The formal spectrum of a tensor-triangulated category

THE F ORMAL SPECTR UM OF A TENSOR-TRIANGULA TED CA TEGOR Y DREW HEARD AND MARIUS NIELSEN Abstract. T o any essentially small tensor-triangulated category K and Thoma- son subset Y ⊆ Sp c ( K ) we asso ciate a ringed space ( Spf ( K , Y ) , O Spf ( K ,Y ) ) , called the formal sp e ctrum of ( K , Y ) . W e establish basic prop erties of this con- struction and compute it in several examples from algebraic geometry , chromatic homotopy theory , equiv ariant homotopy theory , and mo dular representation theory . Contents 1. In tro duction 1 2. F ormal schemes and formal completions 4 3. The Balmer sp ectrum of a tt-category 6 4. Completions of tt-categories 10 5. The formal sp ectrum of a tt-category 10 6. The formal Hopkins–Neeman theorem 14 7. Globalizing to schemes 16 8. A comparison map 18 9. Chromatic homotop y 21 10. F urther examples 23 References 25 1. Intr oduction T ensor–triangular geometry pack ages a tensor–triangulated category into a geo- metric ob ject via the Balmer sp ectrum [ Bal05 ]. Concretely , if K is an essentially small tt-category (or a 2 -ring in our terminology), then its Balmer sp ectrum Sp c ( K ) is a sp ectral space. Its top ology has a basis of closed sets given by supp orts of ob jects in K , and it carries a natural structure sheaf O Spc( K ) v alued in graded rings. When K = D qc ( X ) c for a quasi-compact and quasi-separated (qcqs) scheme X , one recov ers X as a lo cally ringed space from ( Sp c ( K ) , O Spc( K ) ) ; see Example 3.20 . This is Balmer’s reconstruction theorem [ Bal02 ], relying crucially on Thomason’s classification of thick tensor ideals in D qc ( X ) c [ Tho97 ]. In classical algebraic geometry , passing from a sc heme X to its formal com- pletion b X Y along a closed subset Y isolates the infinitesimal neighborho o d of Y and remembers all nilp oten t thick enings at once. This is preferable whenever a problem is in trinsically lo cal around Y , and it is natural to ask for an analogue in tensor–triangular geometry . F or example, the K ( n ) -lo cal stable homotopy category is obtained from the E ( n ) -lo cal category b y a process resembling adic completion 1 2 DREW HEARD AND MARIUS NIELSEN (see the form ulas in [ HS99 , Proposition 7.10]), and it morally behav es like a “residue field”. One migh t therefore expect its sp ectrum to b e a single p oin t. Ho w ever, in tt-geometry one m ust w ork with essentially small sub categories, and the compact ob jects in the K ( n ) -lo cal category do not con tain the unit (hence do not form a tt-category), while the sp ectrum of dualizable ob jects is, p erhaps, unexp ectedly large [ BHN22 ]. The goal of this paper is therefore to develop a formal analogue of the Balmer sp ectrum. Giv en an essentially small tt-category 1 K and a Thomason subset Y ⊆ Spc ( K ) , w e define the formal sp e ctrum Spf ( K , Y ) to b e a ringed space whose underlying top ological space is Y , equipp ed with a comple ted analogue of Balmer’s structure sheaf; see Definition 5.1 for the complete definition. Roughly speaking, Sp c ( K ) pla ys the role of Sp ec ( R ) , and Spf ( K , Y ) is an analogue of the formal sp ectrum Spf ( R, I ) of a ring R along an ideal I (see Definition 2.1 for our con v entions on formal sp ectra of rings, which differ sligh tly from some standard sources). In the classical case, it is straightforw ard to c hec k that the completion map R → R ∧ I induces an isomorphism of formal schemes Spf ( R ∧ I , I R ∧ I ) ∼ − → Spf ( R, I ) . An analogue holds in the tt-w orld, although the pro of is more delicate. Since Bousfield lo calization is most naturally formulated in the presen table setting, it is con venien t to pass from an essen tially small K to the associated stable homotopy theory C : = Ind ( K ) . Inside C w e can form the completion at Y ⊆ Spc ( K ) , obtaining a full subcategory b C Y of Y -complete ob jects together with a symmetric monoidal lo calization d ( − ) Y : C → b C Y . Unlike categories usually considered in tensor–triangular geometry , b C Y is t ypically not rigidly-compactly generated. Our first main result iden tifies the formal sp ectrum computed from the dualizable ob jects of b C Y with the original definition in terms of ( K , Y ) . Theorem A ( Theorem 5.14 ) . L et K b e a 2 -ring and let Y ⊆ Spc ( K ) b e a Thomason subset. Ther e is a natur al isomorphism of ringe d sp ac es ( ϕ, ϕ # ) : Spf ( b C d Y , ϕ − 1 ( Y )) − → Spf ( K , Y ) , wher e ϕ = Sp c( d ( − ) Y ) is the map on Balmer sp e ctr a induc e d by c ompletion. This result is fundamen tal for us: the left-hand side is often the conceptual ob ject of interest, while the righ t-hand side is typically easier to compute. F or the remainder of this in troduction, we abbreviate the left-hand side to Spf ( b C d Y ) , suppressing the implicit Thomason subset ϕ − 1 ( Y ) . —— ∗ ∗ ∗ —— A basic computation in tt-geometry is the Hopkins–Neeman theorem: for any comm utative no etherian ring R one has a canonical iden tification Sp c ( D ( R ) c ) ∼ = Sp ec ( R ) , under which tt-support agrees with ordinary (homological) supp ort. Since formal completion in algebraic geometry is tak en along a closed subset, we fix an ideal I ⊆ R and consider the closed subset V ( I ) ⊆ Sp ec ( R ) ∼ = Sp c ( D ( R ) c ) . Our formal sp ectrum reco v ers the usual affine formal scheme: 1 F or technical reasons, we work with stable ∞ -categories throughout; see Remark 3.3 . THE FORMAL SPECTRUM OF A TENSOR-TRIANGULA TED CA TEGOR Y 3 Theorem B ( Theorem 6.3 ) . L et R b e a c ommutative no etherian ring and I ⊆ R an ide al. Ther e is a natur al isomorphism of formal schemes ρ : Spf  D( R ) c , V ( I )  ∼ − → Spf ( R, I ) . Conse quently, the c ompletion of D( R ) at V ( I ) satisfies Spf  [ D( R ) d V ( I )  ∼ − → Spf ( R ∧ I ) . The completed category [ D( R ) V ( I ) here is the usual complete deriv ed category of a no etherian ring at an ideal; see Example 4.6 . W e next extend this affine computation to noetherian sc hemes. The k ey input is a lo cality statemen t for the formal sp ectra with resp ect to restriction to quasi- compact op ens. If U ⊆ Sp c ( K ) is quasi-compact with closed complement Z , recall that K ( U ) : = ( K / K Z ) ♮ denotes the idempotent-completed V erdier quotien t (cf. Remark 3.10 ). Then w e prov e: Theorem C ( Theorem 5.16 ) . L et K b e a 2 -ring and Y ⊆ Spc ( K ) a Thomason subset. L et U ⊆ Sp c ( K ) b e a quasi-c omp act op en subset and let q U : K → K ( U ) b e the c anonic al functor. Set Y ′ : = Y ∩ U , viewe d as a Thomason subset of Sp c ( K ( U )) ∼ = U . Then r estriction along Y ′  → Y induc es an isomorphism of ringe d sp ac es Spf ( K ( U ) , Y ′ ) ∼ = − → Spf ( K , Y )   Y ′ . With Theorem C in hand, the affine computation globalizes b y co v ering a no e- therian sc heme by finitely many affine opens and chec king that the resulting lo cal iden tifications glue on o verlaps. This yields a formal v ersion of Thomason’s theorem. Theorem D ( Theorem 7.7 ) . L et X b e a no etherian scheme, and let Z ⊆ X b e a close d subset. Then ther e is a natur al isomorphism of formal schemes Φ : Spf  D qc ( X ) c , Z  ∼ − → b X Z , wher e b X Z denotes the formal c ompletion of X along Z . —— ∗ ∗ ∗ —— Our second main family of examples comes from chromatic homotopy theory . Fix a prime p and a height n ≥ 0 , and write S p n for the E ( n ) -lo cal stable homotopy category with compact ob jects S p c n . By work of Hov ey–Strickland, the Balmer sp ectrum Spc( S p c n ) is the finite chain of primes P 0 ⊊ P 1 ⊊ · · · ⊊ P n indexed b y c hromatic height, and the structure sheaf on the basic op ens can b e describ ed in terms of the lo calizations L k S 0 ; see Theorem 9.5 . F or each 0 ≤ h ≤ n − 1 w e consider the Thomason subset Y h +1 = { P h +1 , . . . , P n } ⊆ Sp c( S p c n ) , corresp onding to sp ectra of t yp e at least h + 1 . Completion at Y h +1 reco vers the lo cal category S p K ( h +1) ∨···∨ K ( n ) , and we compute the follo wing: Theorem E ( Theorem 9.8 ) . Fix an inte ger 0 ≤ h ≤ n − 1 , then the formal sp e ctrum Spf  S p d K ( h +1) ∨···∨ K ( n )  has underlying sp ac e Y h +1 , and its structur e she af on b asic op ens is given by O ( U k ) ∼ = π ∗ L K ( h +1) ∨···∨ K ( k ) S 0 ( k ≥ h + 1) , 4 DREW HEARD AND MARIUS NIELSEN for the op ens U k = U ( L n F ( k + 1)) ∩ Y h +1 describ e d in The or em 9.5 . These c hromatic examples illustrate one of the motiv ations for introducing Spf ( K , Y ) : completion can drastically c hange the geometry detected by dualizable ob jects, and the formal sp ectrum pro vides a conv enient receptacle for the resulting lo cal information. In particular, in the sp ecial case where h = n − 1 , we see that the formal sp ectrum of dualizable ob jects in the K ( n ) -lo cal category is indeed a single p