ S n ⊗ S M ≃ M by induction, via the canonical map. □ GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 27 The ﬁnal lemma we will need concerns the sp eciﬁc structure of A 1 , + η , and suggests ho w we will use the previous three lemmas: Lemma 5.6. The sp e ctrum Hom A 1 , + η ( X i,j , X r,s ) is nonzer o only when r − s ≥ i − j . F urthermor e, if r − s = i − j , the map Hom A 1 , + η ( X i,j , X r,s ) → Hom A 1 η ( X i,j , X r,s ) induc e d by p is an e quivalenc e. Pr o of. F or the ﬁrst half, we note that this sp ectrum is a basechange along a certain ring map of the Q -homology of mapping spaces in a cobordism category . It thus suﬃces to show that if r − s < i − j , this mapping space is empty , which we ha ve in Prop osition 3.13 . F or the second half of the claim, note that p has a section deﬁned as the unique 2-ring map sending X 7→ X and so it suﬃces to pro ve the analogous claim for the section. How ever, that section exists already at the lev el of cob ordism cate- gories, and since b oth mapping sp ectra are basec hanges along the same ring map of Q -homologies of the mapping spaces in those cob ordism categories, it suﬃces to prov e the claim at the lev el of cob ordism categories, which is also cov ered in Prop osition 3.13 . □ W e now ha ve all that we need to prov e Prop osition 5.1 which, w e recall, states that p is conserv ativ e: Pr o of of Pr op osition 5.1 . The increasing union of the thic k subcategories generated b y X i,j for i + j ≤ n is equal to A 1 , + η , hence it suﬃces to prov e conserv ativity when restricted to any of these. By the Sch wede-Shipley theorem, this amoun ts to pro ving that basec hange along the E 1 -algebra map End A 1 , + η ( M i + j ≤ n X i,j ) → End A 1 η ( M i + j ≤ n X i,j ) is conserv ative (on perfect mo dules, though our pro of sho ws it in general). Observ e that we can grade naturally the objects L i + j ≤ n X i,j in A 1 , + η and A 1 η resp ectiv ely by putting X i,j in degree i − j , which canonically upgrades their en- domorphism rings to graded ring spectra, and the map betw een them to a map of graded ring sp ectra. By Lemma 5.6 , both these graded rings are concentrated in nonnegative gradings, and furthermore that map is an equiv alence on the grading 0 piece, so that its ﬁb er is concentrated in gradings ≥ 1. W e conclude using Corollary 5.4 and Lemma 5.5 . □ Theorem 5.7. The c ate gory A 1 , + η is lo c al, and the exact-nilp otenc e c ondition fails for A 1 , + η , for the (image of the) universal ﬁb er se quenc e Y → 1 → X . In p articular, the homolo gic al sp e ctrum of A 1 , + η is strictly lar ger than its Balmer sp e ctrum. Pr o of. Since p : A 1 , + η → A 1 η is conserv ative, with the target a tt-ﬁeld, w e ﬁnd that A 1 , + η is lo cal. 28 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI No w, let Y g − → 1 f − → X b e the ﬁber sequence asso ciated to the universal arro w f : 1 → X and supp ose there exists some z ∈ A 1 , + η with either z ⊗ f or z ⊗ g b eing ⊗ -nilp oten t. Recall from Construction 3.16 and Proposition 3.17 the in volution ι : A 1 , + → A 1 , + , whic h sends f to g ∨ . Since it is a 2-ring map and A 1 , + η is deﬁned by inv erting nonzero endomorphisms of the unit, it also induces an in volution on A 1 , + η , whic h w e abusively still denote ι . By replacing z with ι ( z ) if necessary , we may assume that z ⊗ g is ⊗ -nilpotent. This implies then that p ( z ) ⊗ p ( g ) ≃ p ( z ⊗ g ) is ⊗ -nilp oten t as well. Since z was assumed to b e nonzero, Proposition 5.1 implies that p ( z ) is also nonzero. How ever, by deﬁnition of p , p ( g ) is the ﬁb er of 1 0 − → X in A 1 η , and hence it is the pro jection map Ω X ⊕ 1 → 1 , which is split surjective. In particular, p ( z ) ⊗ p ( g ) is split surjectiv e as well, with target p ( z ), and the only wa y a split surjectiv e map can b e ⊗ -nilp otent is if the target is zero, a contradiction. □ Remark 5.8. As in Section 4 , and the discussion in Section 4.B , w e note that a sligh t v arian t of the argumen ts given here also sho ws that for an y map of com- m utative ring sp ectra End( 1 A 1 )[ 1 t − n , n ∈ Z ] → K , where K is a graded ﬁeld, the exact-nilp otence condition also fails for the basechange A 1 , + K := A 1 , + ⊗ End( 1 A 1 ) K . The only p oint where the argument needs to b e changed is when considering the in volution ι : instead, it induces an equiv alence A 1 , + K ≃ A 1 , + K ′ , where K ′ is the same graded ﬁeld as K , but equipp ed with a diﬀerent map End( 1 1 A ) → K GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 29 P art I I. Constructible Sp ectra in Higher Zariski Geometry 6. The E ∞ -Constructible Spectrum In [ BSY22 , Appendix A], a constructible spectrum is constructed for objects in an y suitably nice “category of algebras,” by generalizing the notion of an “alge- braically closed ﬁeld” . In the context of higher Zariski geometry , there is a map from the constructible sp ectrum to the Balmer sp ectrum whose image tracks exactly those primes which can b e detected by maps to p ointlik e rigid 2-rings. F or the con- v enience of the reader, we recall the deﬁnitions/constructions of Nullstellensatzian ob jects and the constructible sp ectrum. Deﬁnition 6.1 ([ BSY22 , Deﬁnition A.1]) . Let A b e a presentable category . Then A is said to b e we akly sp e ctr al if it is compactly generated, the terminal ob ject is compact, and any map from the terminal ob ject is an equiv alence. Deﬁnition 6.2 ([ BSY22 , Deﬁnition 1.1]) . Let A b e presentable and tak e any ob ject R ∈ A . Then w e say that R is Nul lstel lensatzian if R is nonzero and every compact ob ject in A R/ is either zero or admits a map to the initial object. These are the key ingredien ts going in to creating a constructible spectrum in some general category . This is summarized by the follo wing theorem. Theorem 6.3 ([ BSY22 , Theorem A.3/Lemma A.35]) . L et A b e a we akly sp e ctr al c ate gory. Then ther e exists a unique functor Sp ec cons A : A op → T op cpt ,T 1 , cl to c omp act T 1 -top olo gic al sp ac es with close d maps b etwe en them satisfying a numb er of pr op erties. In p articular, p oints of Spec cons A ( R ) c orr esp ond to e quivalenc e classes of maps R → S for S Nul lstel lensatzian, wher e S , S ′ ar e said to b e e quivalent if S ` R S ′  = 0 . The top olo gy is determine d by the stipulation that the image of Sp ec cons A ( S ) → Sp ec cons A ( R ) is close d for al l maps R → S in A . With this in hand, w e can now discuss the relation to the homological sp ectrum in the rational case. Prop osition 6.4. The c ate gory 2 − Ring rig of rigid 2-rings is we akly sp e ctr al, and so to o is the c ate gory 2 − Ring rig Q of r ational rigid 2-rings. Pr o of. The compactly generated assumption follo ws e.g., from the fact that Cat perf is compactly generated, and that the forgetful functor 2 − Ring rig → Cat perf is con- serv ativ e, commutes with ﬁltered colimits, and has a left adjoin t (whic h therefore sends compact ob jects to compact ob jects, whic h generate 2 − Ring rig ). The exis- tence of a map from the terminal ob ject 0 to a rigid 2-ring C implies End C ( 1 ) ≃ 0, whic h in turn implies that C ≃ 0. The ﬁnal claim follows by [ BSY22 , Lemma A.15] and the fact that 2 − Ring rig Q ≃ 2 − Ring rig D b ( Q ) / . □ Deﬁnition 6.5. F or an y rigid 2-ring C , deﬁne its c onstructible sp e ctrum to b e Sp ec cons ( C ) : = Sp ec cons 2 − Ring rig ( C ) . Before we pro ceed, w e record the follo wing help er lemma. Lemma 6.6. L et C b e a rigid 2-ring. Then ther e is an induc e d adjunction 30 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI P erf C ( − ) : CAlg (Ind ( C ))  2 − Ring rig  C / : End − ( 1 ) , with ful ly faithful left adjoint and wher e the right adjoint c ommutes with ﬁlter e d c olimits and ﬁnite c opr o ducts. Pr o of. Since C is rigid, so to o are the categories Perf C ( R ) := Mo d R (Ind( C )) ω for comm utative algebras R in Ind( C ), and the claim ab out the induced adjunction follo ws from [ Lur17 , Corollary 4.8.5.21]. Since ﬁltered colimits commute with taking endomorphisms of the unit, the righ t adjoin t comm utes with ﬁltered colimits. W e reduce to c hecking that for rigid 2-rings D and E under C , End D ⊗ C E ( 1 ) ≃ End D ( 1 ) ⊗ End E ( 1 ) , considered as algebras in Ind( C ). W riting out the pushout diagram and giving names to the relev an t functors, (6.1) Ind( C ) Ind( D ) Ind( E ) Ind( D ) ⊗ Ind( C ) Ind( E ) , f ∗ g ∗ h ∗ k ∗ w e note that w e can rewrite the algebra End D ⊗ C E ( 1 ) in Ind( C ) using the righ t adjoin ts, End D ⊗ C E ( 1 ) ≃ f ∗ h ∗ ( 1 D ⊗ C E ) ≃ g ∗ k ∗ ( 1 D ⊗ C E ) . As in Prop osition 2.21 , w e can use [ Hau21 , Theorem 4.6] to show that the square 6.1 is vertically right adjoin table, whic h tells us that there is an equiv alence f ∗ g ∗ ≃ h ∗ k ∗ . Applying this to the unit, we learn that f ∗ h ∗ ( 1 D ⊗ C E ) ≃ f ∗ h ∗ k ∗ ( 1 E ) ≃ f ∗ f ∗ g ∗ ( 1 E ) . By the pro jection formula, we learn that f ∗ f ∗ g ∗ ( 1 E ) ≃ ( f ∗ f ∗ ( 1 C )) ⊗ g ∗ ( 1 E ) ≃ f ∗ ( 1 D ) ⊗ g ∗ ( 1 E ) . Finally , using again the identiﬁcation of End D ( 1 ) with the image f ∗ ( 1 D ) of the unit under the righ t adjoin t to f ∗ , and similarly for E , this yields the claim. □ With this in hand, w e can make the follo wing observ ation, relating the con- structible sp ectrum of a category C back to the constructible sp ectrum of its unit in the category of commutativ e C -algebras. Prop osition 6.7. L et C ∈ 2 − Ring rig b e a rigid 2-ring. Then ther e is an e quivalenc e of sp ac es Sp ec cons ( C ) ≃ Sp ec cons CAlg(Ind( C )) ( 1 C ) . Pr o of. Using the induced adjunction from Lemma 6.6 , P erf C ( − ) : CAlg (Ind ( C ))  2 − Ring rig  C / : End − ( 1 ) , and the fact that the right adjoint comm utes with ﬁltered colimits, the left adjoin t preserv es compact ob jects. W e claim that the right adjoin t tak es Nullstellensatzian rigid 2- C -algebras to Nullstellensatzian commutativ e algebras in CAlg (Ind ( C )). Indeed, given any Null- stellensatzian rigid 2-ring D with a map C → D , consider a nonzero algebra R compact ov er End D ( 1 ). Using the induced pushout square, GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 31 P erf C (End D ( 1 )) D P erf C ( R ) P erf D ( R ) , w e ﬁnd that P erf D ( R ) is compact ov er D . Since the top horizon tal map is fully faithful, Lemma 2.20 implies the b ottom map is to o, and in particular Perf D ( R ) is nonzero. Since D is Nullstellensatzian, D → Perf D ( R ) must split, and applying End − ( 1 ) yields a splitting of the map End D ( 1 ) → R , as desired. Giv en an y t wo maps from C to Nullstellensatzian rigid 2-rings D and E , these represen t the same point in Sp ec cons ( C ) if and only if there’s a common reﬁnement to a map to a Nullstellensatzian rigid 2-ring F . Applying End − ( 1 ) to suc h a reﬁne- men t sho ws that the Nullstellensatzian C -algebras obtained from these categories determine the same point in Spec cons CAlg(Ind( C )) ( 1 ), giving a w ell-deﬁned map Sp ec cons ( C ) → Sp ec cons CAlg(Ind( C )) ( 1 ) , whic h we claim to b e an isomorphism. First, consider a p oint in Sp ec cons CAlg(Ind( C )) ( 1 C ) represented by a map 1 C → S for some Nullstellensatzian C -algebra S . The category Perf ( S ) is nonzero, so admits a map Perf ( S ) → S ′ for some Nullstellensatzian rigid 2-ring S ′ b y [ BSY22 , Proposition A.31]. This giv es surjectivity of the map. Next, supp ose that w e ha ve Nullstellensatzian rigid 2-rings S and S ′ under C suc h that S ⊗ C S ′ ≃ 0. By Lemma 6.6 , End − ( 1 ) comm utes with ﬁnite copro ducts, so that End S ( 1 ) ⊗ End S ′ ( 1 ) ≃ End S ⊗ C S ′ ( 1 ) ≃ 0 in CAlg (Ind ( C )), giving injectivity . □ T o relate the constructible sp ectrum back to higher Zariski geometry , we make the following simple observ ation. Lemma 6.8. If L is a Nul lstel lensatzian rigid 2-ring, then it has no non-zer o pr op er tt-ide als, and in p articular the Balmer sp e ctrum Spc( L ) ≃ ∗ is a single p oint. Pr o of. If L had a prop er non-zero tt-ideal I , we could tak e some non-zero x ∈ I , and note that L / ⟨ x ⟩ is a compact non-zero algebra under L , so m ust split. An y splitting of L → L / ⟨ x ⟩ shows that x ≃ 0 in L , con tradicting the choice of x . □ Corollary 6.9. F or any rigid 2-ring C , ther e is a c anonic al map Sp ec cons ( C ) → Sp c( C ) natur al in C . Pr o of. This follo ws from Lemma 6.8 , where the map is deﬁned explicitly by sending a p oint in the constructible sp ectrum, represen ted b y a map C → L for some Nullstellensatzian rigid 2-ring L , to the image of the map Spc( L ) → Sp c( C ). □ W e can give a precise meaning to the image of the comparison map: Prop osition 6.10. L et C b e a rigid 2-ring, and c onsider any p oint P ∈ Sp c( C ) . Then, the fol lowing ar e e quivalent. 