oint, as one w ould hope. W e conclude with t wo further examples, one from equiv ariant homotop y theory and one from mo dular representation theory , to illustrate additional b eha vior of the formal sp ectrum. Organization of the pap er. In Section 2 w e recall formal schemes and formal completions in algebraic geometry and fix con v en tions. In Section 3 we review the Balmer sp ectrum and its structure sheaf. In Section 4 we recall completion of stable homotopy theories at Thomason subsets, and in Section 5 w e introduce the formal sp ectrum Spf ( K , Y ) and prov e the completion theorem Theorem 5.14 . In Sections 6 and 7 we establish the formal v ersion of the Hopkins–Neeman and Thomason theorems. Section 8 pro duces a comparison map, analogous to Balmer’s for the ordinary spectrum and in Section 9 w e compute the formal spectra arising in c hromatic homotopy theory . Finally , we finish in Section 10 with the t w o examples from equiv ariant homotop y theory and mo dular represen tation theory . A c kno wledgmen ts. W e thank Anish Chedalav ada and Leovigildo Alonso T arrio for helpful discussions. DH thanks Utrech t Univ ersit y and MPIM Bonn, where parts of this work w ere carried out. 2. F ormal schemes and formal completions W e begin b y recalling the definition of the formal spectrum of a commutativ e no etherian ring. W e do this to fix our conv entions, whic h ma y differ sligh tly from the standard literature. 2.1. Definition. Let R b e a commutativ e no etherian ring and let I ⊆ R b e an ideal. The formal sp ectrum of R (with respect to I ), denoted Spf ( R, I ) , is the lo cally ringed space ( | Spf ( R, I ) | , O Spf ( R,I ) ) where: (a) The underlying top ological space | Spf ( R, I ) | is the closed subset of the Zariski sp ectrum giv en by the v anishing lo cus of I , i.e., | Spf ( R, I ) | : = V ( I ) =  p ∈ Sp ec( R )   I ⊆ p  . It has a basis of op en subsets of the form D ( s ) : = D ( s ) ∩ V ( I ) =  p ∈ Sp ec( R )   I ⊆ p , s / ∈ p  , where s ∈ R and D ( s ) =  p ∈ Sp ec( R )   s / ∈ p  . (b) The structure sheaf O Spf ( R,I ) is defined on the basic op en sets D ( s ) as the I -adic completion of the lo calization R [1 /s ] . THE FORMAL SPECTRUM OF A TENSOR-TRIANGULA TED CA TEGOR Y 5 2.2. Example. Let R b e a no etherian ring, and let I b e a nilp otent ideal (for example, I = (0) ). Then it is immediate from the definitions that Spf ( R, I ) ∼ = Sp ec( R ) , the usual Zariski sp ectrum. 2.3. Example. Let R = Z and I = ( p ) for a prime p . Then Spf ( Z , ( p )) is a one-p oin t lo cally ringed space with ring of global sections Z p . 2.4. R emark. There is an obvious extension to Z -graded commutativ e rings, where w e work with homo gene ous prime ideals. W e denote this by a sup erscript h , i.e., Sp ec h ( R ∗ ) denotes the homogeneous Zariski sp ectrum of a graded ring R ∗ , and Spf h ( R, I ) denotes the formal homogeneous sp ectrum with resp ect to a homogeneous ideal I . 2.5. R emark. W e do not assume that R is complete with respect to the I -adic top ology . Ho w ev er, replacing R b y its completion R ∧ I do es not c hange the formal sp ectrum; there is an isomorphism of lo cally ringed spaces Spf ( R ∧ I , I R ∧ I ) ∼ − → Spf ( R, I ) induced by the completion map R → R ∧ I . In the former case, since the ideal for the formal sp ectrum is clear, we ma y simply write Spf ( R ∧ I ) . 2.6. R emark. Although the definition of the formal spectrum of a ring R do es not strictly require R to b e noeth erian, we fix this assumption due to the p o orly b eha ved nature of I -adic completion in the non-noetherian case. 2.7. R emark. There is a useful w ay to think ab out the top ological space | Spf ( R, I ) | . Recall that the supp ort of a (finitely generated) R -mo dule M is Supp R ( M ) =  p ∈ Sp ec( R )   M p  = 0  . Since S − 1 ( R/I ) = 0 ⇐ ⇒ S ∩ I  = ∅ , w e see that Supp R ( R/I ) =  p ∈ Sp ec( R )   I ⊆ p  . In other words, | Spf ( R, I ) | = Supp R ( R/I ) . In the same wa y that the sp ectrum of a ring globalizes to define schemes, the formal sp ectrum globalizes to define formal schemes. 2.8. Definition. Let ( X, O X ) b e a lo cally ringed space. W e sa y that ( X, O X ) is a no etherian formal scheme if | X | is quasi-compact and there exists a cov ering of | X | b y op en subsets U α suc h that each ringed space ( U α , O X | U α ) is isomorphic to ( | Spf ( R α , I α ) | , O Spf ( R α ,I α ) ) for some no etherian ring R α and ideal I α ⊆ R α . 2.9. R emark. It is clear from the definition that the condition of b eing a no ether- ian formal scheme is lo cal: if there exists a co v ering { U α } of X suc h that eac h ( U α , O X | U α ) is a no etherian formal scheme, then so is ( X, O X ) . 6 DREW HEARD AND MARIUS NIELSEN 2.10. R emark. W e form the category of noetherian formal sc hemes as the full sub category of the category of locally ringed spaces whose ob jects are no etherian formal schemes. In contrast to some other sources, we do not require the structure sheaf to carry a topological structure (i.e., we w ork with locally ringed spaces rather than top ologically ringed spaces). 3. The Balmer spectrum of a tt-ca tegor y W e assume the reader has some familiarity with ordinary tt-geometry , for example, the foundational paper [ Bal05 ]. F or the b enefit of the reader, we briefly recall the basic definitions. W e b egin b y in tro ducing the ‘small’ and ‘big’ v ariants of triangulated categories that will b e used; see also [ BCH + 24 , Section 5] for a more thorough discussion. 3.1. Definition. Let 2 - Ring : = CAlg(Cat perf ∞ ) rig b e the ∞ -category of comm utativ e algebra ob jects in Cat perf ∞ , the ∞ -category of small, stable, idemp otent-complete ∞ -categories and exact functors, equipped with the symmetric monoidal structure describ ed in [ BGT13 , Section 3.1]. The subscript rig denotes the full sub category spanned b y those K for whic h every ob ject of the underlying symmetric monoidal ∞ -category is dualizable. A 2-ring is then an ob ject K ∈ 2 - Ring . W e will also need a ‘big’ v ariant of this notion. 3.2. Definition. A stable homotopy the ory is an ob ject of CAlg ( Pr L st ) , i.e., a pre- sen table, symmetric monoidal stable ∞ -category ( C , ⊗ , 1 ) whose tensor pro duct preserv es all colimits separately in each v ariable. The full sub category K : = C d of dualizable ob jects is then a 2 - Ring in the sense of Definition 3.1 (see [ NP24 , Lemma 2.5]). A stable homotop y theory is rigid ly-c omp actly gener ate d if it is compactly generated and its sub category of dualizable ob jects coincides with its sub category of compact ob jects. 3.3. R emark. The homotopy category of a given K ∈ 2 - Ring is a tensor-triangulated category in the usual sense of [ Bal05 ]. Lik ewise, if C is a stable homotop y theory , then its homotopy category is a tensor-triangulated category with small copro ducts. F or the definition of the Balmer sp ectrum Sp c ( K ) it suffices to work at the level of these underlying triangulated categories. By contrast, our construction of the formal spectrum as a ringe d sp ac e is most naturally form ulated after passing to “big” categories, as we will so on describ e. W e therefore w ork throughout in the setting of stable ∞ -categories: among other adv an tages, Ind-completion is w ell b ehav ed in this setting, whereas on the level of triangulated categories the Ind-completion need not remain triangulated; see [ Kra23 , Remark 5.9] for an example. 3.4. R emark. By taking Ind-categories, any 2-Ring can b e turned into a rigidly- compactly generated stable homotop y theory . Indeed, set C : = Ind ( K ) . As noted in [ NP24 ], this is presen table, stable, and symmetric monoidal b y com bining [ Lur09 , Theorem 5.5.1.1] and [ Lur17 , Corollary 4.8.1.14], and it is rigidly-compactly gener- ated by construction. Moreo ver, the (restricted) Y oneda embedding j : K → C is symmetric monoidal. In fact, if Pr L,ω st denotes the (non-full) symmetric monoidal THE FORMAL SPECTRUM OF A TENSOR-TRIANGULA TED CA TEGOR Y 7 sub category of Pr L st spanned by compactly generated stable ∞ -categories and colimit- preserving functors that preserve compact ob jects (equiv alently , whose right adjoin t preserv es colimits), then Ind -completion defines a symmetric monoidal equiv alence Ind : Cat perf ∞ ∼ − → Pr L,ω st , with in verse giv en by sending D to D ω . See [ BGT13 , Section 3.1] and [ BMCSY25 , Prop osition 2.3]. W e will use this equiv alence implicitly whenev er needed. Ha ving fixed terminology for b oth the small and big settings, we now give a standard example that will serve as a running model in what follo ws. 3.5. Example. The prototypical example of a 2 - Ring is the category Mo d c R of compact R -mo dules, where R is a ring sp ectrum. The associated stable homotopy theory is the category Mo d R of all R -mo dules. Note that if R = H A is an Eilenberg– MacLane sp ectrum, these corresp ond to the derived category of perfect complexes of A -mo dules and the unbounded derived category of A -modules, resp ectively . 3.6. Definition (Balmer [ Bal05 ]) . Let K ∈ 2 - Ring . The sp e ctrum of K is the set Sp c( K ) : = { P | P is a prime thic k ⊗ -ideal of K } . W e top ologize this set by declaring a basis of op en sets { U ( c ) } c ∈ K defined by U ( c ) : =  P ∈ Sp c( K )   c ∈ P  . 3.7. Definition. The supp ort of an ob ject a ∈ K is supp( a ) : =  P ∈ Sp c( K )   a / ∈ P  = U ( a ) ∁ . The sets { supp( a ) } a ∈ K form a basis of closed subsets for the top ology on Sp c( K ) . 3.8. R emark. The spectrum is functorial [ Bal05 , Proposition 3.6]: given a symmetric monoidal exact functor F : K → L , there is an induced contin uous map ϕ : = Sp c( F ) : Sp c( L ) − → Sp c( K ) , P 7− → F − 1 ( P ) . Moreo ver, ϕ − 1 (supp K ( a )) = supp L ( F ( a )) and ϕ − 1 ( U ( c )) = U ( F ( c )) . One ma jor reason to study the spectrum is its classification of thick tensor ideals. 3.9. Theorem (Balmer, [ Bal05 , Theorem 4.10]) . L et K b e a 2-ring. Then supp ort induc es an inclusion-pr eserving bije ction: n thick ⊗ -ide als J ⊆ K o ⇆ n Thomason subsets Y ⊆ Sp c( K ) o J 7− → supp( J ) : = [ x ∈ J supp( x ) K Y : = { x ∈ K | supp( x ) ⊆ Y } ← − [ Y . 3.10. R emark. W e no w describ e Balmer’s structure sheaf on Sp c ( K ) . Giv en a thick ⊗ -ideal J ⊆ K as ab o ve, w e can form the V erdier quotient K / J . If U ⊂ Sp c ( K ) is a quasi-compact op en subset with closed complement Z , set K Z : =  a ∈ K   supp( a ) ⊆ Z  and define K ( U ) : =  K / K Z  ♮ , where ( − ) ♮ denotes idemp otent completion. This is also known as the Karoubi quotien t. There is a natural functor q U : K → K ( U ) , and we write 1 U for the image of the unit 1 ∈ K under q U . 8 DREW HEARD AND MARIUS NIELSEN 3.11. Definition. Let K ∈ 2 - Ring b e an essen tially small tt-category . Balmer’s structure sheaf O Spc( K ) on Sp c( K ) is the sheafification of the presheaf 1 e O Spc( K ) : Op en(Spc( K )) op − → grCAlg ♡ giv en by (3.12) U 7− → π ∗ End K ( U ) ( 1 U ) . 3.13. R emark. Let K b e a 2-ring and C : = Ind ( K ) the asso ciated stable homotop y theory . Let U ⊆ Sp c ( K ) be a quasi-compact op en subset with closed complement Z . Consider the lo calizing sub category L : = Lo c ⟨ K Z ⟩ ⊆ C . Since K Z is a thic k ⊗ -ideal in K and the tensor product in C preserv es colimits separately in eac h v ariable, L is a lo calizing ⊗ -ideal in C . Consequently , there is a unique wa y to equip C ( U ) and the lo calization functor L : C → C ( U ) with symmetric monoidal structures. W e now iden tify the compact ob jects of the lo calization. The restriction of L to dualizable (equiv alently , compact) ob jects, L | K : K → C ( U ) , lands in C ( U ) c (since L preserv es compact ob jects) and v anishes on K Z . By the univ ersal prop ert y of the V erdier quotient, this induces a symmetric monoidal functor ¯ L : K / K Z − → C ( U ) c . The Neeman–Thomason localization theorem [ Nee92b , Theorem 2.1] & [ CHL + 26 , Theorem A.3.12] 2 asserts that ¯ L is fully faithful and that C ( U ) c iden tifies with the idempotent completion of the image of ¯ L . Therefore, passing to idempotent completions yields a symmetric monoidal equiv alence K ( U ) : = ( K / K Z ) ♮ ∼ − → ( C ( U ) c ) ♮ ≃ C ( U ) c . 3.14. R emark. If 1 U denotes the tensor unit of the localized category C ( U ) , then the presheaf from ( 3.12 ) can b e rewritten as (3.15) U 7− → π ∗ Hom C ( 1 , 1 U ) , where in the righ t-hand side, 1 U is implicitly view ed inside C via the fully faithful righ t adjoint to the lo calization functor L . Although ( 3.15 ) lo oks sligh tly less direct than ( 3.12 ) , this am bien t form ulation in C will b e crucial for the construction of the formal Balmer sp ectrum, where we will encoun ter lo calizations that do not preserve compact ob jects, necessitating the use of the large category C . 3.16. R emark. In [ ABC + 25 , Theorem D and Theorem F], it is sho wn that for any 2 -Ring K with asso ciated stable homotopy theory C : = Ind ( K ) , the functor sending a quasi-compact op en U ⊆ Sp c( K ) to the ∞ -category C ( U ) refines to a sheaf O K : Op en(Spc( K )) op − → CAlg (Pr L st ) . 3.17. R emark. It follows from [ Lur17 , Theorem 4.8.5.11] and [ Lur17 , Theorem 4.8.5.16] that the functor Alg( S p ) − → Pr L st , sending an asso ciativ e algebra A in sp ectra to the stable ∞ -category LMo d A ( S p ) of left A -mo dule sp ectra, is symmetric monoidal and admits a right adjoint. In 1 Here, and throughout, grCAlg ♡ denotes the category of (discrete) graded commutative rings with the Koszul sign rule. 2 The cited theorems are essentially the same in different settings. In the setting of triangulated categories in the former and stable ∞ -categories in the latter. THE FORMAL SPECTRUM OF A TENSOR-TRIANGULA TED CA TEGOR Y 9 particular, by Dunn additivit y and passing to comm utativ e algebra ob jects, we obtain a right adjoin t functor (3.18) End( 1 ( − ) ) : CAlg(Pr L st ) − → CAlg ( S p ) , sending a stable homotop y theory C to the comm utativ e algebra in sp ectra giv en b y the endomorphism sp ectrum Hom C ( 1 C , 1 C ) (see also [ ABC + 25 , Construction 4.29]). If w e comp ose O K with End( 1 ( − ) ) we obtain a sheaf O 1 K : Op en(Spc( K )) op − → CAlg ( S p ) of comm utativ e algebras in sp ectra. This sheaf is given on a quasi-compact op en U ⊆ Sp c( K ) by Hom C ( U ) ( 1 U , 1 U ) ≃ Hom C ( 1 , 1 U ) in the notation of Remark 3.13 . In particular, Balmer’s structure sheaf of graded rings O Spc( K ) is the induced sheaf obtained from the functor π ∗ : CAlg( S p ) → grCAlg ♡ . 3.19. R emark (F unctoriality) . The functoriality of Remark 3.8 is compatible with this ringed space structure. That is, given a tt-functor F : K → L , there is a morphism of ringed spaces Sp c( F ) : = ϕ : ( | Sp c( L ) | , O Spc( L ) ) → ( | Sp c( K ) | , O Spc( K ) ) . F or details, see [ BKS07 , Lemma 7.2]. 3.20. Example. The most fundamental example comes from w ork of Hopkins and Neeman in the affine case, as extended to schemes b y Thomason [ Hop87 , Nee92a , Tho97 ]. Let X b e a quasi-compact and quasi-separated scheme. There is an isomorphism of lo cally ringed spaces (and hence schemes) f : X ∼ − → Sp c(D qc ( X ) c ) giv en by f ( x ) =  a ∈ D qc ( X ) c   a x ≃ 0 in D q c ( O X,x ) c  for all x ∈ X. Moreo ver, under this equiv alence, the homological support 1 supph ( a ) ⊆ X of a p er- fect complex a ∈ D qc ( X ) c corresp onds to the closed subset supp ( a ) ⊆ Spc ( D qc ( X ) c ) . F or the translation to tt-geometry in this form, see [ Bal05 , Corollary 5.6 and Theorem 6.3] and [ BKS07 , Theorem 8.5]. 3.21. R emark. When X = Sp ec ( R ) is affine, there is a simple description of the in verse to the abov e isomorphism, given b y the following general construction and theorem of Balmer. 3.22. Theorem (Balmer) . L et K ∈ 2 - Ring b e an essential ly smal l tt-c ate gory. Then ther e is a natur al c ontinuous, inclusion-r eversing map ρ : Sp c( K ) → Sp ec h ( π ∗ End K ( 1 )) of lo c al ly ringe d sp ac es, given by ρ ( P ) =  f   cone( f ) / ∈ P  . 3.23. R emark. It is not alw a ys the case that Sp c ( K ) is a scheme; for instance, taking K = SH c giv es a counterexample. See [ Bal10 , Prop osition 9.7]. 1 That is, the support of the total homology of a . 10 DREW HEARD AND MARIUS NIELSEN 4. Completions of tt-ca tegories The tensor-triangular analog of I -adic completion is completion with resp ect to a closed subset of the Balmer sp ectrum. In this short section we recall the notion, and review the most basic example of the deriv ed category of a commutativ e ring. Nothing in this section is new; the ideas date back to at least Greenlees [ Gre01 ], and we follo w the recent exposition in [ BS25 ]. 4.1. Definition. Let C b e a rigidly compactly generated stable homotop y theory , and let Y ⊆ Sp c( C d ) b e a Thomason subset. W e set C d Y : =  x ∈ C d   supp( x ) ⊆ Y  and C Y : = Lo c ⟨ C d Y ⟩ . 4.2. R emark. The inclusion C Y  → C admits a right adjoin t given b y tensoring with the Balmer–F avi idempotent e Y [ BF11 ]. Our primary in terest, how ever, lies in the double right orthogonal to C Y . Recall that if E ⊆ C is any class of ob jects, we write E ⊥ : =  c ∈ C   Hom C ( s, c ) = 0 for all s ∈ E  . Note that this is closed under susp ension in C . 4.3. Definition. The c ompletion of C at Y ⊆ Sp c( C d ) is the full sub category b C Y : = ( C Y ) ⊥ ⊥ . Ob jects of b C Y are called Y -c omplete . 4.4. R emark. The inclusion b C Y  → C has a symmetric monoidal left adjoin t, giv en b y d ( − ) Y : = hom ( e Y , − ) . The category b C Y is a tensor-triangulated category under the lo calized tensor product. It is not, how ever, rigidly-compactly generated, except in trivial cases (see [ BS24 , Remark 2.12]). Note that via the inclusion w e can also consider d ( − ) Y as an endofunctor on C . 