32 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI (a) Ther e exists a rigid 2-ring D with Spc( D ) ≃ ∗ , to gether with a map C → D of rigid 2-rings, such that the image of Sp c( D ) → Sp c( C ) is exactly P . (b) The p oint P is in the image of the map Sp ec cons ( C ) → Sp c( C ) . Pr o of. (a) = ⇒ (b) Consider a rigid 2-ring D under C as in the statement. Since the Balmer sp ectrum of D is non-empty , D is in particular nonzero, and so there exists a map D → L to some Nullstellensatzian rigid 2-ring L . The point of Spec cons ( C ) represen ted by the rigid 2-ring map C → L maps to the point P . (b) = ⇒ (a) Consider any p oin t in the constructible sp ectrum mapping to P , and tak e a Nullstellensatzian represen tative C → L for said p oint. Since Sp c( L ) ≃ ∗ , and the point represented by L maps to P , w e can tak e D = L in the statement. □ The following is essen tially [ BSY22 , Example A.72]: Corollary 6.11. L et R b e a height n < ∞ E ∞ -ring sp e ctrum. F or any p oint P ∈ Sp c(P erf ( R )) such that P has height n > 0 , ther e is no map of c ommutative 2 -rings Perf ( R ) → C to a 2 -ring C with Sp c( C ) ≃ ∗ which picks out the p oint P . Pr o of. It follows b y [ Hah16 ] (see also [ BSY22 , Theorem 1.5]) that if R is an L f n -lo cal Nullstellensatzian E ∞ -ring, then R is rational, so has height n = 0. □ The height n > 0 phenomenon is nothing new, as there already are no E ∞ maps out of spectra pic king out the points of Sp c(Perf ( S )) corresp onding to the Morav a K -theories K ( n ) , n ∈ [1 , ∞ ). It turns out that the map Spec cons ( C ) → Sp c( C ) can fail to b e surjective in p ositiv e c haracteristic, to o, as the following example sho ws. Example 6.12. Consider the free E ∞ - F 2 -algebra R = F 2 { x } ∞ on a class x in degree 0. It is kno wn (cf. [ La w20 , Example 5.10]) that π ∗ ( R ) is an inﬁnite dimen- sional p olynomial ring on sequences of Dyer–Lashof op erations Q I ( x ) applied to x . Consider the localization S : = R ( x ) to the prime ideal ( x ) in π ∗ ( R ). One can c heck that P erf ( R ) has t wo p oints in its Balmer spectrum by noting that the weak rings R /x 2 and R [1 /x ] are join tly nil-conserv ativ e, so the homological spectrum consists of at least tw o p oin ts 5 , which corresp ond resp ectiv ely to the zero ideal and the ideal generated by R /x (that R/x 2 is a weak ring follo ws from [ Bur22 , Lemma 5.4, Remark 5.5]). W e claim that there is no symmetric monoidal functor Perf ( S ) → D to a rigid 2-ring D with Balmer sp ectrum a p oin t which picks out the p oint P corresp onding to R/x . By Prop osition 6.10 , it suﬃces to c heck that there is no map S → L to a Nullstellensatzian E ∞ - F 2 -algebra L representing a p oint in Sp ec cons ( S ) mapping to P . Nullstellensatzian E ∞ - F 2 -algebras hav e b een studied by Riedel [ Rie25 , Prop o- sition 4.16], where it was shown that if L is Nullstellensatzian, then it is 1-perio dic with π 0 ( L ) a ﬁeld. In particular, if we had an E ∞ -ring map S → L picking out the closed point of π ∗ ( S ), then the image of x must v anish in π ∗ ( L ). How ever, we are considering E ∞ -ring maps and they are compatible with p ow er op erations. Thus w e would hav e to ha ve Q 1 ( x ) 7→ 0 as w ell – by construction, Q 1 ( x ) is inv ertible in S , so this forces L = 0, a contradiction. 5 In fact, using Theorem 8.2 , one can show that these are the only t wo points in the homological spectrum. GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 33 7. Constructible Spectra in the Ra tional Case 7.A. Constructible sp ectra and homological spectra. One cannot exp ect in general that the comparison map Sp ec cons ( C ) → Sp c( C ) to b e surjectiv e for a given rigid 2-ring C , see Corollary 6.11 and Example 6.12 . Both of the examples there are not rational, and in fact the rational case is nice enough that the comparison map is actually alwa ys surjective. In fact we can sa y more, namely we ha ve the follo wing, whic h is the main theorem of this section: Theorem 7.1. L et C b e a r ational rigid 2-ring. Then ther e is a natur al isomor- phism of sets Sp ec cons ( C ) ≃ Spc h ( C ) b etwe en the c onstructible sp e ctrum and the homolo gic al sp e ctrum of C . Remark 7.2. W e hop e that the ab ov e result motiv ates the study of Nullstellen- satzian (rational) rigid 2-rings for tt-geometers. F orthcoming w ork of the third author partially initiates the study of their structure, and in particular of their K -theory . The following formula for free comm utative algebras on p oin ted objects in the rational case will be crucial to proving Theorem 7.1 . Lemma 7.3 ([ KY25 , Corollary B]; [ BD25 , Prop osition 4.6]) . L et T b e a r atio- nal stable pr esentably symmetric monoidal ∞ -c ate gory, and c onsider an E 0 -algebr a (that is, a p ointe d obje ct) 1 → X in T . Then the fr e e E ∞ -algebr a on this E 0 -algebr a is given by the ﬁlter e d c olimit F ree E ∞ / E 0 ( 1 → X ) = lim − → n  ( X ⊗ n ) h Σ n  . This in turn, allo ws us to mak e the following observ ation. Prop osition 7.4. L et T b e a r ational pr esentably symmetric monoidal stable ∞ - c ate gory such that the unit 1 T is c omp act (e.g., if T = Ind ( C ) is the big c ate gory attache d to a r ational rigid 2-ring C ). A map f : 1 → X in T is ⊗ -nilp otent if and only if the fr e e E ∞ -algebr a on f vanishes, which is to say, if and only if F ree E ∞ / E 0 ( 1 → X ) ≃ 0 : 1 → 0 . In p articular, any p ointe d obje ct f : 1 → X for which the p ointing map is not ⊗ - nilp otent admits a nontrivial E 0 -algebr a map to the underlying E 0 -algebr a of an E ∞ -algebr a. Pr o of. Clearly , if f is ⊗ -nilp otent, F ree E ∞ / E 0 ( 1 → X ) ≃ 0 : 1 → 0 , since the multiplication on any E ∞ -algebra R splits the map 1 → X o ver R , and if this map w ere ⊗ -nilpotent, then taking a high enough ⊗ -p o wer for it to b e zero, w e can factor 1 → R as 1 → 0 → R , so R ≃ 0. Con versely , supp ose that F ree E ∞ / E 0 ( f : 1 → X ) ≃ 0 : 1 → 0 . By Lemma 7.3 , w e can identify F ree E ∞ / E 0 ( f : 1 → X ) with the ﬁltered colimit of f ⊗ n h Σ n to rewrite this as lim − →  f ⊗ n h Σ n : 1 → X ⊗ n h Σ n  ≃ 0 : 1 → 0 . 34 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI Since the unit object w as assumed to b e compact, this implies there exists some n suc h that f ⊗ n h Σ n ≃ 0, and we claim also that f ⊗ n ≃ 0. Note that there is a natural comm utative diagram 1 h Σ n ( X ⊗ n ) h Σ n 1 X ⊗ n 1 h Σ n ( X ⊗ n ) h Σ n . ( f ⊗ n ) h Σ n f ⊗ n f ⊗ n h Σ n Since w e are rational, the rightmost vertical comp osite from the homotopy ﬁxed p oin ts to the homotopy orbits of the action of the ﬁnite group Σ n is an equiv alence, with inv erse giv en by a multiple of the norm map Nm (sp eciﬁcally 1 n ! Nm). Similarly , since the action of Σ n on 1 ⊗ n ≃ 1 is trivial, b oth leftmost vertical maps are equiv alences, where in fact the ﬁrst leftmost vertical map can be iden tiﬁed with the iden tity map, and the second one with multiplication b y n !. In particular, we hav e that f ⊗ n factors through ( f ⊗ n ) h Σ n whic h is equiv alent to ( f ⊗ n ) h Σ n , which in turn is nullhomotopic, so that f ⊗ n is nullhomotopic. □ The next step is to ﬁnally relate the homological spectrum to the constructible sp ectrum by means of computing the homological sp ectrum of Nullstellensatzian ob jects. Lemma 7.5. L et C b e a r ational rigid 2-ring, and let A ∈ CAlg (Ind ( C )) b e a Nul lstel lensatzian c ommutative algebr a obje ct. Then the homolo gic al sp e ctrum of the c ate gory of p erfe ct A -mo dules in C , Sp c h (P erf C ( A )) , c onsists of a single p oint. Pr o of. Consider tw o points m 1 and m 2 in Sp c h (P erf C ( A )). T aking the corresp onding “residue ﬁelds” from Remark 2.11 we ﬁnd corresp onding w eak rings E m i o ver A . Since the p oin ted ob ject in A - Mo d(Ind( C )) given by A → E m i is a weak ring, the p oin ting map cannot b e ⊗ -nilp otent. Applying Prop osition 7.4 , w e can ﬁnd nonzero E ∞ - A -algebras L i in Ind( C ) together with maps of p ointed ob jects A → E m i → L i , Since L i are nonzero as A -algebras, they are also nonzero in CAlg (Ind ( C )), so there exists Nullstellensatzian objects S i in CAlg (Ind ( C )) with maps L i → S i . Since A was chosen to b e Nullstellensatzian, its constructible spectrum consists of a single p oint by [ BSY22 , Proposition A.31], and so w e ﬁnd that S 1 ⊗ A S 2  = 0. This implies in turn that L 1 ⊗ A L 2  = 0, and th us that E m 1 ⊗ A E m 2  = 0. Applying Prop osition 2.12 , this in turn implies that m 1 = m 2 . Since they w ere arbitrary p oin ts, this prov es the claim. □ Corollary 7.6. L et C b e a r ational rigid 2-ring. F or any Nul lstel lensatzian algebr a A ∈ CAlg (Ind ( C )) , every non-zer o obje ct of the c ate gory Perf C ( A ) is ⊗ -faithful; e quivalently, its c o evaluation map splits. Pr o of. Note ﬁrst that given any non-zero X ∈ Perf C ( A ), the morphism co ev X : A → X ⊗ X ∨ GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 35 is split after tensoring with X , so cannot be ⊗ -nilp oten t. Using Prop osition 7.4 , the free E ∞ - A -algebra on co ev X is a nonzero compact A -algebra, so admits a section bac k to A since A is Nullstellensatzian. F or a given c hoice of section, the comp osite A → X ⊗ X ∨ → F ree E ∞ / E 0 (co ev X ) → A splits co ev aluation of X , and so X must b e ⊗ -faithful. □ Corollary 7.7. L et C b e a r ational rigid 2-ring. F or any Nul lstel lensatzian algebr a A ∈ CAlg (Ind ( C )) , every rigid 2-ring map F : Perf C ( A ) → D with D  = 0 is c onservative for maps b etwe en c omp act obje cts, me aning that any non-zer o map f : Y → X b etwe en c omp act obje cts in Perf C ( A ) has F ( f )  = 0 in D . Pr o of. Without loss of generality , A = 1 (and 1 is Nullstellensatzian). Using rigidit y , up to replacing f by its mate Y ⊗ X ∨ → 1 , we may assume without loss of generalit y that f has target 1 . If f is not zero, then the map r : = coﬁb( f ) : 1 → Z is not split injective. Since F ree E ∞ / E 0 ( r ) is a compact comm utative algebra, and 1 is Nullstellensatzian, it follows that F ree E ∞ / E 0 ( r ) ≃ 0. Applying Lemma 7.3 , this implies that r is ⊗ -nilp oten t, and hence so is F ( r ). Since D is nonzero and F ( r ) has source 1 D , it follo ws that F ( r ) is not split injective, and hence F ( f ) is non zero, as was to b e sho wn. □ Remark 7.8. Although it seems reasonable to exp ect, forthcoming w ork of the third author will show that Nullstellensatzian rigid 2-rings are not tt-ﬁelds in the sense of [ BKS19 ]! In fact, they are not ev en pure-semisimple in the sense of [ BKS19 , Theorem 5.7] and drastically fail condition (iii) in lo c. cit. Without further ado, w e can prov e the main theorem of the section. Pr o of of The or em 7.1 . By Prop osition 6.7 , instead of working with the constructible sp ectrum of C , Spec cons ( C ), we can work equally w ell with the constructible sp ec- trum of 1 C deﬁned internally to comm utative algebras in the big category Ind( C ), Sp ec cons CAlg(Ind( C )) ( 1 C ) . An y Nullstellensatzian algebra S in CAlg (Ind ( C )) determines a functor C −⊗ S − − − → Perf C ( S ) . By Lemma 7.5 , we hav e that Sp c h (P erf C ( S )) is a singleton, so we can deﬁne a map Sp ec cons CAlg(Ind( C )) ( 1 C ) → Sp c h ( C ) , b y taking a Nullstellensatzian commutativ e algebra S represen ting a p oin t in the constructible sp ectrum to the image of the induced map ∗ ≃ Sp c h (P erf C ( S )) → Sp c h ( C ) . If S 1 and S 2 are t wo Nullstellensatzian comm utative algebras representing the same p oin t in the constructible sp ectrum, then again by [ BSY22 , Lemma A.35], there exists another Nullstellensatzian comm utative algebra S 3 with maps S 1 → S 3 ← S 2 . F unctoriality of Sp c h ( − ) shows that p oint of the homological sp ectrum attached to S 1 agrees with the one attac hed to S 2 , and so our total comparison map is indeed w ell-deﬁned. 