4.5. R emark. W e are often in terested in the case where Y = supp ( A ) for some A ∈ C d . In this case, the completion functor d ( − ) supp( A ) coincides with Bousfield lo calization with resp ect to A (see, for example, [ BHV18 , Prop osition 2.34]). 4.6. Example. The most basic example is as follows. Let R b e a no etherian ring, I ⊆ R an ideal, and consider the closed subset Y = V ( I ) ⊆ Spec ( R ) ∼ = Sp c ( D ( R ) c ) (see Example 3.20 ). Then the functor d ( − ) I : D( R ) − → [ D( R ) I : = [ D( R ) V ( I ) is the deriv ed I -adic completion functor; see, for instance, [ GM92 , DG02 ]. Note that ev en if M is a discrete R -mo dule, the deriv ed completion M ∧ I need not agree with the classical I -adic completion. F or example, with R = Z , I = ( p ) , and M = L n ≥ 1 Z /p n , the tw o notions differ [ MP12 , Example 10.1.16]. 5. The formal spectrum of a tt-ca tegor y W e now turn to the definition of the formal sp ectrum in the tensor-triangular setting. 5.1. Definition. Let K ∈ 2 - Ring b e a 2-ring and let J = K Y b e the thick ⊗ -ideal in K corresp onding to a Thomason subset Y ⊆ Sp c ( K ) (note that every thic k ⊗ -ideal is of this form b y Theorem 3.9 ). W rite C for the stable homotop y theory asso ciated to THE FORMAL SPECTRUM OF A TENSOR-TRIANGULA TED CA TEGOR Y 11 K ( Remark 3.4 ). The formal sp e ctrum of K (with respect to Y ), denoted Spf ( K , Y ) , is the ringed space ( | Spf ( K , Y ) | , O Spf ( K ,Y ) ) defined as follows: (a) The underlying top ological space is | Spf ( K , Y ) | : = Y ⊆ Spc( K ) equipp ed with the subspace top ology . A basis of op en sets is given b y U ( c ) : = U ( c ) ∩ Y =  P ∈ Y   c ∈ P  . (b) The structure sheaf is given on basic op ens U ( c ) by the sheafification (in grCAlg ♡ ) of the presheaf U ( c ) 7− → π ∗ Hom C  1 , \ ( 1 [ c − 1 ]) Y  , where 1 [ c − 1 ] denotes the tensor unit of the lo calization C [ c − 1 ] = C ( U ( c )) , view ed as an ob ject of C via the right adjoin t to the lo calization functor, and d ( − ) Y denotes Y -completion as introduced in Section 4 . 5.2. Question. Is Spf ( K , Y ) alwa ys a lo c al ly ringed space? 5.3. R emark. Recall that d ( − ) Y ≃ hom ( e Y , − ) . Therefore, the presheaf can equiv a- len tly b e written as U ( c ) 7− → π ∗ Hom C  e Y , 1 [ c − 1 ]  . 5.4. R emark. Recall from Remark 3.14 there is a sheaf O 1 K : Op en(Spc( K )) op − → CAlg ( S p ) of comm utativ e algebras in spectra. On a quasi-compact open U the v alue of the sheaf is given b y Hom C ( 1 , 1 U ) . If Y ⊆ Spc ( K ) is a Thomason subset and e Y is the asso ciated Balmer-F a vi idemp oten t, then the association sending a quasi-compact op en U ⊆ Sp c( K ) to Hom C ( 1 , [ ( 1 U ) Y ) defines a functor O K Y : Op en(Spc( K )) op − → CAlg ( S p ) . W e claim that O K Y is a sheaf. By com bining [ Lur17 , Theorem 4.8.5.11], [ Lur17 , Theorem 4.8.5.16] and [ ABC + 25 , Corollary 5.41] we see that the functor Op en(Spc( K )) op − → CAlg ( C ) sending a quasi-compact op en U ⊆ Sp c ( K ) to 1 U is a sheaf. In particular, it suffices to see that the functor sending U to [ ( 1 U ) Y is a sheaf. This follo wing from [ Lur09 , Theorem 7.3.5.2], since d ( − ) Y : C → C preserv es limits and Spc( K ) is sp ectral. W e claim this sheaf is extended from its restriction to | Spf ( K , Y ) | = Y . It suffices to see that for an y quasi-compact open U ⊆ Spc ( K ) suc h that U ∩ Y = ∅ the comm utative algebra in sp ectra of sections O K Y ( U ) ≃ 0 is trivial. This follo ws from the calculus of idempotents coming from Thomason subsets of Sp c ( K ) presented in [ BF11 , Theorem 5.18]. Indeed, let Z b e the closed complemen t of U , so 1 U ≃ f Z . If U ∩ Y = ∅ then Y ⊆ Z , hence e Y ⊗ e Z ≃ e Y . T ensoring the triangle e Z → 1 → f Z with e Y sho ws e Y ⊗ f Z ≃ 0 , so Hom C ( e Y , f Z ) ≃ Hom C ( e Y ⊗ f Z , f Z ) ≃ 0 , i.e. O K Y ( U ) ≃ 0 . W e deduce, like in Remark 3.17 , that the structure sheaf of graded commutativ e rings defined in Definition 5.1 is induced from O K Y | Y via π ∗ : CAlg ( S p ) → grCAlg ♡ . 12 DREW HEARD AND MARIUS NIELSEN 5.5. R emark. Unlike in Remark 3.16 , w e are curren tly unable to construct a sheaf v alued in CAlg (Pr L st ) . W e hop e to pursue this further in future work. 5.6. R emark. T aking c = 0 (which lies in ev ery prime), the v alue on U (0) = Y is Y 7− → π ∗ Hom C  1 , b 1 Y  ∼ = π ∗ Hom C  e Y , 1  . 5.7. Example. T aking J = thic k ⊗ ⟨ 1 ⟩ (equiv alently Y = Sp c ( K ) ), the construction reduces to the usual Balmer sp ectrum: Spf ( K , Sp c( K )) ∼ = Sp c( K ) as ringed spaces (compare with ( 3.15 )). 5.8. R emark. There is a subtle difference from the algebraic situation: for rings, one t ypically forms formal sp ectra only along close d subsets of Sp ec ( R ) , whereas here w e allow arbitrary Thomason subsets Y ⊆ Sp c( K ) . The formal sp ectrum is functorial in the obvious w a y: 5.9. Lemma. L et K , L b e 2-rings and let Y ⊆ Spc ( K ) and Y ′ ⊆ Spc ( L ) b e Thomason subsets. Supp ose F : K → L is an exact symmetric monoidal functor and write ϕ : = Sp c ( F ) : Sp c ( L ) → Spc ( K ) for the induc e d map. If ϕ ( Y ′ ) ⊆ Y , then ther e is a morphism of ringe d sp ac es ( ϕ, ϕ # ) : Spf ( L , Y ′ ) − → Spf ( K , Y ) . Pr o of. On underlying spaces this is the restriction ϕ | Y ′ : Y ′ → Y . F or the sheaf map, w e set C : = Ind ( K ) and D : = Ind ( L ) ; the functor F extends to a colimit-preserving symmetric monoidal functor F : C → D . Let U ( c ) : = U ( c ) ∩ Y b e a basic open of Spf ( K , Y ) , with c ∈ K compact. Using Remark 5.3 we ha ve O Spf ( K ,Y ) ( U ( c )) ∼ = π ∗ Hom C ( e Y , 1 [ c − 1 ]) . Define on the basis the comp osite π ∗ Hom C ( e Y , 1 [ c − 1 ]) − → π ∗ Hom D ( F ( e Y ) , F ( 1 [ c − 1 ])) ∼ − − → π ∗ Hom D ( e φ − 1 ( Y ) , 1 [ F ( c ) − 1 ]) res − − → π ∗ Hom D ( e Y ′ , 1 [ F ( c ) − 1 ]) , where F ( e Y ) ≃ e φ − 1 ( Y ) and F ( 1 [ c − 1 ]) ≃ 1 [ F ( c ) − 1 ] b y the functorialit y of idempo- ten ts (see [ BS17 , Proposition 5.11]), and the last map is induced b y the mor- phism of idempotents e Y ′ → e φ − 1 ( Y ) coming from Y ′ ⊆ ϕ − 1 ( Y ) . Note that the target is precisely O Spf ( L ,Y ′ ) ( U ( F ( c ))) , where U ( F ( c )) : = U ( F ( c )) ∩ Y ′ and U ( F ( c )) = ϕ − 1 ( U ( c )) as op en sets of Y ′ . The constructions are all natural in c and compatible with restriction maps, and hence define a morphism ϕ # : O Spf ( K ,Y ) − → ϕ ∗ O Spf ( L ,Y ′ ) . Therefore, ( ϕ, ϕ # ) is a morphism of ringed spaces Spf ( L , Y ′ ) → Spf ( K , Y ) . □ 5.10. Example. As a sp ecial case of Lemma 5.9 , tak e F = id and Y = Sp c ( K ) . T ogether with Example 5.7 , this shows that for any Thomason subset Y ′ ⊆ Spc ( K ) , there is a canonical morphism of ringed spaces Spf ( K , Y ′ ) − → Sp c( K ) , THE FORMAL SPECTRUM OF A TENSOR-TRIANGULA TED CA TEGOR Y 13 whic h realizes the inclusion Y ′  → Sp c ( K ) on underlying spaces. Here the morphism of sheav es is induced by the map e Y ′ → 1 . 5.11. Notation. Let K b e a 2-ring, and write C for the asso ciated stable homotop y theory . F or a Thomason subset Y ⊆ Sp c( K ) , consider the functor d ( − ) Y : C → b C Y . This is symmetric monoidal, and so restricts to a functor of 2-rings (5.12) d ( − ) Y : C d = K → b C d Y . By Lemma 5.9 we therefore get a morphism of ringed spaces (5.13) ( ϕ, ϕ # ) : Spf ( b C d Y , ϕ − 1 ( Y )) − → Spf ( K , Y ) . The crucial ingredient for the following result is due to Balmer and Sanders [ BS25 ]. Note that this is the analog of Remark 2.5 . 5.14. Theorem. L et K b e a 2-ring and let Y ⊆ Spc ( K ) b e a Thomason subset. Then the c anonic al morphism ( ϕ, ϕ # ) : Spf ( b C d Y , ϕ − 1 ( Y )) − → Spf ( K , Y ) is an isomorphism of ringe d sp ac es. Pr o of. On underlying topological spaces, ϕ is a homeomorphism b y [ BS25 , Theo- rem 4.6]. It remains to show that ϕ # is an isomorphism of shea ves. Since b oth structure sheav es are defined as sheafifications of preshea ves on the basis { U ( c ) } , it suffices to chec k that ϕ # is an isomorphism on each basic open set. Fix a basic op en U ( c ) and set X : = 1 [ c − 1 ] ∈ C . Using Remark 5.3 , the sec- tions of O Spf ( K ,Y ) o ver U ( c ) is π ∗ Hom C ( e Y , X ) . On the other hand, in b C Y w e ha ve L = d ( − ) Y ≃ hom ( e Y , − ) , and hence the corresp onding ring of sections is π ∗ Hom b C Y ( L ( e Y ) , L ( X )) . The map ϕ # U ( c ) : π ∗ Hom C ( e Y , X ) − → π ∗ Hom b C Y ( L ( e Y ) , L ( X )) is induced by the symmetric monoidal functor L . W e claim that the underlying map of mapping spectra Hom C ( e Y , X ) − → Hom b C Y ( L ( e Y ) , L ( X )) is an equiv alence. Indeed, b y the adjunction L ⊣ ι and the iden tification ιL ( − ) ≃ hom ( e Y , − ) , there are natural equiv alences Hom b C Y ( L ( e Y ) , L ( X )) ≃ Hom C ( e Y , ιL ( X )) ≃ Hom C ( e Y , hom ( e Y , X )) ≃ Hom C ( e Y ⊗ e Y , X ) ≃ Hom C ( e Y , X ) , where the third equiv alence is the tensor–hom adjunction in C and the last uses the idemp otence e Y ⊗ e Y ≃ e Y . Under these identifications, the map induced by L corresp onds to the comp osite abov e, hence is an equiv alence. Applying π ∗ sho ws that ϕ # U ( c ) is an isomorphism for every basic op en U ( c ) , and therefore ϕ # is an isomorphism of sheav es. □ 14 DREW HEARD AND MARIUS NIELSEN 5.15. R emark. Because of this theorem, we will usually write Spf ( b C d Y ) instead of the more cumbersome Spf ( b C d Y , ϕ − 1 ( Y )) . W e finish this section with the follo wing locality theorem. 