36 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI T o see that the map is surjective, pic k an y homological prime m ∈ Sp c h ( C ), and consider once again the residue ﬁeld E m at m . Applying Prop osition 7.4 , there exists a nonzero E ∞ -algebra L with a map of p ointed ob jects E m → L . Cho ose some Nullstellensatzian E ∞ -algebra S in CAlg (Ind ( C )) with a map L → S . Then, for all homological primes p  = m ∈ Spec h ( C ), since we hav e that E p ⊗ E m ≃ 0 by [ Bal20b , Prop osition 5.3], it follows that S ⊗ E p ≃ 0 for all primes p  = m . Since the kernel of S ⊗ − contains E p for all p  = m , the only p ossible prime that the map Sp c h (P erf C ( S )) → Sp c h ( C ) can possibly hit is m itself. Since m w as arbitrary , the map is surjective. Finally , we must sho w injectivity of the comparison map. Cho ose an arbitrary set-theoretic section Sp c h ( C ) → Sp ec cons CAlg(Ind( C )) ( 1 C ) , and lift this arbitrarily to a collection S m of Nullstellensatzian comm utative algebras in CAlg (Ind ( C )) with S m represen ting the p oint ab o ve m in the c hosen section. If the map is not injectiv e, then there exists some p oin t not in the image of this section, which corresp onds to a Nullstellensatzian commutativ e algebra S such that S ⊗ S m ≃ 0 for all m ∈ Sp c h ( C ). No w, [ Bar+24 , Theorem 1.9] tells us that the family { C → P erf C ( S m ) } m ∈ Spc h ( C ) is join tly nil-conserv ativ e. By Prop osition 2.21 , the basechange { P erf C ( S ) → Perf C ( S ) ⊗ C P erf C ( S m ) } m is also jointly nil-conserv ativ e. Ho wev er, we hav e that P erf C ( S ) ⊗ C P erf C ( S m ) ≃ Perf C ( S ⊗ S m ) ≃ 0 , for all m ∈ Sp c h ( C ). Since the map from Perf C ( S ) → 0 is nil-conserv ativ e, we ﬁnd that S itself must b e zero, contradicting the fact that S is Nullstellensatzian. □ 7.B. Consequences for residue ﬁelds. Using Theorem 7.1 , we can provide cri- teria for certain rational rigid 2-rings to admit enough tt-ﬁelds. Corollary 7.9. L et C b e a r ational rigid 2-ring such that for al l Nul lstel lensatzian c ommutative C -algebr as L ∈ CAlg (Ind ( C )) , Perf C ( L ) is a tt-ﬁeld in the sense of [ BKS19 ]. Then for any c ommutative algebr a R ∈ CAlg (Ind ( C )) , the c ate gory P erf C ( R ) has enough tt-ﬁelds. Pr o of. F or any p oint in the homological sp ectrum of Perf C ( R ), c ho ose a map R → L to a Nullstellensatzian commutativ e C -algebra L representing that point, whic h exists by Theorem 7.1 . The induced functor P erf C ( R ) → P erf C ( L ) acts as a tt-ﬁeld at the given p oint. □ Corollary 7.10. Supp ose that C is a r ational rigid 2-ring such that al l Nul lstel- lensatzian c ommutative algebr as in Ind( C ) have mo dule c ate gories which ar e tt- ﬁelds. Then for al l c ommutative algebr as R ∈ CAlg (Ind ( C )) , and for any obje ct X ∈ Ind((P erf C ( R )) , the genuine and naive homolo gic al supp ort of X agr e e, Supp h ( X ) = Supp n ( X ) . Pr o of. This follows by Corollary 7.9 and [ Bar+26 , Prop osition 1.16]. □ GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 37 Corollary 7.11. If R is a r ational E ∞ -ring, then Perf ( R ) has enough tt-ﬁelds, and p oints in the homolo gic al sp e ctrum of Perf ( R ) ar e al l witnesse d by maps fr om R into r ational 2-p erio dic ﬁelds. In p articular, for any mo dule over a r ational E ∞ -ring R , its naive homolo gic al supp ort agr e es with its genuine homolo gic al supp ort. Pr o of. This follows from Corollary 7.9 , using the description ([ BSY22 , Theorem A]) of Nullstellensatzian rational E ∞ -rings as even 2-p erio dic ﬁelds with π 0 algebraically closed. The ﬁnal claim follows b y Corollary 7.10 . □ 7.C. The c-top ology. In the context of the general theory of constructible spec- tra, a certain canonical top ology w as in tro duced in [ BSY22 ], whic h w e no w recall. Deﬁnition 7.12 ([ BSY22 , Deﬁnition A.44]) . A map C → D of rational rigid 2- rings will b e said to b e a c-c over if it induces a surjection on constructible sp ectra, and the top ology generated by c-co vers is called the c-top olo gy . Remark 7.13. The c-top olo gy has another more classical name, app earing in [ MM94 , pg. 115, (e)] under the name of the dense top olo gy (or later as the double ne gation top olo gy ). Remark 7.14. Combining Theorem 7.1 and Theorem 2.19 w e ﬁnd that c-cov ers of rational rigid 2-rings are exactly nil-conserv ative functors. In our sp eciﬁc case, we are able to use Theorem 7.1 to deduce some very nice prop erties ab out co vers for the c-top ology , at least lo cally . Prop osition 7.15. L et C b e a r ational rigid 2-ring, I a set, and L i an I -indexe d c ol le ction of Nul lstel lensatzian algebr as L i ∈ CAlg (Ind ( C )) . A ny c-c over Q I L i → Q J L ′ j by a pr o duct of Nul lstel lensatzians r eﬁnes to a c-c over Y I L i → Y J L ′ j → Y I L ′′ i , with the c omp osite induc e d by maps of Nul lstel lensatzian algebr as L i → L ′′ i indexe d by the original set I . Pr o of. First, note that since Q I L i → Q J L ′ j is a c-cov er, it is nil-conserv ative, so that L i ⊗ Q I L i Y J L ′ j  = 0 . Deﬁne L ′′ i as some Nullstellensatzian commutativ e algebra in C under this tensor pro duct L i ⊗ Q I L i Q J L ′ j . W e ﬁnd that these maps L i → L ′′ i assem ble to giv e a factorization Y I L i → Y J L ′ j → Y I L ′′ i , and we are left to show that the comp osite is a c-co ver. F or this, recall from Corollary 7.7 that an y functor out of a Nullstellensatzian rational rigid 2-ring is conserv ative on morphisms b etw een compacts, and in par- ticular this is true for basechange along L i → L ′′ i . As this prop erty is stable under taking pro ducts of rigid 2-rings (which the categories of perfect mo dules ov er the pro ducts of algebras embed fully faithfully into), basechange along Y I L i → Y I L ′′ i 38 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI is conserv ativ e on morphisms betw een compact ob jects. Finally , using that a 2- ring map which is conserv ative on morphisms b etw een compacts is in particular nil-conserv ativ e (e.g. b y writing an y w eak ring as a ﬁltered colimit of compact E 0 -algebras and using that 1 → 0 is compact), we ﬁnd that Q I L i → Q I L ′′ i is a c-co ver, as desired. □ Before deducing more consequences of Theorem 7.1 , we make a small digression to compute the Balmer sp ectrum of a product of Nullstellensatzian ob jects. Lemma 7.16. L et C b e a r ational rigid 2-ring, I a set, and L i an I -indexe d c ol le c- tion of Nul lstel lensatzian algebr as L i ∈ CAlg (Ind ( C )) . Then the Balmer sp e ctrum Sp c(P erf C ( Q I L i )) is isomorphic to the Stone– ˇ Ce ch c omp actiﬁc ation β I of the set I . Pr o of. Note that there is a fully faithful inclusion P erf C ( Y I L i )  → Y I P erf C ( L i ) , whic h is therefore nil-conserv ativ e and hence induces a surjectiv e map on Balmer sp ectra b y [ Bar+24 , Theorem 1.4]. By naturality of the comparison map from Prop osition 2.2 , and the fact that the endomorphism ring of the unit in the tw o categories agree, it suﬃces to sho w that β I ∼ = Sp c( Y I P erf C ( L i )) ∼ = Sp ec h ( π ∗ (End Q I Perf C ( L i ) ( 1 ))) . Since each L i is a Nullstellensatzian commutativ e algebra in C , its endomorphism ring is also Nullstellensatzian, and in particular by [ BSY22 , Theorem 6.3] is an even 2-p eriodic algebraically closed ﬁeld. Since taking endomorphism rings of the unit comm utes with products, w e ha ve that End Q I Perf C ( L i ) ( 1 ) ≃ Y I End Perf C ( L i ) ( 1 ) is an I -indexed pro duct of even 2-p erio dic algebraically closed ﬁelds, and in partic- ular has graded homogeneous spectrum exactly the Stone– ˇ Cec h compactiﬁcation β I of the set I . The Balmer sp ectrum of Q I P erf C ( L i ) clearly surjects on to β I , since for any ultraﬁlter U on I , the ultrapro duct Q U P erf C ( L i ) is nonzero, living ov er the point U ∈ β I . Note that this lo calization also has Balmer sp ectrum consisting of a single p oin t b y Lemma 2.7 and Corollary 7.6 since every nonzero ob ject is ⊗ -faithful, and the lo calization corresp onds to the prime P U = { ( x i ) i ∈ I ∈ Y I P erf C ( L i ) : { i : x i ≃ 0 } ∈ U } . W e claim that these are the only prime ideals. Indeed, note ﬁrst that for any ( x i ) i ∈ I ∈ Q I P erf C ( L i ), the ideal generated by x i agrees with the ideal generated by the ob ject ( δ x i  =0 ) i ∈ I with a 1 in the position of every nonzero x i and zero elsewhere. T o see this, note that this second ob ject (which is in fact idemp otent) tensors with ( x i ) i ∈ I to itself, and also splits oﬀ of ( x i ) i ∈ I ⊗ ( x ∨ i ) i ∈ I , again b y Corollary 7.6 . Therefore any ideal I ⊆ Q I P erf C ( L i ) is determined b y the set F ( I ) : = { S ⊆ I : ( δ i ∈ S ) i ∈ I / ∈ I } . GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 39 In general, one can chec k that for an y proper ideal I , the set F ( I ) is a ﬁlter on the set I , and this giv es a bijective corresp ondence b etw een ﬁlters on I and prop er ideals in Q I P erf C ( L i ) . Under this corresp ondence, an ideal I is prime if and only if, giv en any tw o sets S 1 , S 2 ⊆ I , S 1 ∪ S 2 ∈ F ( I ) implies that S 1 ∈ F ( I ) or S 2 ∈ F ( I ), which is exactly the condition that F ( I ) is an ultraﬁlter, in which case the prime ideal I is nothing but P F ( I ) , which must therefore be the only primes, as claimed. □ Corollary 7.17. L et C b e a r ational rigid 2-ring. When r estricte d to the ful l sub-c ate gory NS Π (CAlg (Ind ( C ))) of pr o ducts of Nul lstel lensatzian c ommutative algebr as in C , the Balmer sp e ctrum functor R 7→ Sp c(P erf C ( R )) , c onsider e d as a pr eshe af value d in Set op , is a she af for the c-top olo gy. Pr o of. T o b egin, note that if we hav e a co ver of the form Q I L i → Q I L ′ i for L i , L ′ i Nullstellensatzian comm utative algebras and maps L i → L ′ i , then given distinct ultraﬁlters U 1 and U 2 on I , we hav e Y U 1 L ′ i ⊗ Q I L i Y U 2 L ′ i ≃ 0 . This implies that the tw o induced maps on Balmer spectra under Q I L ′ i Q I L ′ i ⊗ Q I L i Q I L ′ i actually agree, whic h forces Sp c(Perf C ( − )) to satisfy the sheaf condition on these co vers. W e m ust still chec k, giv en a cov er of the form Y I L i → Y J L ′ j , that Spc(Perf C ( Q I L i )) is the quotient of Sp c(Perf C ( Q J L ′ j )) under the tw o maps from Sp c(Perf C ( Q J L ′ j ⊗ Q I L i Q J L ′ j )). Using Prop osition 7.15 , ﬁx some reﬁnement of this cov er of the form Q I L i → Q J L ′ j → Q I L ′′ i . Fix some ultraﬁlter U on I , and denote b y U ′ the ultraﬁlter on J such that Q J L ′ j → Q U L ′′ i factors o ver Q U ′ L ′ j , that is, the image of U in Sp c(Perf C ( Q J L ′ j )). T ake any other ultraﬁlter V on J which maps to U in Sp c(P erf C ( Q I L i )), w e claim that there is a p oin t mapping to b oth V and U ′ under the t wo maps w e wish to co-equalize. Using once again (the pro of of ) Prop osition 7.15 , we ﬁnd that the map Y U L i → Y U L ′′ i is nil-conserv ative, and b y construction, this factors ov er the map (7.1) Y U L i → Y U ′ L ′ j , whic h must therefore be nil-conserv ative as well. Finally , w e note that to say V maps to U , is to say that the map Y I L i → Y V L ′ j 40 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI factors ov er Q U L i , and by nil-conserv ativit y of 7.1 , w e hav e that Y V L ′ j ⊗ Q U L i Y U ′ L ′ j  = 0 , and this giv es a p oint of Sp c(P erf C ( Q J L ′ j ⊗ Q I L i Q J L ′ j )) which co-equalizes U ′ and V , as desired. □ This allows us to construct a sheaﬁﬁcation for the Balmer spectrum functor in the c-top ology , whic h tak es v alues in a category which is not presentable, but that w e can nevertheless work with. F or this, we ﬁrst recall some basic facts about shea ves in not necessarily presen table 1-categories. Prop osition 7.18. L et C b e a Gr othendie ck site, and i : B ⊂ C a subb asis for the top olo gy: that is, a ful l sub c ate gory such that e ach c ∈ C admits a c over by an obje ct in B . In this c ase, r estriction along the inclusion i induc es an e quivalenc e on c ate gories of she aves with values in any c omplete 1 -c ate gory, with inverse given by right K an extension. Pr o of. F or the target category b eing Set, this is [ AGV63 , Exp os´ e II I, Th´ eor` eme 4.1]. F or a general complete target category , use the Y oneda lemma to reduce to Set. The claim ab out the inv erse follo ws [ AGV63 , Exp os´ e I I I, Proposition 2.3, 3)], with the same reduction to the case of sets. □ Remark 7.19. F or ∞ -categories of co eﬃcients, one would ha ve to restrict to hy- p ershea ves. Corollary 7.20. L et C b e a Gr othendie ck site and B a subb asis on C . L et D b e a c omplete 1 -c ate gory and F : C op → D b e a pr eshe af. If the r estriction of F to B is a she af, then the she af ˜ F on C c orr esp onding to F | B under the e quivalenc e fr om the pr evious Pr op osition is the she aﬁﬁc ation of F ; in p articular F admits a she aﬁﬁc ation. Pr o of. Let G b e a sheaf. By the righ t Kan extension prop ert y of Prop osition 7.18 applied to G , restriction to B induces an isomorphism hom( F , G ) ∼ = hom( F | B , G | B ) No w since b oth are sheav es, Proposition 7.18 implies that this is isomorphic (again via restriction to B ) to hom( ˜ F , G ), as was to b e sho wn. □ In particular, concretely , for any ob ject x ∈ C , the v alue of the sheaﬁﬁcation of F at x is giv en by ﬁnding a h yp ercov er y 1 ⇒ y 0 → x , with y 0 , y 1 ∈ B , and taking eq ( F ( y 0 ) ⇒ F ( y 1 )). This implies in particular that these results remain v alid for large sites. Corollary 7.21. L et C b e a Gr othendie ck site and B a subb asis on C . L et D b e a c omplete 1 -c ate gory and F : C op → D b e a pr eshe af. If the r estriction of F to B is a she af, and F → G is a map to a she af which is an isomorphism lo c al ly on B , then it is a she aﬁﬁc ation. One can lo ok at the comparison map from the Balmer sp ectrum to the sp ectrum of graded homogeneous prime ideals in the graded endomorphism ring of the unit, GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 41 rationally . Although these functors will often diﬀer, as a corollary of what w e hav e seen thus far, we ﬁnd that when we sheaﬁfy them with respect to the c-top ology , the sheaﬁﬁcations will in particular exist, and they will b ecome equal. Corollary 7.22. L et C b e a r ational rigid 2-ring. The she aﬁﬁc ation of the Balmer sp e ctrum functor Sp c(P erf C ( − )) : CAlg (Ind ( C )) → Set op with r esp e ct to the c-top olo gy exists, and agr e es with its value on pr o ducts of Nul l- stel lensatzian obje cts. F urthermor e, this she aﬁﬁc ation is e qual to the c-she aﬁﬁc ation of the functor sending R to the sp e ctrum of gr ade d homo gene ous prime ide als in the underlying E ∞ -ring of R : R 7→ Sp ec hom ( π ∗ (Hom C ( 1 , R ))) . Pr o of. When restricted to the sub-category of pro ducts of Nullstellensatzian ob- jects, these t wo functors agree b y Lemma 7.16 . No w, since this restriction is a c-sheaf b y Corollary 7.17 , and this full sub-category forms a sub-basis for the c- top ology , the sheaﬁﬁcations of b oth functors exist and agree by Corollary 7.20 . □ In nice cases, w e can say more about this constructible top ology . Theorem 7.23. L et C b e a r ational rigid 2-ring, and c onsider the fol lowing state- ments. (a) Ther e exists some n such that for al l Nul lstel lensatzian c ommutative algebr as L ∈ CAlg (Ind ( C )) , the c ate gory Perf C ( L ) satisﬁes the exact-nilp otenc e c ondition to or der n . (b) Ultr apr o ducts of Nul lstel lensatzian obje cts in CAlg (Ind ( C )) have c onstructible sp e ctrum c onsisting of a single p oint, which is to say that ultr apr o ducts ar e p ointlike. In p articular, in this c ase the c ate gory CAlg (Ind ( C )) is sp e ctr al in the sense of [ BSY22 , Deﬁnition A.63]. (c) F or al l R ∈ CAlg (Ind ( C )) , the she aﬁﬁc ation of the Balmer sp e ctrum functor R 7→ Spc(Perf C ( R )) landing in (Set) op with r esp e ct to the c-top olo gy agr e es with Sp ec cons (P erf C ( R )) . (d) F or al l R ∈ CAlg (Ind ( C )) , the c onstructible top olo gy on Sp ec cons (P erf C ( R )) is Hausdorﬀ. Then, (a) = ⇒ (b) ⇐ ⇒ (c) = ⇒ (d). Pr o of. (a) = ⇒ (b) Consider an ultraﬁlter U on a set I , and an I -indexed family L i of Nullstellensatzian algebras in C . W e wish to sho w that the constructible sp ectrum of the ultrapro duct Q U L i is a single p oin t. By Theorem 7.1 , it suﬃces to show that the homological sp ectrum of P erf C ( Q U L i ) is a single point. Using full faithfulness (and hence nil-conserv ativit y) of the inclusion P erf C ( Y U L i ) → Y U P erf C ( L i ) , it suﬃces to show that the homological sp ectrum of Q U P erf C ( L i ) is a single p oin t. No w, by assumption, the categories P erf C ( L i ) satisfy ENC( n ), and so to o does their ultrapro duct, whose homological sp ectrum therefore agrees with its Balmer sp ectrum. Lemma 7.16 further shows that the Balmer spectrum of the ultraproduct is a single point. 42 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI (b) = ⇒ (c) By [ BSY22 , Corollary A.46], the target of the natural comparison map Sp c(P erf C ( R )) → Sp ec cons (P erf C ( R )) of presheav es v alued in (Set) op is such that the target is actually a sheaf. By Corollary 7.21 , in order to chec k that the target is a sheaﬁﬁcation of the source, it suﬃces to chec k that this map is an isomorphism locally for the c-top ology . T aking a c-cov er R → Y q ∈ Spec cons ( R ) ( L q ) giv en by a pro duct of Nullstellensatzian represen tatives L q for eac h p oint in the constructible sp ectrum of R , w e ma y assume that R = Q I L i is a pro duct of Null- stellensatzian C -algebras. Using that ultrapro ducts are p oin tlike and CAlg (Ind ( C )) is op-disjunctive by [ BSY22 , Lemma A.58], [ BSY22 , Lemma A.61] implies that the constructible spectrum of a pro duct Sp ec cons ( Q I L q ) is given by the Stone– ˇ Cec h compactiﬁcation β I of the set I . Similarly , Lemma 7.16 sho ws that the Balmer sp ectrum of P erf C ( Q I L i ) is also the Stone– ˇ Cec h compactiﬁcation of I , with the lo cal category at the prime U ∈ β I given by the ultrapro duct P erf C ( Q U L i ), whic h tells us that the comparison map Sp ec cons ( Y I L i ) → Sp c(Perf C ( Y I L i )) is an equiv alence, and the claim is sho wn. (c) = ⇒ (b) Using Corollary 7.22 , the sheaﬁﬁcation of the Balmer sp ectrum functor with resp ect to the c-top ology agrees with the Balmer sp ectrum functor on pro ducts of Nullstellensatzian comm utative algebras. T o say that this sheaﬁﬁcation is the constructible sp ectrum implies that, for such a product Q I L i , the comparison map induces an equiv alence Sp ec cons (P erf C ( Y I L i )) ≃ Sp c(Perf C ( Y I L i )) . That is to sa y , the nerves of steel condition holds for the category P erf C ( Q I L i ). T ranslating to the lo cal rings ov er a p oint in the Balmer sp ectrum, we ﬁnd that, for any ultraﬁlter U on a set I , and an I -indexed set of Nullstellensatzians, we ha ve Sp ec cons (P erf C ( Y U L i )) ≃ Sp c(Perf C ( Y U L i )) ≃ ∗ , whic h is to say , ultrapro ducts in CAlg (Ind ( C )) are p ointlik e. (b) = ⇒ (d) This follo ws from [ BSY22 , Proposition A.62]. □ Corollary 7.24. L et C b e a r ational rigid 2-ring. Then the fol lowing ar e e quivalent: (a) The we akly sp e ctr al c ate gory CAlg (Ind ( C )) is sp e ctr al. (b) Given an ultr aﬁlter U on a set I , and an I -indexe d c ol le ction L i ∈ CAlg (Ind ( C )) of Nul lstel lensatzian obje cts, the nerves of ste el c ondition holds for the c ate gory P erf C ( Q U L i ) . (c) Given a set I , and an I -indexe d c ol le ction L i ∈ CAlg (Ind ( C )) of Nul lstel len- satzian obje cts, the nerves of ste el c ondition holds for the c ate gory Perf C ( Q I L i ) . Pr o of. (a) ⇐ ⇒ (b) Note that CAlg (Ind ( C )) is sp ectral if and only if ultrapro ducts are p oin tlike, which is to sa y , given an ultraﬁlter U on a set I , and an I -indexed GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 43 collection L i of Nullstellensatzian ob jects, the sp ectrum Spec cons C ( Q U L i ) is a sin- gle p oint. Once again, b y Corollary 7.6 , ev ery non-zero ob ject in Perf C ( Q U L i ) is ⊗ -faithful, and in particular the Balmer sp ectrum has only a single p oint. In particular, the nerves of steel condition holds for this category if and only if Sp ec cons C ( Q U L i ) is a single p oint, if and only if CAlg (Ind ( C )) is spectral. (b) ⇐ ⇒ (c) Again, it follows from Corollary 7.6 that the Balmer sp ectrum of P erf C ( Q I L i ) is given by the Stone– ˇ Cec h compactiﬁcation β I of I . In particular, b y the ﬁb er-wise criterion for the nerves of steel condition, this category satisﬁes NoS if and only if all of its localizations to closed p oints do. Since the p oin ts of β I are given by ultraﬁlters U on I , with corresp onding lo cal rigid 2-ring Perf C ( Q U L i ) at the prime represen ted b y U , the claim follows. □ If we are allow ed to work with all rational rigid 2-rings, the conditions of Theo- rem 7.23 strengthen to all be equiv alen t, and in fact one can add a few more: Prop osition 7.25. The fol lowing ar e e quivalent (a) F or al l r ational rigid 2-rings C , the sp ac e Sp ec cons ( C ) is Hausdorﬀ. (b) The sp ac e Sp ec cons ( A 1 , + η ) is Hausdorﬀ. (c) Ther e exists some n ≫ 0 such that the algebr as F ree E ∞ / E 0 (coﬁb( f ∨ , ⊗ n )) and F ree E ∞ / E 0 (coﬁb( g ⊗ n )) in A 1 , + η ar e jointly nil-c onservative, with f the fr e e map fr om the unit to the fr e e p ointe d obje ct, and g its ﬁb er. (d) Ther e exists n ≫ 0 such that for al l Nul lstel lensatzian r ational rigid 2-rings L which c ontain an obje ct Z such that A 1 Z − → L factors over A 1 η , L satisﬁes ENC(n). (e) Ther e exists some n ≫ 0 such that al l Nul lstel lensatzian r ational rigid 2-rings satisfy ENC(n). (f ) The c ate gory 2 − Ring rig Q is sp e ctr al. Pr o of. (a) = ⇒ (b) is clear. (b) = ⇒ (c) The p oints in A 1 , + η determined b y p : A 1 , + η → A 1 η and p ◦ ι hav e residue ﬁelds pick ed out by the E ∞ -algebras F ree E ∞ / E 0 ( f ) and F ree E ∞ / E 0 ( g ∨ ), resp ectively . If Sp ec cons ( A 1 , + η ) is Hausdorﬀ, then there exists disjoint op en neighborho o ds of these t wo p oints. These open subsets are represen ted