5.16. Theorem. L et K b e a 2-ring and Y ⊆ Sp c ( K ) a Thomason subset. L et U ⊆ Sp c ( K ) b e a quasi-c omp act op en subset, and let q U : K → K ( U ) b e the functor fr om R emark 3.10 . Set Y ′ : = Y ∩ U ⊆ Sp c ( K ( U )) . Then the r estriction of Spf ( K , Y ) to Y ′ identifies with Spf ( K ( U ) , Y ′ ) as a ringe d sp ac e: Spf ( K ( U ) , Y ′ ) ∼ = − → Spf ( K , Y )   Y ′ . Pr o of. Since U is quasi-compact, there exists d ∈ K suc h that U = U ( d ) [ Bal05 , Prop osition 2.14]. Consequen tly , the lo calization K ( U ) is identified with the idem- p oten t completion of the V erdier quotient, ( K [ d − 1 ]) ♮ . By functorialit y ( Lemma 5.9 ) applied to q U , we obtain a morphism ( ϕ, ϕ # ) : Spf ( K ( U ) , Y ′ ) − → Spf ( K , Y ) . On underlying top ological spaces, ϕ is the inclusion Y ′  → Y . Since the image of ϕ lies in the op en set Y ′ , this morphism factors through the restriction Spf ( K , Y ) | Y ′ . T o pro v e that this factorization is an isomorphism, it remains to show that the induced map on structure sheav es is an isomorphism. It suffices to v erify this on a basis of the op en sets of Y ′ . Let V ⊆ Y ′ b e a basic op en set. W e can write V = U ( c ) ∩ Y ′ for some c ∈ K . By replacing c with c ⊕ d (whic h satisfies U ( c ⊕ d ) = U ( c ) ∩ U ( d ) ), we ma y assume without loss of generality that U ( c ) ⊆ U ( d ) = U . Let C : = Ind ( K ) and let L : C → C ( U ) be the lo calization functor, with fully faithful right adjoin t ι . W e compare the sections o v er V : (a) On Spf ( K , Y ) : Since U ( c ) ∩ Y = V (as U ( c ) ⊆ U ), the section is π ∗ Hom C ( e Y , 1 C [ c − 1 ]) . (b) On Spf ( K ( U ) , Y ′ ) : The section is π ∗ Hom C ( U ) ( e Y ′ , 1 C ( U ) [ c − 1 ]) . Since U ( c ) ⊆ U ( d ) , the ob ject 1 C [ c − 1 ] is U -lo cal. Thus, the canonical map 1 C [ c − 1 ] → ι ( 1 C ( U ) [ c − 1 ]) is an equiv alence. Using the adjunction L ⊣ ι and the fact that L ( e Y ) ≃ e Y ∩ U = e Y ′ (see [ BS17 , Prop osition 5.11]), we obtain natural isomorphisms: Hom C ( e Y , 1 C [ c − 1 ]) ∼ = Hom C ( U ) ( L ( e Y ) , 1 C ( U ) [ c − 1 ]) ∼ = Hom C ( U ) ( e Y ′ , 1 C ( U ) [ c − 1 ]) . This shows the map of shea ves is an isomorphism on basic op en sets, completing the pro of. □ 6. The formal Hopkins–Neeman theorem Let R b e a no etherian ring and let I ⊆ R b e an ideal. In Example 4.6 , w e in tro duced the deriv ed category of I -complete R -mo dules, [ D( R ) I , as the completion of D ( R ) along the closed subset V ( I ) ⊆ Sp c ( D ( R ) c ) . In this section, we compute the formal spectrum Spf ( D ( R ) c , V ( I )) and iden tify it with the classical affine formal sc heme Spf ( R ∧ I ) . THE FORMAL SPECTRUM OF A TENSOR-TRIANGULA TED CA TEGOR Y 15 6.1. R emark. W e first recall some properties of the deriv ed I -completion functor d ( − ) I : D ( R ) → [ D( R ) I from Example 4.6 . F or a complex M ∈ D ( R ) , the ob ject c M I is the derive d I -adic completion of M . Ev en if M is a discrete mo dule, c M I ma y differ from the classical I -adic completion. In general, [ GM92 , Prop osition 1.1] pro vides a short exact sequence for any discrete mo dule M : 0 → lim ← − k 1 T or R i +1 ( R/I k , M ) → π i c M I → lim ← − k T or R i ( R/I k , M ) → 0 . This implies the following useful lemma: 6.2. Lemma. If M is a flat discr ete R -mo dule, then the natur al map c M I → M ∧ I is an isomorphism, wher e M ∧ I denotes the or dinary I -adic c ompletion. Our v ersion of the Hopkins–Neeman theorem is the follo wing: 6.3. Theorem. L et R b e a c ommutative no etherian ring and let I ⊆ R b e an ide al. Then ther e is an isomorphism of lo c al ly ringe d sp ac es (and henc e, formal schemes) ρ : Spf  D( R ) c , V ( I )  ∼ − → Spf ( R ∧ I ) induc e d by the c omp arison map of The or em 3.22 . Conse quently, we have an isomor- phism Spf  [ D( R ) d I  ∼ − → Spf ( R ∧ I ) . Pr o of. By Remark 2.5 and Theorem 5.14 , it suffices to establish the isomorphism ρ : Spf  D( R ) c , V ( I )  ∼ − → Spf ( R, I ) . In this context, the stable homotop y theory associated to D( R ) c is D( R ) . On the lev el of top ological spaces, the result follows from Example 3.20 : the isomorphism Sp c ( P erf ( R )) ∼ = Sp ec ( R ) identifies the tensor-triangular support with the classical supp ort of R -mo dules, mapping V ( I ) homeomorphically to itself. It remains to identify the structure shea v es. W e chec k this on the basis of open sets U = D ( s ) = D ( s ) ∩ V ( I ) . Recall from [ Bal10 , Theorem 5.3] that under ρ , the preimage of D ( s ) is the op en set U (cone( s )) . Thus, ρ − 1  D ( s )  = U (cone( s )) ∩ V ( I ) = U (cone( s )) . The ring of sections of the structure sheaf of Spf ( R, I ) ov er this op en set is simply O Spf ( R,I )  D ( s )  = R [1 /s ] ∧ I . On the tensor-triangular side, since R is the tensor unit of D ( R ) , the ring of sections is given b y O Spf (D( R ) c ,V ( I ))  U (cone( s ))  = π ∗ Hom D( R )  R, \ ( 1 [cone( s ) − 1 ]) I  ∼ = π ∗  \ ( 1 [cone( s ) − 1 ]) I  . Here, 1 [ cone ( s ) − 1 ] denotes the tensor unit in the lo calization of D ( R ) at the op en set U ( cone ( s )) . This lo calization corresp onds to the V erdier quotien t by the sub category generated by cone ( s ) . As shown in [ HPS97 , Theorem 3.3.7], this quotient identifies with D ( R [1 /s ]) as tensor-triangulated categories. Under this equiv alence, the lo calized unit 1 [ cone ( s ) − 1 ] corresp onds to the ring R [1 /s ] . Therefore, w e hav e an equiv alence of completions \ ( 1 [cone( s ) − 1 ]) I ≃ \ R [1 /s ] I . 16 DREW HEARD AND MARIUS NIELSEN Since R [1 /s ] is a flat R -mo dule, its deriv ed completion coincides with its classical completion by Lemma 6.2 . Combining these results yields a natural isomorphism O Spf (D( R ) c ,V ( I ))  U (cone( s ))  ∼ = R [1 /s ] ∧ I ∼ = O Spf ( R,I )  D ( s )  . These local isomorphisms are compatible with restriction, so they glue to an isomor- phism of structure sheav es. Th us, ρ is an isomorphism of ringed spaces. Finally , since Spf ( R, I ) is a lo cally ringed space, the isomorphism means that Spf ( D ( R ) c , V ( I )) is as well, and b y definition, this means it is an isomorphism of formal schemes. □ 6.4. R emark. This result will also follow from results in Section 8 , in particular Prop osition 8.6 . Ho w ev er we believe it is still instructiv e to do this sp ecial case. 7. Globalizing to schemes W e can now globalize the formal Hopkins–Neeman theorem ( Theorem 6.3 ) to arbitrary no etherian schemes. W e b egin by recalling the notion of completion in the setting of schemes. 7.1. Definition. Let X b e a no etherian scheme, and let Y ⊆ | X | b e a closed subset. An ide al of definition for Y is a quasi-coherent ideal sheaf J ⊆ O X of finite type suc h that | V ( J ) | = Y . 7.2. Definition. Let X b e a quasi-compact quasi-separated no etherian scheme, and Y ⊆ | X | a closed subset with ideal of definition J . F or n ≥ 0 , set X n : = ( V ( J n +1 ) , O X / J n +1 ) . The colimit in the category of lo cally ringed spaces b X Y : = lim − → n X n is called the formal c ompletion of X along Y . 7.3. R emark. Explicitly , b X Y is the lo cally ringed space whose underlying top ological space is Y , equipp ed with the structure sheaf O b X Y : = lim ← − n O X / J n +1 . By [ GD71 , Prop osition 10.8.5], b X Y is a formal scheme. 7.4. Example. Let X = Sp ec ( R ) and let Y ⊆ | Sp ec ( R ) | b e a closed subset. Then there exists an ideal I ⊆ R suc h that V ( I ) = Y , and one has natural isomorphisms b X Y ∼ = Spf ( R, I ) ∼ = Spf ( R ∧ I ) . 7.5. Definition. A formal scheme is algebr aizable if it is of the form b X Y for some no etherian sc heme X and a closed subset Y ⊆ | X | . 7.6. R emark. In the rest of this section, we will restrict our atten tion on algebraizable formal sc hemes. Ev en in the no etherian case, there exist non-algebraizable formal sc hemes, although they are difficult to construct; see [ HM68 , Section 5]. 