by functors A 1 , + η → C and A 1 , + η → D , such that F ree E ∞ / E 0 ( f ) = 0 in C , F ree E ∞ / E 0 ( g ∨ ) = 0 in D , and A 1 , + η → C × D is nil-conserv ativ e. Since F ree E ∞ / E 0 ( f ) ≃ 0 in C , Proposition 7.4 implies that f ⊗ m ≃ 0 in C for some m ≫ 0. This means in particular that F ree E ∞ / E 0 (coﬁb( f ∨ , ⊗ m )) admits a splitting back to the unit after base-c hanging it in to C , which in turn implies that A 1 , + η → C factors o ver A 1 , + η → Perf A 1 , + η (F ree E ∞ / E 0 (coﬁb( f ∨ , ⊗ m ))) → C . Similarly , there exists some k such that A 1 , + η → D factors as A 1 , + η → Perf A 1 , + η (F ree E ∞ / E 0 (coﬁb( g ⊗ k ))) → D . Up to increasing k or m as needed, we may assume without loss of generality that k = m = n , for some n ≫ 0. No w, since A 1 , + η → C and A 1 , + η → D are jointly nil-conserv ativ e, so to o are the functors A 1 , + η → Perf A 1 , + η (F ree E ∞ / E 0 (coﬁb( g ⊗ n ))) and A 1 , + η → Perf A 1 , + η (F ree E ∞ / E 0 (coﬁb( f ∨ , ⊗ n ))) , 44 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI as claimed. (c) = ⇒ (d) Let L b e a rational Nullstellensatzian rigid 2-ring containing such an ob ject Z , and consider any ﬁb er sequence Y g − → 1 f − → X in L . Up to replacing f : 1 → X by f ⊕ 0 ⊕ 0 : 1 → X ⊕ Σ X ⊕ Z if needed, w e may assume without loss of generality that the functor A 1 , + f − → L factors o ver A 1 , + η . By (c), there exists some n such that the algebras F ree E ∞ / E 0 (coﬁb( f ∨ , ⊗ n )) and F ree E ∞ / E 0 (coﬁb( g ⊗ n )) are jointly nil-conserv ativ e on A 1 , + η , and so too then will b e their basec hange to L . Since these are free E ∞ -algebras in L on c omp act E 0 - algebras, and L is Nullstellensatzian, the map from the unit splits as so on as one of them is nonzero- and one must b e nonzero b ecause they are jointly nil-conserv ative. By symmetry of cases, assume that, in L , F ree E ∞ / E 0 (coﬁb( g ⊗ n ))  = 0, suc h that this algebra splits back to 1 L . Then the map coﬁb( g ⊗ n ) : 1 → W splits, whic h implies that g ⊗ n ≃ 0, and L satisﬁes ENC( n ). (d) = ⇒ (e) Using Corollary 7.7 (applied e.g. to C = L and A = 1 ), w e ﬁnd that an y functor out of a Nullstellensatzian rational rigid 2-ring L is conserv ative for morphisms b et ween compact ob jects. T aken together with Corollary 7.6 , whic h tells us that we are free to alwa ys take Z = 1 when chec king the exact-nilp otence condi- tion in Nullstellensatzian rational rigid 2-ring, w e ﬁnd that any failure of ENC( n ) in suc h a category L will provide a similar failure for every Nullstellensatzian rational rigid 2-ring L ′ admitting a map from L . It remains to sho w that any Nullstellensatzian L admits a map to another Null- stellensatzian rigid 2-ring L ′ satisfying the h ypothesis in (d). F or this, use that D b ( Q ) → A 1 η is nil-conserv ativ e, and so to o then is L → L ⊗ A 1 η , and in particular the target is nonzero. F or any c hoice of map L ⊗ A 1 η → L ′ to a Nullstellensatzian rigid 2-ring L ′ yields a map L → L ′ with L ′ satisfying the hypothesis in (d). (e) = ⇒ (f ) Since 2 − Ring rig Q can b e sho wn to be op-disjunctiv e, it remains to show that ultraproducts are pointlik e. Using Theorem 7.1 , this is equiv alen t to showing that for an y ultraﬁlter U on a set I , and I -indexed collection L i of Nullstellensatzian rigid 2-rings, the ultraproduct Q U L i has homological spectrum consisting of a single p oint. Since its Balmer spectrum is a single point (as once again every ob ject is ⊗ -faithful), this is equiv alen t to the Nerv es of steel condition holding for this pro duct, which is itself, b y locality , equiv alent to the exact-nilp otence condition in this category . Since the ultraproduct of lo cal categories satisfying ENC( n ) is still lo cal, and still satisﬁes ENC( n ), the claim follo ws. (f ) = ⇒ (a) This follows by [ BSY22 , Theorem A.65]. □ Remark 7.26. Since it seems lik ely that Nullstellensatzian rational rigid 2-rings should b e called tt-ﬁelds, in the case that ultrapro ducts in rational rigid 2-rings are not p ointlik e, it would mean that one do es not exp ect ultrapro ducts of a go o d notion of “tt-ﬁelds” to remain tt-ﬁelds. This would be a p oint of divergence from classical algebraic geometry , where of course, ultraproducts of ﬁelds remain ﬁelds. GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 45 8. The E n -Constructible Spectrum 8.A. E n -constructible sp ectra. W e ha ve seen that the E ∞ -constructible spec- trum of the unit in a rigid 2-ring may not necessarily ev en surject onto the Balmer sp ectrum. In contrast, it turns out that ﬁxing any ﬁnite n ≥ 1 and working with the E n -constructible sp ectrum, this problem can essentially be ﬁxed b ecause of Burklund’s w ork [ Bur22 ]. The natural generalit y to w ork in for the purp oses of this section is with rigid E n -2-rings, n ≥ 3. W e make the follo wing deﬁnition. Deﬁnition 8.1. Let Ind( C ) be the Ind category of a rigid E m -2-ring. F or an y 1 ≤ n ≤ m , and any A ∈ Alg E n (Ind( C )), let Sp ec cons E n ( A ) : = Sp ec cons Alg E n (Ind( C )) ( A ) b e the constructible spectrum of A in the weakly spectral 6 category of E n -algebra ob jects in Ind( C ). The main goal of this subsection is to prov e the following theorem. Theorem 8.2. L et Ind( C ) b e the Ind c ate gory of a rigid E m -2-ring. Then for al l 1 ≤ n < m , ther e is a natur al e quivalenc e of sets Sp c h ( C ) ≃ Sp ec cons E n ( 1 C ) . As a consequence of this theorem, we can give a new top ology to the homolog- ical sp ectrum: the constructible top ology . In its usual top ology , the homological sp ectrum is not necessarily ev en T 0 , but in the constructible top ology it b ecomes T 1 ! Remark 8.3. The topology deﬁned via the constructible top ology lik ely agrees with the one deﬁnable from [ BW25 ] through the iden tiﬁcation of the homological sp ectrum with their “closed Zeigler sp ectrum” KZg ⊗ Cl ( − ), which naturally carries the top ology of a T 1 -space. Before giving the proof, w e require a few generalities on homological sp ectra. Prop osition 8.4. L et Ind( C ) b e the Ind c ate gory of a rigid E m -2-ring, and take any 1 ≤ n ≤ m . L et m ∈ Spc h ( C ) b e a homolo gic al prime, and let E m b e the c orr esp onding homolo gic al r esidue ﬁeld. W rite E m = coﬁb( v : I m → 1 ) . Then the obje ct E n m : = coﬁb( v n +1 : I ⊗ n +1 m → 1 ) is an E n -algebr a with homolo gic al supp ort Supp h ( E n m ) = { m } . Pr o of. Since E m is a w eak ring, it admits a righ t unital mul tiplication. Fixing a c hoice of right unital multiplication, w e ﬁnd that E n m inherits an E n -algebra structure b y [ Bur22 , Theorem 1.5]. Since E m ⊗ v is nullhomotopic, w e hav e that E m ⊗ E n m  = 0, so m ∈ Supp h ( E n m ). On the other hand, if w e had any m ′ ∈ Sp c h ( C ) diﬀerent from m , E m ′ ⊗ E m ≃ 0 implies that E m ′ ⊗ v is an equiv alence, and then so to o is E m ′ ⊗ v n +1 , and hence E m ′ ⊗ E n m ≃ 0 as w ell. □ W e mak e use of the following simple observ ation. 6 See [ BSY22 , Remark A.10]. 46 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI Lemma 8.5. If A, B ar e we ak rings in Ind( C ) , and ther e is a map A → B of p ointe d obje cts, then Supp h ( B ) ⊆ Supp h ( A ) . In p articular, if B  = 0 and Supp h ( A ) is a single p oint, then Supp h ( B ) = Supp h ( A ) . Pr o of. Suppose that we are given m / ∈ Supp h ( A ), so that E m ⊗ A ≃ 0. Note that E m ⊗ B is a tensor pro duct of w eak rings, so is itself a w eak ring. Using that the map 1 → E m ⊗ B factors ov er 1 → E m ⊗ A ≃ 0 → E m ⊗ B the zero map, w e see that w e m ust hav e E m ⊗ B ≃ 0 as well. □ Prop osition 8.6. L et A, B b e E n -algebr as in the Ind c ate gory of a rigid E m -2-ring Ind( C ) , with 1 ≤ n < m . Then, we have that Supp h ( A a B ) = Supp h ( A ) ∩ Supp h ( B ) = Supp h ( A ⊗ B ) , and in p articular, A ` B  = 0 if and only if A ⊗ B  = 0 . Pr o of. Since n < m , A ⊗ B can b e given a canonical E n -algebra structure, and the maps A → A ⊗ B , B → A ⊗ B can b e made canonically E n . Th us, there exists an E n -algebra map A a B → A ⊗ B b y the univ ersal property , so Lemma 8.5 ensures that Supp h ( A ⊗ B ) ⊆ Supp h ( A a B ) . Similarly , the E n -algebra maps A → A a B and B → A a B sho w that Supp h ( A a B ) ⊆ Supp h ( A ) ∩ Supp h ( B ) . Since Supp h ( A ⊗ B ) = Supp h ( A ) ∩ Supp h ( B ) b y Proposition 2.14 , w e get the desired equalit y . The ﬁnal claim follo ws from Prop osition 2.16 . □ No w w e can ﬁnally b egin to introduce the constructible sp ectrum to the picture, b y wa y of describing homological supp ort of Nullstellensatzian E n -algebras: Prop osition 8.7. L et L b e a Nul lstel lensatzian E n -algebr a in Ind( C ) , with n < m . Then the homolo gic al supp ort Supp h ( L ) of L c onsists of a single p oint in the homolo gic al sp e ctrum. Pr o of. Indeed, suppose that m and m ′ are tw o distinct p oints in Supp h ( L ). By Prop osition 8.6 , the following tw o E n -algebras under L E n m a L, E n m ′ a L are nonzero, but ( E n m a L ) a ( E n m ′ a L ) ≃ 0 . GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 47 This in turn implies that ( E n m a L ) a L ( E n m ′ a L ) ≃ 0 as well. This tells us that the images of the t wo maps on constructible spectra of induced b y these algebras are disjoin t. Ho wev er, we assumed that L was Nullstellensatzian, so that Sp ec cons E n ( L ) is a single p oin t by [ BSY22 , Prop osition A.31(2)]. Therefore, the constructible sp ectrum of one of these algebras must actually b e empty , and that algebra must then b e zero, a contradiction. □ Finally , w e can prov e the theorem. Pr o of of The or em 8.2 . W e claim that the map Sp ec cons E n ( 1 C ) → Sp c h ( C ) taking any p oin t of the source, represented b y a Nullstellensatzian E n -algebra L , to the p oin t Supp h ( L ), is a bijection. F or injectivit y , note that if L 1 and L 2 are t w o Nullstellensatzian E n -algebras with Supp h ( L 1 ) = Supp h ( L 2 ), then Prop osition 8.6 implies Supp h ( L 1 a L 2 ) = Supp h ( L 1 ) ∩ Supp h ( L 2 )  = ∅ , and in particular L 1 ` L 2  = 0, so that L 1 , L 2 represen t the same p oin t in the constructible sp ectrum. F or surjectivity , ﬁx an arbitrary m ∈ Sp c h ( C ), take the E n -ring E n m from Prop o- sition 8.4 , and c hoose some Nullstellensatzian E n -algebra L under it. Since L is non-zero, and Supp h ( E n m ) = { m } is a singleton, Lemma 8.5 sho ws that Supp h ( L ) = { m } as well. Since the prime m w as arbitrary , surjectivit y follo ws. □ Remark 8.8. Giv en an E n -algebra R , one can also consider R as an E k -algebra for an y 1 ≤ k ≤ n , and deﬁne a constructible sp ectrum that wa y . This will in general b e diﬀerent from the homological sp ectrum as sets. F or example, consider a ﬁeld K , and take the polynomial ring on K in one v ariable, K [ t ]. The map K [ t ] → End K M N K ! sending t to the endomorphism e i 7→ e 2 i induces a map the other w ay on E 1 - constructible sp ectra. W e claim that the image of this map misses the usual (com- m utative) sp ectrum of K [ t ]. First, note that since t maps to an element whic h has a left inv erse, w e cannot ﬁnd a common factorization of the map to this algebra with the residue ﬁeld where t = 0. In the same vein, we can c ho ose tw o left inv erses, e.g. f : e 2 i 7→ e i , e 2 i +1 7→ 0, and g : e 2 i 7→ e i , e 2 i +1 7→ e i , suc h that g − f has a right inv erse, and so there is no algebra map out of the target where f = g . This tells us that there is no map to a non-zero algebra where t obtains a righ t in verse. In particular, w e cannot 48 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI ha ve a common reﬁnement of this asso ciativ e ring map together with a residue ﬁeld of K [ t ] where t  = 0, so the constructible sp ectrum of K [ t ] considered as an asso ciativ e algebra is strictly larger than its constructible sp ectrum in comm utative rings (which agrees with its homological spectrum). 