7.7. Theorem. L et X b e a no etherian scheme, and let Z ⊆ X b e a close d subset. L et b X Z denote the formal c ompletion of X along Z . Then ther e is an isomorphism of lo c al ly ringe d sp ac es (and henc e formal schemes) Φ : Spf  D qc ( X ) c , Z  ∼ − → b X Z . THE FORMAL SPECTRUM OF A TENSOR-TRIANGULA TED CA TEGOR Y 17 Pr o of. Let { U α } b e a finite affine open co v er of X , where U α = Sp ec ( R α ) . Since U α is affine, it is a quasi-compact op en subset of X . Let Z α : = Z ∩ U α b e the restriction of the closed subset to the affine patch; note that Z α = V ( I α ) for some ideal I α ⊆ R α . By the definition of the formal completion of a scheme, b X Z is obtained by gluing the affine formal schemes ( b X Z ) | U α ∼ = Spf ( R α , I α ) . On the tensor-triangular side, we iden tify Sp c ( D qc ( X ) c ) ∼ = X as top ological spaces ( Example 3.20 ). Since U α is a quasi-compact open set, we ma y apply the lo cality theorem ( Theorem 5.16 ): Spf  D qc ( X ) c , Z    Z α ∼ = Spf  D qc ( X ) c ( U α ) , Z α  . where Z α : = Z ∩ U α . Recall from Remark 3.10 that D qc ( X ) c ( U α ) denotes the Karoubi quotien t. By the Thomason–T robaugh lo calization theorem [ TT90 , Theorem 5.2.2] (see also [ Bal02 , Theorem 2.13]), there is an equiv alence D qc ( X ) c ( U α ) ≃ D qc ( U α ) c ≃ D( R α ) c . Th us, we ha ve an isomorphism of formal schemes Spf  D qc ( X ) c , Z    Z α ∼ = Spf  D( R α ) c , V ( I α )  . No w we apply the affine formal Hopkins–Neeman theorem ( Theorem 6.3 ). This pro vides an isomorphism of ringed spaces ρ α : Spf  D( R α ) c , V ( I α )  ∼ − → Spf ( R α , I α ) . Com bining these identifications, w e obtain local isomorphisms φ α : Spf  D qc ( X ) c , Z    Z α ∼ − → ( b X Z ) | U α . T o verify that these glue, consider an in tersection U αβ = U α ∩ U β . Since X is qcqs, this in tersection is quasi-compact. The comparison map ρ in Theorem 6.3 is natural with resp ect to localization, and the lo cality theorem is functorial with respect to restriction. Consequen tly , the isomorphisms φ α and φ β satisfy the cocycle condition on the ov erlaps U αβ . Therefore, they glue to a global isomorphism of ringed spaces. The argument is then completed the same w a y as in Theorem 6.3 . □ 7.8. R emark. One can ask for extensions b eyond schemes (e.g., to certain alge- braic stac ks or other lo cally affine tensor-triangular situations). Conceptually , the argumen t abov e only uses (i) an affine iden tification (a formal Hopkins–Neeman statemen t), (ii) a locality statemen t for Spf under restriction to quasi-compact op ens, via Theorem 5.16 and (iii) a Thomason–T robaugh type lo calization theorem iden tifying the relev an t V erdier quotients with the corresp onding “affine” categories on op ens. Any setting in whic h analogues of (i)–(iii) hold admits the same gluing argumen t. 7.9. Definition. Let b X Z denote the formal completion of X along Z as ab o v e. W e set D ∧ qc ( b X Z ) : = \ D qc ( X ) Z , the completion of D qc ( X ) at the Thomason subset Z ⊆ Sp c ( D qc ( X ) c ) ∼ = X in the sense of Definition 4.3 . 18 DREW HEARD AND MARIUS NIELSEN 7.10. Corollary . L et X b e a no etherian scheme and let Z ⊆ X b e a close d subset. L et b X Z denote the formal c ompletion of X along Z . Then ther e is an isomorphism of ringe d sp ac es (and henc e formal schemes) Φ : Spf  D ∧ qc ( b X Z ) d  ∼ − → b X Z . 7.11. R emark. Recall from Remark 4.4 that completion at a Thomason subset Z ⊆ Sp c(D qc ( X ) c ) ∼ = X is given b y the endofunctor d ( − ) Z ≃ hom ( e Z , − ) , where e Z is the Balmer–F avi idempotent associated to Z . When X is separated, one can identify this idempotent with a familiar geometric ob ject: there is an equiv alence e Z ≃ R Γ Z ( O X ) in D qc ( X ) , where R Γ Z denotes the righ t derived functor of Γ Z , the subsheaf with supp orts in Z , see [ Ste18 , Lemma 5.4]. In particular, in this case the abstract completion functor hom ( e Z , − ) agrees with the derived completion (lo cal homology) functor of Alonso T arrío–Jeremías Lóp ez–Lipman: by [ A TJLL97 , Theorem (0.3)], the righ t adjoin t to R Γ Z is naturally isomorphic to L Λ Z , and under the identification e Z ≃ R Γ Z ( O X ) this yields a natural equiv alence of endofunctors on D qc ( X ) d ( − ) Z ≃ hom ( R Γ Z ( O X ) , − ) ≃ L Λ Z ( − ) . 7.12. R emark. W e briefly indicate how Definition 7.9 compares with the triangulated category denoted D ∧ qc ( b X Z ) in [ A TJLL99 , §6.3], at least when X is separated. This comparison is not used elsewhere in the pap er. W rite e D ∧ qc ( b X Z ) for the category constructed in [ A TJLL99 , Remark 6.3.1]. Alonso– Jeremías–Lipman show that on the formal scheme b X Z the “complete” and “torsion” v ariants agree, and identify the torsion category on b X Z with the Z -supp orted sub category on X : e D ∧ qc ( b X Z ) ≃ D qct ( b X Z ) ≃ D qc Z ( X ) , where D qc Z ( X ) denotes the essential image of R Γ Z on D qc ( X ) . (F or precise state- men ts and hypotheses, see [ A TJLL99 , §6.3] and [ A TJLL99 , Prop osition 5.2.4].) On the other hand, Remark 7.11 iden tifies our completion functor d ( − ) Z with L Λ Z (when X is separated). A form of Greenlees–Ma y dualit y iden tifies the torsion and complete pieces as equiv alent subcategories of D qc ( X ) , so the essen tial image of L Λ Z agrees with the completion sub category \ D qc ( X ) Z of Definition 7.9 . Under these identifications, one obtains an equiv alence e D ∧ qc ( b X Z ) ≃ \ D qc ( X ) Z . In fact, the form ulas in [ A TJLL99 , Prop osition 5.24] implies that this equiv alence is giv en by L Λ Z ◦ k ∗ where k : b X Z → X is the completion map. 8. A comp arison map One of the most important tools in computing the Balmer sp ectrum of a tensor- triangulated category is Balmer’s comparison map ( Theorem 3.22 ). A general strategy for computing Sp c ( K ) is to first determine Sp ec h ( π ∗ Hom K ( 1 , 1 )) and then compute the fibers of the comparison map; see [ BS17 ] and [ ABHS25 ] for THE FORMAL SPECTRUM OF A TENSOR-TRIANGULA TED CA TEGOR Y 19 examples. It is therefore natural to seek an analogue of the comparison map for the formal sp ectrum in tro duced in this paper. Throughout this section w e assume R ∗ : = π ∗ Hom K ( 1 , 1 ) is noetherian, whic h implies that ρ is surjective [ Bal10 , Theorem 7.3]. In this section w e will construct a comparison map for ‘algebraic’ completions, i.e., those coming from an ideal in R ∗ . 8.1. Definition. Let K b e a 2 - Ring and let I ⊆ R ∗ b e a homogeneous ideal. W rite Y ( I ) : = ρ − 1 ( V ( I )) for the preimage of the closed subset V ( I ) ⊆ Spec h ( R ∗ ) under the comparison map of Theorem 3.22 . Define b C I : = b C Y ( I ) to be the completion of C at the closed subset Y ( I ) ⊆ Sp c ( K ) in the sense of Definition 4.3 . 8.2. Example. If K = P erf ( R ) for a commutativ e noetherian ring R , then ρ is a homeomorphism, and we are exactly in the situation of Example 4.6 . 8.3. R emark. The spectrum R : = Hom C ( 1 , 1 ) is a comm utativ e algebra in spectra and, for any X ∈ C , Hom C ( 1 , X ) is an R -mo dule. By Morita theory there is a symmetric monoidal adjunction (8.4) F : Mo d R C : G with F ( M ) = M ⊗ R 1 and G ( X ) = Hom C ( 1 , X ) . Since C = Ind ( K ) is rigidly- compactly generated and 1 is compact, F is fully faithful and G exhibits Mo d R as a colo calization of C . Let ρ K : Sp c ( K ) → Spec h ( R ∗ ) and ρ R : Sp c ( Mo d c R ) → Spec h ( R ∗ ) b e the resp ec- tiv e comparison maps. F unctorialit y of the comparison map [ Bal10 , Theorem 5.3] yields a commutativ e diagram Sp c( K ) Sp c(Mod c R ) Sp ec h ( R ∗ ) Sp ec h ( R ∗ ) ρ K Spc( F ) ρ R and hence, for any homogeneous ideal I ⊆ R ∗ , the asso ciated Thomason subsets Y ( I ) : = ρ − 1 K  V ( I )  ⊆ Spc( K ) , Y R ( I ) : = ρ − 1 R  V ( I )  ⊆ Spc(Mo d c R ) satisfy Y ( I ) = Sp c( F ) − 1  Y R ( I )  . Let e Y ( I ) ∈ C and e Y R ( I ) ∈ Mo d R b e the associated tensor idemp oten ts. By functorialit y [ BS17 , Prop osition 5.11] we ha v e F ( e Y R ( I ) ) ∼ = e Y ( I ) , and since F is fully faithful, G ( e Y ( I ) ) ∼ = e Y R ( I ) in Mo d R . 8.5. Prop osition. F or every X ∈ C ther e is a natur al isomorphism of R -mo dules Hom C ( e Y ( I ) , X ) ∼ = Hom Mod R  e Y R ( I ) , Hom C ( 1 , X )  . Pr o of. Using F ( e Y R ( I ) ) ∼ = e Y ( I ) and the adjunction ( 8.4 ), Hom C ( e Y ( I ) , X ) ∼ = Hom C  F ( e Y R ( I ) ) , X  ∼ = Hom Mod R  e Y R ( I ) , Hom C ( 1 , X )  as required. □ 20 DREW HEARD AND MARIUS NIELSEN 8.6. Prop osition. L et K ∈ 2 - Ring with R ∗ no etherian, and let I ⊆ R ∗ b e a homo- gene ous ide al. Set Y ( I ) : = ρ − 1 ( V ( I )) ⊆ Spc ( K ) , wher e ρ : Sp c ( K ) → Spec h ( R ∗ ) is Balmer’s c omp arison map of The or em 3.22 . Then ther e is a natur al morphism of lo c al ly ringe d sp ac es ρ : Spf ( K , Y ( I )) − → Spf h ( π ∗ R, I ) , whose underlying map of sp ac es is the r estriction ρ | Y ( I ) : Y ( I ) → V ( I ) . In p articular, if ρ is an isomorphism, then so is ρ . Pr o of. On spaces, ρ ( Y ( I )) = ρ ( ρ − 1 ( V ( I ))) = V ( I ) . F or the sheav es, it suffices to construct compatible maps on a basis of opens of Spf h ( R ∗ , I ) , namely D ( s ) = D ( s ) ∩ V ( I ) with s ∈ R ∗ . By [ Bal10 , Theorem 5.3(b)], ρ − 1 ( D ( s )) = U ( cone ( s )) , hence ρ − 1  D ( s )  = Y ( I ) ∩ ρ − 1 ( D ( s )) = U (cone( s )) . By definition, O Spf h ( R ∗ ,I ) ( D ( s )) = R ∗ [1 /s ] ∧ I . On the other hand, by Remark 5.3 and Prop osition 8.5 , O Spf ( K ,Y ( I ))  U (cone( s ))  ∼ = π ∗ Hom Mod R  e Y R ( I ) , Hom C ( 1 , 1 [1 /s ])  . The right-hand side is the deriv ed I -adic completion of the R -mo dule Hom C ( 1 , 1 [1 /s ]) , computed by the Greenlees–Ma y sp ectral sequence [ GM95 , Eq. (3.3), Theorem 4.2] with E p,q 2 = H I − p, − q  π ∗ Hom C ( 1 , 1 [1 /s ])  Note that H I − p, ∗ ( − ) denote the lo cal homology groups of [ GM92 ]. Since π ∗ Hom C ( 1 , 1 [1 /s ]) ∼ = R ∗ [1 /s ] is R ∗ -flat, [ GM92 , Theorem 2.5 and 4.1] imply that these terms v anish for p  = 0 , so that the spectral sequence collapses and O Spf ( K ,Y ( I ))  U (cone( s ))  ∼ = R ∗ [1 /s ] ∧ I = O Spf h ( R ∗ ,I ) ( D ( s )) . These identifications are natural in s and compatible with restriction, hence glue to ρ # : O Spf h ( R ∗ ,I ) → ρ ∗ O Spf ( K ,Y ( I )) . If ρ is an isomorphism, the same local id en tifica- tions show ρ is an isomorphism of ringed spaces. □ 8.7. Corollary . Under the hyp otheses ab ove, ther e is a natur al morphism of ringe d sp ac es b ρ : Spf  b C d Y  − → Spf h  ( R ∗ ) ∧ I  . Pr o of. Com bine Theorem 5.14 with Remark 2.5 and Prop osition 8.6 . □ 8.8. Example. Let A b e an even p eriodic comm utativ e ring spectrum with π 0 ( A ) a complete regular lo cal ring with maximal ideal m . By [ DS16 , Theorem 1.1] the comparison map ρ : Sp c(Perf ( A )) − → Sp ec h ( π ∗ A ) ∼ = Sp ec( π 0 A ) is a homeomorphism. In fact, b y [ Bal10 , Prop osition 6.11], this is ev en an isomor- phism of ringed spaces. Let κ ( A ) : = A/ m (so that π 0 κ ( A ) ∼ = π 0 ( A ) / m ). This is an A -mo dule with supp ( κ ( A )) = V ( m ) . Applying Proposition 8.6 and Corollary 8.7 with I = m w e obtain a homeomorphism Spf  ( L κ ( A ) Mo d A ) d  ∼ = − → Spf ( π 0 A ) . Here w e use Remark 4.5 to iden tify the completion functor as a Bousfield lo calization in this case. THE FORMAL SPECTRUM OF A TENSOR-TRIANGULA TED CA TEGOR Y 21 9. Chr oma tic homotopy W e no w turn to examples coming from c hromatic homotop y theory . As usual, w e fix a prime num b er p throughout. 9.1. R emark. Recall that the Brown–P eterson spectrum B P has co efficient ring B P ∗ ∼ = Z ( p ) [ v 1 , v 2 , . . . ] . Johnson–Wilson theory E ( n ) is obtained from B P b y first quotienting b y the ideal ( v n +1 , v n +2 , . . . ) and then in v erting v n (using, for example, highly structured ring sp ectra [ EKMM97 ]). Its coefficients are E ( n ) ∗ ∼ = v − 1 n B P ∗ / ( v n +1 , v n +2 , . . . ) . More generally , for an inv ariant regular sequence J k = ( p i 0 , v i 1 1 , . . . , v i k − 1 k − 1 ) , w e can form a quotient B P /J k with π ∗ ( B P /J k ) ∼ = B P ∗ /J k , and for 0 ≤ n ≤ k define a sp ectrum E ( n, J k ) with E ( n, J k ) ∗ ∼ = v − 1 n  ( B P /J k ) ∗ / ( v n +1 , v n +2 , . . . )  . F or k = 0 this reco vers E ( n ) , while for J k = ( p, v 1 , . . . , v n − 1 ) we ha v e E ( n, J k ) ≃ K ( n ) , Morav a K -theory . Moreov er, by [ Hea23 , Prop osition 2.10], there is an equiv alence of Bousfield classes ⟨ E ( n, J k ) ⟩ = ⟨ K ( k ) ∨ · · · ∨ K ( n ) ⟩ , hence S p E ( n,J k ) ≃ S p K ( k ) ∨···∨ K ( n ) . 9.2. Notation. W rite S p n : = S p E ( n ) for the E ( n ) -lo cal stable homotopy category , S p c n for its subcategory of compact ob jects, and L n for Bousfield localization at E ( n ) . F or h ≥ 0 let K ( h ) denote Morav a K -theory and fix a finite t ype ( h + 1) -sp ectrum F ( h + 1) . 9.3. Definition. F or 0 ≤ h ≤ n , define the ⊗ -ideal P h : =  x ∈ S p c n   K ( h ) ∗ ( x ) = 0  . 9.4. R emark. Since K ( h ) ∗ ( x ⊗ y ) ∼ = K ( h ) ∗ ( x ) ⊗ K ( h ) ∗ K ( h ) ∗ ( y ) , each P h is prime. Moreo ver, P h = thic kid ⟨ L n F ( h + 1) ⟩ . 9.5. Theorem (Hov ey–Stric kland) . The sp e ctrum of the E ( n ) -lo c al c ate gory is the finite chain Sp c( S p c n ) = { P 0 |{z} generic ⊊ P 1 ⊊ · · · ⊊ P n |{z} close d } . The top olo gy is define d by the closur e op er ator { P h } =  P k   h ≤ k ≤ n  . F or 0 ≤ k ≤ n , c onsider the b asic op en set define d by the typ e ( k + 1) -sp e ctrum F ( k + 1) : U k : = U  L n F ( k + 1)  =  P i   L n F ( k + 1) ∈ P i  =  P i   0 ≤ i ≤ k  . The value of the structur e she af on this op en set is O Spc( S p c n ) ( U k ) ∼ = π ∗ L k S 0 . 22 DREW HEARD AND MARIUS NIELSEN Pr o of. The top ological description is due to [ HS99 , Theorem 6.9]; see also [ BHN22 , Prop osition 3.5] for a translation into this language. F or the sheaf, note that by [ HS99 , Prop osition 6.10] the finite localization of S p n a wa y from L n F ( k + 1) identifies the lo calized category S p n [ L n F ( k + 1) − 1 ] with S p k . Under this identification the localized tensor unit corresp onds to L k S 0 . Therefore, O Spc( S p c n ) ( U k ) ∼ = π ∗ End S p k ( L k S 0 ) ∼ = π ∗ L k S 0 . □ 9.6. Definition. F or a fixed heigh t h < n , let Y h +1 b e the closed subset corresp onding to sp ectra of t yp e ≥ h + 1 : Y h +1 : = supp( L n F ( h + 1)) = { P h +1 , . . . , P n } . 9.7. R emark. By Remark 4.5 , the completion ( d S p n ) Y h +1 is equiv alent to Bousfield lo calization of S p n at L n F ( h + 1) . By [ BHV18 , Lemma 6.16], this is equiv alent to lo calization of S p at F ( h + 1) ⊗ E ( n ) , which b y [ Hea23 , Prop osition 2.10] is equiv alent to S p K ( h +1) ∨···∨ K ( n ) (the sp ecial case where h = n − 1 is considered in [ BS25 , Prop osition 8.11]). Moreo v er, [ Hea23 , Corollary 2.29] gives L n F ( h + 1) ≃ L K ( h +1) ∨···∨ K ( n ) F ( h + 1) . By Theorem 5.14 , we therefore obtain an isomorphism of ringed spaces Spf ( S p d n , Y h +1 ) ∼ = Spf  S p d K ( h +1) ∨···∨ K ( n )  . 9.8. Theorem. F or 0 ≤ h ≤ n − 1 , the formal sp e ctrum Spf  S p d K ( h +1) ∨···∨ K ( n )  has underlying top olo gic al sp ac e Y h +1 = { P h +1 ⊊ P h +2 ⊊ · · · ⊊ P n } ⊆ Sp c( S p c n ) . F or k ≥ h + 1 , on the b asic op en set U k : = U ( L n F ( k + 1)) ∩ Y h +1 =  P i   h + 1 ≤ i ≤ k  , the structur e she af takes the value 1 O  U k  ∼ = π ∗ L K ( h +1) ∨···∨ K ( k ) S 0 . Pr o of. The underlying space statemen t follows from Theorem 9.5 and the definition of Y h +1 . F or k ≥ h + 1 , the Bousfield class argument in Remark 9.7 sho ws that O  U k  ∼ = π ∗ Hom S p n  L n S 0 , L K ( h +1) ∨···∨ K ( n ) L k S 0  ∼ = π ∗ L K ( h +1) ∨···∨ K ( n ) L k S 0 . It remains to show that L K ( h +1) ∨···∨ K ( n ) L k S 0 ≃ L K ( h +1) ∨···∨ K ( k ) S 0 . By [ Hea23 , Remark 2.27], there is a pullback square L K ( h +1) ∨···∨ K ( n ) L k S 0 L K ( h +1) ∨···∨ K ( k ) L k S 0 L K ( k +1) ∨···∨ K ( n ) L k S 0 L K ( h +1) ∨···∨ K ( k ) L K ( k +1) ∨···∨ K ( n ) L k S 0 . Since ⟨ E ( k ) ⟩ = ⟨ K (0) ∨ · · · ∨ K ( k ) ⟩ and K ( i ) ∧ E ( k ) ≃ 0 for i > k , w e hav e L K ( k +1) ∨···∨ K ( n ) L k S 0 ≃ 0 . Hence the b ottom-right corner also v anishes, b eing a 1 W e omit the subscript on the O for reasons of space. THE FORMAL SPECTRUM OF A TENSOR-TRIANGULA TED CA TEGOR Y 23 further lo calization of the b ottom-left corner. Therefore, the pullback iden tifies the upp er-left corner with the upp er-righ t corner: L K ( h +1) ∨···∨ K ( n ) L k S 0 ≃ L K ( h +1) ∨···∨ K ( k ) L k S 0 . Finally , since ⟨ K ( h + 1) ∨ · · · ∨ K ( k ) ⟩ ≤ ⟨ E ( k ) ⟩ , lo calization at E ( k ) do es not affect K ( h + 1) ∨ · · · ∨ K ( k ) -lo calization, and w e obtain L K ( h +1) ∨···∨ K ( k ) L k S 0 ≃ L K ( h +1) ∨···∨ K ( k ) S 0 . Com bining these equiv alences yields the claimed iden tification, and hence O  U k  ∼ = π ∗ L K ( h +1) ∨···∨ K ( k ) S 0 , as desired. □ 9.9. R emark. Unsurprisingly , the identification of the sheaf of sp ectra considered in Remark 5.4 takes the form O  U k  ∼ = L K ( h +1) ∨···∨ K ( k ) S 0 , 10. Fur ther examples In this short section, w e consider tw o further examples, from equiv ariant homotopy theory and mo dular represen tation theory . 10.1. Notation. Let G b e a finite group and p a prime. W e write S p cofree G, ( p ) for the Bousfield lo calization of S p G, ( p ) at G + , or equiv alen tly , the completion of S p G, ( p ) at the Thomason subset Y G : = supp( G + ) ⊆ Spc( S p c G, ( p ) ) . As a set, the sp ectrum Sp c ( S p c G, ( p ) ) and the subset Y G are computed in [ BS17 ]. Moreo ver, [ BS17 , Corollary 4.13] shows that restriction induces a homeomorphism from Sp c( S p c ( p ) ) onto Y G . 