8.B. Theorem 8.2 is sharp. W e will no w include a sp eciﬁc example to show that the requirement of n < m in Prop osition 8.6 is sharp, and w e will use this to show the same is true in Theorem 8.2 . F or the purp ose of this section, ﬁx some n ≥ 4, and consider the free E n - F 2 -algebra R : = F 2 { x, y } n on t wo generators x, y in degree 0. The category Perf ( R ) is naturally a rigid E n − 1 -2-ring, which we now pro ceed to analyze. Notation 8.9. Fix some basis E of the free F 2 -Lie algebra on t wo generators x, y , suc h that (a) The generators x, y are in E . (b) The brack et b et ween the generators, [ x, y ], is in E . (c) Ev ery basis vector in E can b e written as an expression inv olving only brac kets, x s and y s. The exact choice of basis itself isn’t so imp ortant, as long as it satisﬁes the prop erties ab ov e. Prop osition 8.10. The homotopy ring of R is isomorphic to a gr ade d p olynomial ring: π ∗ ( R ) ≃ F 2 [ Q I ( e ) : e ∈ E ] for admissible se quenc es I = ( a 1 , . . . , a n − 1 ) of non-ne gative inte gers, and b asis ve ctors e ∈ E . The gr ading on a b asis ve ctor e is determine d by the rules | x | = | y | = 0 , and | [ − , − ] | = n − 1 . Pr o of. This follows from [ La w20 , Theorem 5.5]. □ Construction 8.11. Consider the E n − 1 -coﬁb er of x : F 2 → R , denoted by R  n − 1 x . By [ Rie25 , Lemma 2.2], we hav e that R  n − 1 x ≃ R ⊗ F 2 { x } n F 2 , and in particular, this algebra has homotopy groups given b y the quotient of R by the regular sequence { Q I ( x ) } ov er all admissible sequences I . Prop osition 8.12. Ther e exists E n − 1 - R -algebr as A, B such that A ⊗ R B  = 0 , but A ` R B ≃ 0 in the c ate gory of E n − 1 - R -algebr as. In p articular, the r e quir ement that n < m in the statement of Pr op osition 8.6 is ne c essary. Pr o of. T ake A : = R  n − 1 x [([ x, y ] − 1 )] as a localization of the algebra from Construc- tion 8.11 , and let B : = R  n − 1 y . Then, by the explicit description from Prop osi- tion 8.10 , we learn that A ⊗ R B ≃ A/ ( Q I ( y )) I admissible  = 0 . On the other hand, taking the pushout in E n − 1 - R -algebras, and once again using [ Rie25 , Lemma 2.2], w e learn that A a R B ≃ R  n − 1 ( x, y )[([ x, y ]) − 1 ] ≃ R [([ x, y ]) − 1 ] ⊗ F 2 { x,y } n F 2 ≃ F 2 [0 − 1 ] ≃ 0 , GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 49 as desired. □ Our goal no w is to extend this in to showing that Theorem 8.2 fails for P erf ( R ) if we were to look at the E n − 1 constructible sp ectrum of this rigid E n − 1 -2-ring. T o do this, we will hav e to analyze a little bit about certain residue ﬁelds for R . W e will assume for now that n ≥ 6 to make the details of the example easier, though with more care one should b e able to mak e it work for n = 4 , 5 as w ell. Prop osition 8.13. The E n − 2 - R -algebr a k : = F rac( R  n − 1 x ⊗ R R  n − 1 y ) deﬁne d by inverting al l non-zer o homo gene ous p olynomials in the E n − 2 - R -algebr a given by the tensor pr o duct of the E n − 1 -quotients by x and y , r esp e ctively, is a r esidue ﬁeld for some p oint in the homolo gic al sp e ctrum of P erf ( R ) . Pr o of. Since n ≥ 6, this algebra k is at least E 4 , so that Perf ( k ) is a rigid E n − 3 -2- ring (with n − 3 ≥ 3), generated by the unit, which has π ∗ ( k ) is a graded ﬁeld, is in fact a tt-ﬁeld. Since π ∗ ( k ) is indecomp osable as a π ∗ ( R )-mo dule, the result follows b y [ BC21 , Theorem 3.1]. □ Our goal now is to construct an E n − 1 -algebra under R  n − 1 x [([ x, y ]) − 1 ] whic h has homological supp ort precisely the p oint pic ked out b y this residue ﬁeld k . By symmetry , this will also construct such an E n − 1 -algebra under R  n − 1 y [([ x, y ]) − 1 ], whic h will necessarily hav e copro duct with the former algebra b eing zero. This will sho w that there are at least tw o distinct p oin ts in the E n − 1 constructible sp ectrum of R whose homological supp ort is the point k , whic h will provide the desired failure of Theorem 8.2 . T ow ards this end, we need to ﬁnd a wa y to tell when an algebra has homological supp ort exactly k , whic h we shall do b y ﬁnding explicit jointly nil-conserv ative co vers of R . T o b egin, w e note the following. Prop osition 8.14. L et S b e any E n - F 2 -algebr a, and c onsider a class x ∈ π 0 ( S ) . Then, the family of maps { S → S [ Q I ( x ) − 1 ] : I admissible } ∪ { S → S  n − 1 x } is jointly nil-c onservative. Pr o of. Using Theorem 8.2 , it suﬃces to test nil-conserv ativity with resp ect to E n − 2 - S -algebras A , whic h helps us to work in a case where w e can actually talk ab out categories of A -mo dules as rigid E n − 2 -2-rings (recall we are assuming n ≥ 6 for the momen t). Supp ose that A [ Q I ( x ) − 1 ] ≃ 0 for all admissible sequences I . Then we m ust hav e that Q I ( x ) acts nilp oten tly on A for all such I . In particular, the ﬁber of A → A/Q I ( x ) is ⊗ -nilp otent for all I . Using once again that S  n − 1 x ≃ S ⊗ F 2 { x } n F 2 ≃ O I S/Q I ( x ) , w e ﬁnd that we can write the E n − 2 -algebra map A → A ⊗ S S  n − 1 x ≃ O I A/Q I ( x ) 50 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI as a ﬁltered colimit of E 0 - A -algebra maps with ⊗ -nilp oten t ﬁber. In particular, A → A ⊗ S S  n − 1 x is nil-conserv ativ e, and so if A was non-zero, the target is non-zero as well. □ Applying this to our case with S = R twice, taking b oth x and y , and ﬁnding the common reﬁnement of the resulting nil-conserv ativ e cov ers, w e ﬁnd, Corollary 8.15. The family of maps { R → R [ Q I ( x ) − 1 ] } ∪ { R → R [ Q I ( y ) − 1 ] } ∪ { R → R  n − 1 x ⊗ R R  n − 1 y } is jointly nil-c onservative. Finally , w e will require one last little claim. Prop osition 8.16. The family of we ak rings under R given by { R → R [ Q I ( x ) − 1 ] }∪{ R → R [ Q I ( y ) − 1 ] }∪{ R → ( R  n − 1 x ⊗ R R  n − 1 y ) /p ( e ) 4 }∪{ R → k } , wher e p ( e ) r anges over homo gene ous p olynomials in the variables Q I ( e ) with e ∈ E \{ x, y } , is jointly nil-c onservative. Pr o of. T ake a non-zero E n − 2 -algebra A under R . Using Corollary 8.15 , w e ma y assume without loss of generality that A ⊗ R ( R  n − 1 x ⊗ R R  n − 1 y )  = 0 . Since the residue ﬁeld k is a lo calization of the algebra we are tensoring with at its non-zero homogeneous elements, w e ﬁnd that A ⊗ R k ≃ 0 if and only if there exists some homogeneous p olynomial p ( e ) ∈ π ∗ ( R  n − 1 x ⊗ R R  n − 1 y ) whic h acts nilpotently on A ⊗ R ( R  n − 1 x ⊗ R R  n − 1 y ). In this case, we must then ha ve that the ﬁb er of A ⊗ R ( R  n − 1 x ⊗ R R  n − 1 y ) → A ⊗ R ( R  n − 1 x ⊗ R R  n − 1 y ) /p ( e ) 4 is ⊗ -nilp otent. In particular, since A ⊗ R ( R  n − 1 x ⊗ R R  n − 1 y ) is a nonzero E n − 2 -algebra, and the target is a weak ring b y [ Bur22 , Lemma 5.4], the target m ust b e non-zero as w ell, and the claim is shown. □ W e are now in the p osition to provide our coun terexample. Theorem 8.17. L et R : = F 2 { x, y } n b e the fr e e E n - F 2 -algebr a on two de gr e e zer o gener ators x and y , for any n ≥ 6 . Consider the E n − 1 - R -algebr as deﬁne d as C : = R  n − 1 ( x, Q I ( y ) 2 )[ p ( e ) − 1 ] and D : = R  n − 1 ( y , Q I ( x ) 2 )[ p ( e ) − 1 ] as I r anges acr oss admissible se quenc es and p ( e ) r anges acr oss homo gene ous p oly- nomials in the variables Q I ( e ) ∈ E \{ x, y } . These algebr as ar e b oth non-zer o with homolo gic al supp ort exactly { k } , yet C ` R D ≃ 0 . In p articular, for any choic e of Nul lstel lensatzian E n − 1 - R -algebr as L 1 under C and L 2 under D , L i has homolo gic al supp ort { k } , but L 1 and L 2 determine diﬀer ent p oints in Sp ec cons E n − 1 ( R ) . This is al l to say that the r e quir ement n < m in The or em 8.2 is ne c essary. GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 51 Pr o of. Since the brac ket [ x, z ] = 0 v anishes when z = w 2 is a square of a class, z = Q i ( w ) is a Dyer–Lashof operation applied to a class with i < n − 1, and satisﬁes [ x, Q n − 1 w ] = [ w , [ w, x ]] (see [ La w20 , Theorem 5.2]), it follows that the image of F 2 { x, Q I ( y ) 2 } n → R misses the m ultiplicative set generated by homogeneous p olynomials p ( e ) in Q I ( e ) for e ∈ E \{ x, y } , and in particular C  = 0. Since C admits an algebra map from R  n − 1 x [([ x, y ]) − 1 ], whic h has pushout with R  n − 1 y (which admits an algebra map to D ) equal to zero, it suﬃces to sho w that Supp h ( C ) = { k } . Note that, b y construction, C [ Q I ( x ) − 1 ] = 0 , C [ Q I ( y ) − 1 ] = 0 , and C /p ( e ) ≃ 0 . Using Proposition 8.16 , we necessarily m ust ha ve that C ⊗ k  = 0, and since C ⊗ L = 0 for all other weak rings L in the nil-conserv ative cov er from Proposition 8.16 , Supp h ( C ) ⊆ Supp h ( k ) = { k } , as desired. □ 52 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI 9. Geometric Points f or Homological Spectra In general, w e ha ve seen one cannot exp ect there to b e a rigid 2-ring map from a rigid 2-ring C to a rigid 2-ring D whose Balmer sp ectrum is a p oint, which pic ks out any given p oint in Sp c( C ). Theorem 8.2 almost provides this at the level of constructible spectra, at least if one only asks for rigid E n -2-ring maps, but suﬀers from the follo wing defect: if C is a rigid E n -2-ring, it makes p oin ts in its homological sp ectrum corresp ond to Nullstellensatzian E n -algebras in Ind( C ). Ho wev er, if L is such a Nullstel- lensatzian E n -algebra, it do es not hav e an y reason to be Nullstellensatzian in Alg E n − 1 (Ind(P erf C ( L ))), and so we cannot guarantee via Theorem 8.2 that its ho- mological sp ectrum is a p oin t. As it turns out, using the metho ds that w ent into the pro of of Theorem 8.2 , we can ﬁx this “defect” by proving that, at least for k large enough, the constructible sp ectra of E k m are p oin ts, thus obtaining a suﬃcien t supply of “highly structured geometric p oints” . 9.A. Geometric p oints of the homological sp ectrum for rigid E n -2-rings. The main theorem of this section is: Theorem 9.1. L et C b e a rigid E m -2-ring for some m ≥ 4 (p ossibly m = ∞ ), and let m ∈ Sp c h ( C ) b e a homolo gic al prime. Then, for any 3 ≤ n < m , ther e exists a rigid E n -2-ring K , e quipp e d with an E n -2-ring map C → K , such that Sp c h ( K ) = ∗ is a single p oint, and such that the map Sp c h ( K ) → Sp c h ( C ) has image exactly { m } . T o pro ve this theorem, we will need the follo wing, which relies on inputs that w e will delay un til after proving the main theorem of this section. The term “ v - compatible” comes from [ Bur22 ] and will b e recalled later. Theorem 9.2. L et C b e an E m -monoidal stable ∞ -c ate gory, for some m ≥ 3 , and supp ose we ar e given a map v : I → 1 in C such that the c oﬁb er 1 /v admits a right unital multiplic ation. Then, for any, 3 ≤ n ≤ m , 1 ≤ k ≤ n − 1 , q ≥ n + 1 , and any w ≥ q + k the unique v -c omp atible E n -algebr a structur es on 1 /v q and 1 /v w ar e such that the E k -algebr a map given by inclusion on the left factor 1 /v q → 1 /v q ⊗ 1 /v w is a splitting of a split squar e-zer o E k -algebr a extension. W e will p ostp one the pro of of the ab o ve, opting to ﬁrst pro ve the main theorem of this section assuming the result. Before this though, we note the following quick lemma. Lemma 9.3. L et C b e a rigid E m -ring, and let f : R → S b e a map of E n -algebr as in Ind( C ) , 3 ≤ n ≤ m . If f is a squar e zer o extension, Sp ec h ( f ) : Sp ec h (P erf C ( S )) → Sp ec h (P erf C ( R )) is surje ctive. If f is a split squar e zer o extension, it is a bije ction. GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 53 Pr o of. In the case of a split square zero extension, Sp ec h ( f ) admits a retraction and hence is injectiv e. Th us, the second claim follows from the ﬁrst. F or the ﬁrst claim, we apply Theorem 2.19 : it suﬃces to prov e that S ⊗ R − is nil-conserv ativ e. F or this w e note that it is conserv ative: if S ⊗ R M = 0, then since ﬁb( f ) is an S -mo dule in R -mo dules, ﬁb( f ) ⊗ R M = 0 and so M = R ⊗ R M = 0 as an extension of t wo 0 mo dules. □ Remark 9.4. The keen-ey ed reader will note that in the n = 3 case of Lemma 9.3 , w e actually only had rigid E 2 -rings which we applied Theorem 2.19 to, meaning our homotopy category was merely a “braided tt-category ,” as opp osed to a sym- metric monoidal one. Nevertheless, one can prov e the analogue of Theorem 2.19 with essen tially the same argumen ts as the symmetric monoidal case. One can al- ternativ ely proceed without ever leaving the symmetric monoidal w orld, w ere it so desired, but at the cost of requiring m ≥ 5 in the follo wing theorem, with minimal mo diﬁcations to the pro of. With these preliminaries, w e can prov e the main theorem of this section: Pr o of of The or em 9.1 . Fix a residue ﬁeld E m at the p oint m . A natural candidate for the category K we w ant to consider will b e the category of mo dules o ver E q m for some q ≥ n + 1, which, as a category of mo dules o ver an E n +1 -algebra ob ject, inherits an E n -monoidal structure. T ow ards this end, let K q : = P erf C ( E q m ) for q ≥ n + 1. It is clear that the image of Sp c h ( K q ) → Sp c h ( C ) is exactly m , by construction. Our goal no w is to show that for some q ≫ 0, Sp c h ( K q ) is a single p oin t. In fact, w e claim that taking q ≥ max { n + 1 , 7 } is enough. First, we note that using Theorem 9.2 , the E 3 -algebra map E q − 3 m → E q − 3 m ⊗ E q m is a split square-zero extension of the source, and hence induces an equiv alence on homological sp ectra by Lemma 9.3 . In particular, the multiplication map (which serv es as a retraction for this map), given by the composite E q − 3 m ⊗ E q m → E q − 3 m ⊗ E q − 3 m → E q − 3 m also induces an equiv alence on homological sp ectra, and is th us nil-conserv ativ e. Using this, w e note also that E q m → E q − 3 m is nil-conserv ative- since an y non- zero w eak ring A under E q m has homological supp ort { m } in C , which implies that A ⊗ E q − 3 m a non-zero weak ring in E q m ⊗ E q − 3 m -mo dules, and A ⊗ E q m E q − 3 m ≃ ( A ⊗ E q − 3 m ) ⊗ E q m ⊗ E q − 3 m E q − 3 m  = 0 . No w, ﬁx some given q ≥ max { n + 1 , 7 } , and supp ose that w e had tw o non-zero w eak rings A, B in K q suc h that A ⊗ E q m B ≃ 0. Without loss of generalit y , we may replace B by B ⊗ E q m E q − 3 m to assume that the weak ring structure on B is induced from a w eak E q − 3 m -ring structure on an ob ject which we abusively also denote by B . No w, again since A and B , considered as ob jects of C , hav e homological supp ort 54 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI { m } , A ⊗ B  = 0 in C . In particular, A ⊗ B is a non-zero weak ring in the category of E q m ⊗ E q − 3 m -mo dules. W e hav e ( A ⊗ B ) ⊗ E q m ⊗ E q − 3 m E q − 3 m ≃ (( A ⊗ E q m E q − 3 m ) ⊗ B ) ⊗ E q − 3 m ⊗ E q − 3 m E q − 3 m ≃ ( A ⊗ E q m E q − 3 m ) ⊗ E q − 3 m B ≃ A ⊗ E q m B ≃ 0 , whic h, as A ⊗ B was non-zero, contradicts nil-conserv ativit y of the m ultiplication map. □ 9.B. Recollections on deformations. Before proving the tec hnical lemmas used in the previous subsection, we recall a bit of background from [ PP25 ], [ BP25 ] and [ Bur22 ]. Prop osition 9.5 ([ PP25 , pp. 5.34, 5.37, 5.47, 5.60]) . L et E b e an epimorphism class in an essential ly smal l stable ∞ -c ate gory C . Then ther e exists a stable ∞ -c ate gory D ( C , E ) , a (non-exact) functor ν : C → D ( C , E ) and a natur al tr ansformation τ : Σ ν (Ω − ) → ν ( − ) such that (a) The functor ν is ful ly faithful. (b) The image of ν gener ates D ( C , E ) . (c) The functor ν pr eserves ﬁb er se quenc es x → y → z such that the map y → z is in the epimorphism class E . (d) Ther e is an exact automorphism ( − )[1] on D ( C , E ) determine d by the pr op erty that ν ( X )[1] ≃ ν (Σ X ) . (e) Inverting the natur al tr ansformation τ yields an (exact) functor ( − )[ τ − 1 ] : D ( C , E ) → C such that ( ν ( − ))[ τ − 1 ] ≃ id C . (f ) If I is an E -inje ctive obje ct of C , then for al l obje cts X ∈ C , [Σ − s ν ( X ) , ν ( I )] ≃ 0 for al l s > 0 . Remark 9.6. In contrast to [ PP25 ], where they deal primarily with prestable deformations which they call D ( C , E ), we will nev er ha ve reason to leav e the sta- ble w orld. In particular, w e implicitly work with the stable en velope (that is, the Spanier-Whitehead ∞ -category , see [ Lur18 , Construction C.1.1.1, Prop osi- tion C.1.1.7, Proposition C.1.2.2]) of the prestable categories constructed by Patc hkoria- Pstr ¸ ago wski. Prop osition 9.7 ([ BP25 , Prop osition A.11]) . Given two stable ∞ -c ate gories C , D , with epimorphism classes E , F , in C and D r esp e ctively, as wel l as an exact functor C → D which sends E to F , we obtain an exact functor D ( C , E ) → D ( D , F ) c omp atible with ν and ( − )[ τ − 1 ] . GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 55 Deﬁnition 9.8 ([ Bur22 , Deﬁnition 4.2]) . An epimorphism class E on a stably E n - monoidal ∞ -category C is said to b e ⊗ -c omp atible if E is preserved under tensoring with arbitrary ob jects of C . Prop osition 9.9 ([ Bur22 , Prop osition 4.3]) . If C is a stably E n -monoidal ∞ - c ate gory with ⊗ -c omp atible epimorphism class E , then the deformation D ( C , E ) in- herits the structur e of a stably E n -monoidal c ate gory in such a way that ν and ( − )[ τ − 1 ] ar e E n -monoidal functors. Prop osition 9.10. If C and D ar e two stably E n -monoidal ∞ -c ate gories e quipp e d with ⊗ -c omp atible epimorphism classes E and F , to gether with an E n -monoidal functor C → D mapping E to F , then the induc e d functor D ( C , E ) → D ( D , F ) c an natur al ly b e pr omote d to an E n -monoidal functor. Pr o of. The E n -monoidal functor C → D yields an E n -monoidal functor of prestable ∞ -categories PSh Σ ( C , S ) → PSh Σ ( D , S ) b et ween categories of spherical preshea ves via left Kan extension. The prestable deformation category D ≥ 0 ( C , E ) of C with resp ect to E is given by the category of p erfect sheav es on C with resp ect to the E -epimorphism top ology (see [ PP25 , Deﬁnition 5.32]). In particular, this arises as a lo calization of PSh Σ ( C , S ), where the ⊗ -compatibility of E is used to giv e the lo calization the structure of an E n - lo calization. The diagram PSh Σ ( C , S ) PSh Σ ( D , S ) D ≥ 0 ( C , E ) D ≥ 0 ( D , F ) is induced from a monoidal lo calization of an E n -monoidal map of E n -monoidal categories, and so the b ottom map is itself E n -monoidal. The result now follows b y passing to stabilizations. □ W e no w sp ecialize to the main case of interest. Let C b e a stably E n -monoidal ∞ -category , let v : I → 1 b e a map from some ob ject to the unit such that the coﬁb er 1 /v admits a w eak ring structure. In this case, there is a ⊗ -compatible epimorphism class on C , denoted E ( v ), consisting of those morphisms whic h are split surjective after tensoring with 1 /v . In this case, the ﬁber sequence 1 → 1 /v → Σ I has the second map in the epimorphism class. In particular, this gives rise to a ﬁb er sequence ν ( 1 ) → ν ( 1 /v ) → ν ( I )[ 1] . F ollowing the con ven tions set in [ Bur22 ], we name the ﬁb er of the ﬁrst map here ˜ v : Σ − 1 ν ( I )[1] → ν ( 1 ) . Before restating the main result of [ Bur22 ], w e recall one ﬁnal deﬁnition. 56 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI Deﬁnition 9.11 ([ Bur22 , Deﬁnition 5.1]) . If w e are given a weak ring 1 /v in a stably E n -monoidal ∞ -category C , then for 1 ≤ k ≤ n , an E k -algebra structure on 1 /v q is said to be v -c omp atible if it arises as the τ -inv ersion of an E k -algebra structure on ν ( 1 ) / ˜ v q in D ( C , E ( v )). Theorem 9.12 ([ Bur22 , Theorem 5.2]) . Supp ose we ar e given a we ak ring 1 /v in a stably E n -monoidal ∞ -c ate gory C . Then for any 1 ≤ k ≤ n , and any q ≥ k + 1 , ther e is a unique v -c omp atible E k -algebr a structur e on 1 /v q . This theorem is pro ved through deﬁning an obstruction theory for E n -algebra structures on coﬁb ers. W e will need to make use of the explicit obstruction theory , summarized by the follo wing statemen t. Theorem 9.13 ([ Bur22 , Prop osition 2.4, Remark 2.5, Corollary 2.7]) . L et C b e a stably E m -monoidal ∞ -c ate gory, and c onsider a c oﬁb er se quenc e I v − → 1 → 1 /v involving the unit. F or any ﬁxe d 1 ≤ k ≤ m , ther e is a se quenc e of inductively deﬁne d classes θ r,α ∈ π 0 Hom C (Σ − 2 − k − c α (Σ k +1 I ) ⊗ r , 1 /v ) for r ≥ 2 and 0 ≤ c α ≤ ( r − 1)( k − 1) which pr ovide obstructions to the existenc e of an E k -algebr a structur e on 1 /v . Mor e over, given that an E k -algebr a structur e c an b e deﬁne d on 1 /v , ther e exists a se quenc e of inductively deﬁne d classes γ r,α ∈ π 0 Hom C (Σ − 1 − k − c α (Σ k +1 I ) ⊗ r , 1 /v ) for r ≥ 1 and 0 ≤ c α ≤ ( r − 1)( k − 1) which pr ovide obstructions to the uniqueness of the E k -algebr a structur e on 1 /v . 9.C. T ec hnical lemmas. W e no w pro ceed with the ingredients that w ent in to the pro of of Theorem 9.2 . F or the remainder of this section, ﬁx the follo wing data (a) A small, idemp otent complete E m -monoidal stable ∞ -category C for some m ≥ 3. (b) A map v : I → 1 in C such that 1 /v admits a right unital m ultiplication. (c) In tegers n, k , q , w with 3 ≤ n ≤ m , 1 ≤ k ≤ n − 1, q ≥ n + 1, and w ≥ q + k . Using the epimorphism class E of 1 /v -split epimorphisms in C , we will work in the deformation category D ( C , E ) with respect to this epimorphism class. Notation 9.14. F or the remainder of this section, ﬁx the follo wing notation. (a) W e will write e I : = Σ − 1 ν (Σ I ) for the ob ject written on the right in D ( C , E ). (b) Similarly , w e denote by ˜ v : e I → 1 the ﬁb er of the map 1 ≃ ν ( 1 ) → ν ( 1 /v ), (which has the stated source since the map it is a ﬁb er of is a 1 /v -split monic). T o begin, let’s pro ve the following easy help er lemma. GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 57 Lemma 9.15. The obje ct 1 / ˜ v q ⊗ 1 / ˜ v w has π 0 Hom(Σ − s ν ( X ) , 1 / ˜ v q ⊗ 1 / ˜ v w ) ≃ 0 for al l s ≥ q + w − 1 and X ∈ C . Pr o of. Since 1 / ˜ v q admits a weak ring structure and w ≥ q , w e ha ve that 1 / ˜ v q ⊗ 1 / ˜ v w ≃ 1 / ˜ v q ⊕ Σ e I ⊗ w ⊗ 1 / ˜ v q , and we reduce the claim to proving it for the individual summands. F or the ﬁrst summand, the claim follo ws in fact for s ≥ q b y [ Bur22 , Lemma 4.8]. Mo ving on to the second summand, we pro ceed exactly as in the pro of of [ Bur22 , Lemma 4.8], working b y induction on q . When q = 1, Σ e I ⊗ w ⊗ ν ( 1 /v ) ≃ Σ 1 − w ν (Σ w I ⊗ w ⊗ 1 /v ) is a 1 − w -shift of an E -injectiv e ob ject, and the claim follows b y Prop osition 9.5 (f ). In general, using the ﬁb er sequence Σ e I ⊗ w ⊗ e I ⊗ q − 1 ⊗ 1 / ˜ v → Σ e I ⊗ w ⊗ 1 / ˜ v q → Σ e I ⊗ w ⊗ 1 / ˜ v q − 1 , together with induction and the fact that the leftmost ob ject is a 2 − w − q -shift of an E -injectiv e ob ject, the claim follows. □ No w, we can prov e. Prop osition 9.16. Ther e is a unique E k - 1 / ˜ v q -algebr a structur e on the obje ct 1 / ˜ v q ⊗ 1 / ˜ v w in D ( C , E ) . In p articular, the E k -algebr a map given by inclusion on the left factor 1 / ˜ v q → 1 / ˜ v q ⊗ 1 / ˜ v w admits the structur e of a splitting for a choic e of split squar e-zer o E k - 1 / ˜ v q -algebr a structur e on the tar get. Pr o of. W e note that the ﬁb er sequence 1 / ˜ v q ⊗ e I ⊗ w → 1 / ˜ v q → 1 / ˜ v q ⊗ 1 / ˜ v w , allo ws us to present 1 / ˜ v q ⊗ 1 / ˜ v w as a quotient of the unit (in 1 / ˜ v q -mo dules) by the ideal “ 1 / ˜ v q ⊗ e I ⊗ w ”). Since there is an E k - 1 / ˜ v q -algebra structure on 1 / ˜ v q ⊗ 1 / ˜ v w , Theorem 9.13 applies to give a sequence of inductively deﬁned obstructions to this E k -algebra structure to b e unique. These obstructions take v alues in (9.1) π 0 Hom 1 / ˜ v q − Mod( D ( C , E )) (Σ − 1 − k − c α (Σ k +1 1 / ˜ v q ⊗ e I ⊗ w ) ⊗ 1 / ˜ v q r , 1 / ˜ v q ⊗ 1 / ˜ v w ) , for r ≥ 1 and 0 ≤ c α ≤ ( k − 1)( r − 1). When r = 1, just as in the proof of [ Bur22 , Theorem 5.2], the obstruction is the comp osite 1 / ˜ v q ⊗ e I ⊗ w → 1 / ˜ v q → 1 / ˜ v q ⊗ 1 / ˜ v w , whic h we c ho ose a null-homotop y of in order to make this into a ﬁber sequence. Using tensor-hom, we ma y rewrite the group in eq. (9.1) as π 0 Hom D ( C , E ) (Σ − 1 − k − c α (Σ k +1 e I ⊗ w ) ⊗ r , 1 / ˜ v q ⊗ 1 / ˜ v w ) . Expanding out the deﬁnition of the source, w e ﬁnd that we can write this as Σ − 1 − k − c α (Σ k +1 e I ⊗ w ) ⊗ r ≃ Σ − 1 − k − c α + r ( k +1) − r w ν (Σ rw I ⊗ rw ) , 58 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI where, as c α ≥ 0, and k + 1 ≤ q + k ≤ w , allows us to write − 1 − k + r ( k + 1) − c α − r w ≤ ( r − 1)( k + 1) − r w ≤ k + 1 − 2 w ≤ 1 − w − q . In particular, w e see that the group eq. (9.1) v anishes b y Lemma 9.15 . Since all of the obstructions v anish, this implies the E k - 1 / ˜ v q -algebra structure on 1 / ˜ v q ⊗ 1 / ˜ v w is unique, as desired. The ﬁnal claim follo ws since we can write 1 / ˜ v q ⊗ 1 / ˜ v w ≃ 1 / ˜ v q ⊕ Σ 1 / ˜ v q ⊗ e I ⊗ w , and giv e this the structure of a split E k - 1 / ˜ v q -algebra, whic h must then agree with an y other c hoice of E k - 1 / ˜ v q -algebra structure by uniqueness. □ Pr o of of The or em 9.2 . Using Prop osition 9.16 , w e ﬁnd that 1 / ˜ v q ⊗ 1 / ˜ v w is a split square-zero extension of 1 / ˜ v q in D ( C , E ), and the result follows from inv erting τ to pass back to C . □ GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 59 Appendix A. The Semi-Simplicity of Rep( GL t ) The goal of this appendix is to give a brief sketc h of Deligne’s proof of the semi-simplicit y of Rep( GL t ) from [ Del07 ], for the reader’s conv enience. T o keep things simple, we will actually only prov e a sp ecial case of Deligne’s result (which is suﬃcien t for our main results), namely that when t is specialized to an element whic h is not an algebr aic integer. The generalization to t not an in teger in volv es a more precise analysis, whic h w ould tak e us to o far aﬁeld. F or our purp oses, and b y wa y of [ Del07 , Prop osition 10.3], we ma y deﬁne Rep( GL t ) as follows (though this is not Deligne’s original deﬁnition): Deﬁnition A.1. Deligne’s category Rep( GL t ) is the the free idemp otent complete additiv ely symmetric monoidal category on a dualizable ob ject x . The endomorphism ring of the unit is Z [ t ], and for every comm utative ring A equipp ed with a choice of element δ ∈ A , corresp onding to a map Z [ t ] → A , w e let Rep( GL t ; A ) denote the basec hange of Rep( GL t ) along Z [ t ] → A . By Theorem 3.2 , it follo ws that Rep( GL t ) is the idemp otent completion of Z [ho(Cob)], whic h explains Deligne’s original deﬁnition [ Del07 , Deﬁnition 10.2]. In our language, Deligne’s result (in the restricted generality described ab o ve) is the following: Theorem A.2 (Deligne, [ Del07 , Th ´ eor` eme 10.5]) . L et K b e a ﬁeld of char acteristic 0 and t ∈ K b e an element which is not an algebr aic inte ger. The c ate gory C K := Rep( GL t ; K ) is ab elian semi-simple. Recall that ho(Cob) consists of ob jects X i,j : = X ⊗ i ⊗ X ∨ , ⊗ j where X is the univ ersal dualizable ob ject, and therefore C K is generated under ﬁnite direct sums and retracts by the Y oneda images of these generators. W e organize what w e need ab out these generators in the following lemma: Lemma A.3. L et i, j, r , s ∈ N . • If i − j  = r − s , then Hom Cob ( X i,j , X r,s ) = ∅ and so Hom C K ( X i,j , X r,s ) = 0 ; • If i − j = r − s and i ≤ r , then ther e ar e maps X i,j → X r,s , X r,s → X i,j whose c omp osite is the identity of X i,j multiplie d with a numb er of cir cles in Cob , and so, the identity multiplie d by some p ower of t in C K . In p articular, in C K , X i,j is a r etr act of X r,s . Pr o of. The ﬁrst is visible in the cob ordism category: a cob ordism out of X i,j consists of circles which do not c hange i − j , maps from + to + or − to − whic h also do not change i − j , and ﬁnally ev aluations or co ev aluations which also do not c hange i − j . F or the second one, it suﬃces to prov e that these maps exist for 1 and X k,k , but then one can simply pick ev aluation and co ev aluation, where the composite is k circles. Since t is in vertible in K , the retraction claim follo ws. □ W e no w organize how semi-simplicit y b ehav es with orthogonality and retracts in the following elemen tary lemma: Lemma A.4. L et C b e an additive 1 -c ate gory. (a) Supp ose x ∈ C is such that End C ( x ) is semi-simple. Then the same holds for ⊕ n x for any n ≥ 0 ; 60 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI (b) Supp ose x, y ∈ C ar e such that End C ( x ) is semi-simple, and y is a r etr act of x . Then End C ( y ) is semi-simple. (c) Supp ose x, y ∈ C ar e such that End C ( x ) , End C ( y ) ar e semi-simple and that hom C ( x, y ) = hom C ( y , x ) = 0 . Then End C ( x ⊕ y ) is semi-simple. Pr o of. 1. End C ( ⊕ n x ) ∼ = M n (End C ( x )) and semi-simple rings are closed under matrix rings. 2. Without loss of generality , C is idemp otent-complete and generated under ﬁnite direct sums and retracts by x . Th us it is equiv alent to Proj f . g . End C ( x ) and End C ( x ) is a pro duct of matrix rings ov er division algebras by Artin–W edderburn. In such a category , the endomorphism ring of any ob ject is a pro duct of matrix rings, and hence is semi-simple, whic h pro ves the claim. 3. The conditions guarantee that End C ( x ⊕ y ) ∼ = End C ( x ) × End C ( y ) and semi- simple rings are closed under ﬁnite pro ducts. □ By induction, the ab o ve tw o lemmas reduce us to interrogating when End( X i,j ) is semi-simple for a single ( i, j ). W e need some base case of the induction to prov e that certain things are semi-simple. The following is the key lemma: Lemma A.5. Supp ose x, y ∈ C ar e in a rigid additive idemp otent-c omplete c at- e gory whose unit has a ﬁeld L of endomorphisms, and End C ( x ) is semi-simple. A ssume al l hom’s in C ar e ﬁnite dimensional over L . Final ly, assume the p airing hom C ( x, y ) ⊗ hom C ( y , x ) → End C ( x ) is a p erfe ct p airing and witnesses hom C ( x, y ) and hom C ( y , x ) as End C ( x ) dual to one another. Then y splits as hom C ( x, y ) ⊗ End C ( x ) x ⊕ z wher e hom C ( x, z ) = 0 , hom C ( z , x ) = 0 . Pr o of. By combining ﬁnite dimensionality and semi-simplicit y of End C ( x ) with idemp oten t-completeness of C , the tensor product hom C ( x, y ) ⊗ End C ( x ) x exists in C , and it comes with a canonical map hom C ( x, y ) ⊗ End C ( x ) x → y . Similarly , we hav e a map y → hom End C ( x ) (hom C ( y , x ) , x ). If we can prov e that the comp osite hom C ( x, y ) ⊗ End C ( x ) x → y → hom End C ( x ) (hom C ( y , x ) , x ) is an isomorphism, w e will b e done since this will b e the desired retraction and the prop erties of z are then automatic. Both source and target are in the thick additive category generated by x , so this map b eing an isomorphism can b e c heck ed b y applying hom C ( x, − ), where the map becomes hom C ( x, y ) → hom End C ( x ) (hom C ( y , x ) , End C ( x )), which is an isomorphism by assumption. □ W e are now equipp ed to prov e Theorem A.2 : Pr o of of The or em A.2 . By Lemma A.3 and Lemma A.4 , we reduce by induction to pro ving that End C K ( X i,j ) is semi-simple. W e pro ve that statement by induction as well, The induction hypothesis is b oth that End C K ( X r,s ) is semi-simple for smaller r , s , and also that the trace pairing End C K ( X r,s ) ⊗ K End C K ( X r,s ) → K given b y m ultiplication follow ed by the trace map 7 End C k ( X r,s ) → K is perfect. 7 The trace map coming from dualizability in C K , not the usual trace map of a ﬁnite dimensional K -algebra. GEOMETRIC POINTS IN TENSOR TRIANGULAR GEOMETR Y 61 If either i = 0 or j = 0, End C K ( X i,j ) is a group algebra K [Σ j ] or K [Σ i ], and is therefore semi-simple since K is c haracteristic 0. F urthermore, the trace pairing is p erfect in this case: indeed, by direct calcu- lation, for an y σ ∈ Σ i , tr( σ ) = t n ( σ ) where n ( σ ) is the num ber of cycles in σ , see [ Ram25 , Lemma 4.7], exampliﬁed b elow in a drawing. ∅ + − + − + − + − ∅ + − + − + − + − The trace of the endomorphism in K [ Σ 4 ] given by the 3-cycle σ = (123). Up to replacing the pairing ( σ , τ ) 7→ tr( σ τ ) b y ( σ, τ ) 7→ tr( σ − 1 τ ) (whic h is de- generate exactly if the other one is) the corresp onding matrix therefore has t i ’s on the diagonal and t k , k < i elsewhere. Therefore, expanding out the determinant of this matrix we ﬁnd a monic p olynomial with integer co eﬃcients of degree ii ! in the v ariable t , whic h is nonzero since t is not an algebraic integer. W e now w ant to chec k the hypotheses of Lemma A.5 for x = X i − 1 ,j − 1 , y = X i,j . Since the trace pairing for End C K ( x ) is assumed to b e p erfect by induction, the pairing Hom C K ( x, y ) ⊗ K Hom C K ( y , x ) → End C K ( x ) is perfect if and only if its comp osition with the trace pairing is p erfect. But now the trace pairing Hom C K ( X i − 1 ,j − 1 , X i,j ) ⊗ K Hom C K ( X i,j , X i − 1 ,j − 1 ) → K is equal, under the ob vious identiﬁcations, to the trace pairing on End C K ( X i − 1 ,j ), whic h is perfect b y induction. So we can apply the lemma and ﬁnd that X i,j = Hom C K ( X i − 1 ,j − 1 , X i,j ) ⊗ End C K ( X i − 1 ,j − 1 ) X i − 1 ,j − 1 ⊕ Z for some Z orthogonal to X i − 1 ,j − 1 . In particular, End C K ( Z ) is the quotien t of End C K ( X i,j ) by the ideal of mor- phisms factoring through X i − 1 ,j − 1 , which is clearly isomorphic to K [ Σ i × Σ j ] and is in particular semi-simple. On this term, the trace pairing is ( σ, τ ) 7→ t n ( σ ) n ( τ ) , whic h is p erfect for the same reason as ab ov e: up to a change of basis, the corresp onding matrix has t ij on the diagonal, and t k , k < ij a wa y from the diagonal so that again, p erfectness is guaran teed by t not b eing an algebraic integer. □ Corollary A.6. Supp ose K = Q ( t ) is a function ﬁeld in one variable t . F or any simple obje ct Z in C K arising as a summand of an obje ct of the form X i,j , with i minimal with this pr op erty. Then the dimension of Z is a p olynomial with de gr e e i + j with p ositive le ading c o eﬃcient. In p articular, the p arity of the de gr e e of the dimension of Z is e qual to the p arity of i − j . 62 TOBIAS BAR THEL, LOGAN HYSLOP , MAXIME RAMZI Pr o of. The object X i,j has dimension t i + j , and all of its summands of the form X r,s with r < i hav e dimension a polynomial of degree strictly less than i + j . In particular, killing oﬀ all summands of X i,j whic h were summands of X r,s with r < s , the resulting ob ject has dimension a monic p olynomial of degree i + j . The pro of of Theorem A.2 sho ws that the endomorphism algebra of X i,j mo dulo all of these summands agrees with the group algebra K [Σ i × Σ j ]. In particular, simple summands correspond to minimal idempotents in this group algebra, and in turn to irreducible represen tations of Σ i × Σ j . 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