10.2. Prop osition. L et G b e a finite gr oup and p a prime. R estriction along the inclusion e ≤ G defines an exact symmetric monoidal functor res G e : S p G, ( p ) − → S p ( p ) . L et Y G : = supp ( G + ) ⊆ Spc ( S p c G, ( p ) ) . Then res G e induc es a morphism of ringe d sp ac es ( ϕ, ϕ # ) : Sp c( S p c ( p ) ) − → Spf ( S p cofree , d G, ( p ) ) . Mor e over, ϕ is a home omorphism of the underlying top olo gic al sp ac es. However, for G = C 2 and p = 2 , this morphism is not an isomorphism of ringe d sp ac es. Pr o of. Apply Lemma 5.9 to the restriction functor betw een comp act ob jects: res G e : S p c G, ( p ) − → S p c ( p ) . Let Y : = Sp c( S p c ( p ) ) . Then Spc(res G e ) is a contin uous map Sp c(res G e ) : Sp c( S p c ( p ) ) − → Sp c( S p c G, ( p ) ) , and w e set Y ′ : = Sp c ( res G e )( Y ) ⊆ Sp c ( S p c G, ( p ) ) . By [ BS17 , Corollary 4.13], this image is precisely Y ′ = supp( G + ) = Y G . 24 DREW HEARD AND MARIUS NIELSEN Therefore, Lemma 5.9 yields a morphism of ringed spaces ( ϕ, ϕ # ) : Spf ( S p c ( p ) , Y ) − → Spf ( S p c G, ( p ) , Y G ) . By Example 5.7 , the domain is Spf ( S p c ( p ) , Y ) ∼ = Sp c ( S p c ( p ) ) , and by Theorem 5.14 , the target iden tifies with Spf ( S p cofree , d G, ( p ) ) , yielding the stated map. The fact that ϕ is a homeomorphism on underlying spaces is exactly [ BS17 , Corollary 4.13]. Finally , we sho w this is not an isomorphism for G = C 2 and p = 2 by ev aluating global sections. On the nonequiv ariant side (the source), w e ha ve O Spc( S p c (2) )  Sp c( S p c (2) )  ∼ = π ∗ End S p (2) ( S (2) ) ∼ = π ∗ S (2) . In degree 0 , this is the 2 -lo cal integers Z (2) . On the cofree side (the target), using Remark 5.6 (and the identification of the whole space with Y G ), we ha ve O Spf ( S p cofree C 2 , (2) ) ( Y G ) ∼ = π ∗ Hom S p C 2 , (2) ( e Y G , S (2) ) . The Segal conjecture for C 2 (sp ecifically Lin’s theorem [ Lin80 , Theorem 1.1]) iden ti- fies the degree 0 part of this ring with the I -adic completion of the Burnside ring A ( C 2 ) , where I is the augmen tation ideal. Since the t w o rings are not isomorphic (for example, the latter contains the 2 -adic integers Z 2 as a subring, and so is uncoun table), the induced map on degree- 0 global sections is not an isomorphism. Therefore, ( ϕ, ϕ # ) is not an isomorphism of ringed spaces. □ 10.3. R emark. Our final example comes from mo dular representation theory . Fix a finite group G and a field k of characteristic p dividing | G | . Let T : = K ( Inj k G ) b e the homotop y category of complexes of injectiv e k G -mo dules (equiv alen tly , one may mo del the corresp onding stable homotop y theory by the ∞ -category of mo dules T ≃ Mo d S p G ( k ) , where k denotes the Borel-equiv ariant G -sp ectrum associated to k .). This is a symmetric monoid al, rigidly-compactly generated category , and its compact/dualizable ob jects iden tify with the bounded derived category: T d ≃ D b (Mo d k G ) . By [ BCR97 ] and [ Bal10 , Prop osition 8.5], the comparison map ρ : Sp c  D b (Mo d k G )  − → Sp ec h  H • ( G, k )  is a homeomorphism. The homogeneous sp ectrum Sp ec h ( H • ( G, k )) has a unique closed point, corresponding to the homogeneous maximal ideal m : = H > 0 ( G, k ) . Let Y m ⊆ Spc(D b (Mo d k G )) denote the corresp onding singleton. The tt-completion of T at Y m iden tifies with the unbounded derived category D ( Mo d k G ) equipp ed with the diagonal tensor product; see [ BS25 , Example 3.29]. In this category , the dualizable ob jects are again D b ( Mo d k G ) , although the compact ob jects need not coincide with the dualizable ones. In particular, the completed category is typically not rigidly-compactly generated. Using Prop osition 8.6 we deduce the following: 10.4. Theorem. With notation as ab ove, the formal sp e ctrum Spf  D b ( Mo d k G )  is a one-p oint ringe d sp ac e, and its ring of glob al se ctions is H • ( G, k ) ∧ m . THE FORMAL SPECTRUM OF A TENSOR-TRIANGULA TED CA TEGOR Y 25 References [ABC + 25] Ko Aoki, T obias Barthel, Anish Chedala v ada, T omer Schlank, and Greg Stevenson. Higher zariski geometry , 2025. [ABHS25] Gregory Arone, T obias Barthel, Drew Heard, and Beren Sanders. The spectrum of excisive functors. Invent. Math. , 241(2):363–464, 2025. [A TJLL97] Leovigildo Alonso T arrío, Ana Jeremías Lóp ez, and Joseph Lipman. Lo cal homology and cohomology on schemes. Ann. Sci. Éc ole Norm. Sup. (4) , 30(1):1–39, 1997. [A TJLL99] Leovigildo Alonso T arrío, Ana Jeremías Lóp ez, and Joseph Lipman. Duality and flat base change on formal schemes. In Studies in duality on no etherian formal schemes and non-no etherian or dinary schemes , volume 244 of Contemp or ary Mathematics , pages 3–90. American Mathematical So ciety , Pro vidence, RI, 1999. A v ailable at http://www.ams.org/books/conm/244/ . [Bal02] P . Balmer. Presheav es of triangulated categories and reconstruction of schemes. Math. Ann. , 324(3):557–580, 2002. [Bal05] Paul Balmer. The sp ectrum of prime ideals in tensor triangulated categories. J. R eine Angew. Math. , 588:149–168, 2005. [Bal10] Paul Balmer. Spectra, sp ectra, sp ectra – tensor triangular sp ectra versus Zariski spectra of endomorphism rings. Algebr. Ge om. T opol. , 10(3):1521–1563, 2010. [BCH + 24] T obias Barthel, Natàlia Castellana, Drew Heard, Niko Naumann, Luca P ol, and Beren Sanders. Descent in tensor triangular geometry . In T riangulated c ategories in repr esentation theory and beyond—the Ab el Symp osium 2022 , volume 17 of Ab el Symp. , pages 1–56. Springer, Cham, 2024. [BCR97] Dav e Benson, Jon F. Carlson, and Jeremy Rick ard. Thick sub categories of the stable module category . F und. Math. , 153(1):59–80, 1997. [BF11] Paul Balmer and Giordano F avi. Generalized tensor idemp otents and the telescop e conjecture. Pr oc. L ond. Math. Soc. (3) , 102(6):1161–1185, 2011. [BGT13] Andrew J. Blumberg, David Gepner, and Gonçalo T abuada. A universal characteriza- tion of higher algebraic K -theory . Ge om. T op ol. , 17(2):733–838, 2013. [BHN22] T obias Barthel, Drew Heard, and Nik o Naumann. On conjectures of Hov ey-Strickland and Chai. Sele cta Math. (N.S.) , 28(3), 2022. [BHV18] T obias Barthel, Drew Heard, and Gabriel V alenzuela. Lo cal duality in algebra and topology . A dv. Math. , 335:563–663, 2018. [BKS07] Aslak Bakke Buan, Henning Krause, and Øyvind Solb erg. Supp ort v arieties: an ideal approach. Homolo gy, Homotopy Appl. , 9(1):45–74, 2007. [BMCSY25] Shay Ben-Moshe, Shachar Carmeli, T omer Schlank, and Lior Y anovski. Descent and cyclotomic redshift for chromatically lo calized algebraic K -theory . J. A mer. Math. So c. , 38(2):521–583, 2025. [BS17] Paul Balmer and Beren Sanders. The sp ectrum of the equiv ariant stable homotopy category of a finite group. Invent. Math. , 208(1):283–326, 2017. [BS24] Paul Balmer and Beren Sanders. P erfect complexes and completion. Preprint, 20 pages, av ailable online at , 2024. [BS25] Paul Balmer and Beren Sanders. The Tate Intermediate Value Theorem. A dvances in Mathematics , 483:110675, 2025. [CHL + 26] Baptiste Calmès, Y onatan Harpaz, Markus Land, Denis Nardin, W olfgang Steimle, Emanuele Dotto, F abian Heb estreit, Kristian Moi, and Thomas Nikolaus. Hermitian K-theory for stable ∞ -categories I I: Cobordism categories and additivity . A cta Math. , 235(2):149–400, 2026. [DG02] W. G. Dwyer and J. P . C. Greenlees. Complete mo dules and torsion mo dules. Amer. J. Math. , 124(1):199–220, 2002. [DS16] Ivo Dell’Ambrogio and Donald Stanley . Affine weakly regular tensor triangulated categories. Pacific J. Math. , 285(1):93–109, 2016. [EKMM97] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P . May . R ings, mo dules, and algebr as in stable homotopy the ory , volume 47 of Mathematic al Surveys and Mono gr aphs . American Mathematical So ciety , Providence, RI, 1997. With an app endix by M. Cole. [GD71] A. Grothendieck and J. A. Dieudonné. Éléments de gé ométrie algébrique. I , volume 166 of Grund lehren der mathematischen Wissenschaften [F undamental Principles of Mathematic al Scienc es] . Springer-V erlag, Berlin, 1971. 26 DREW HEARD AND MARIUS NIELSEN [GM92] J. P . C. Greenlees and J. P . May . Deriv ed functors of I -adic completion and lo cal homology . J. Algebr a , 149(2):438–453, 1992. [GM95] J. P . C. Greenlees and J. P . May . Completions in algebra and top ology . In Handb o ok of algebraic top olo gy , pages 255–276. North-Holland, Amsterdam, 1995. [Gre01] J. P . C. Greenlees. T ate cohomology in axiomatic stable homotop y theory . In Co- homolo gical methods in homotopy the ory (Bel laterr a, 1998) , volume 196 of Pr o gr. Math. , pages 149–176. Birkhäuser, Basel, 2001. [Hea23] Drew Heard. The Sp k,n -local stable homotopy category . Algebr. Ge om. T op ol. , 23(8):3655–3706, 2023. [HM68] Heisuke Hironak a and Hideyuki Matsumura. F ormal functions and formal em beddings. J. Math. So c. Jap an , 20:52–82, 1968. [Hop87] Michael J. Hopkins. Global methods in homotopy theory . In Homotopy the ory (Durham, 1985) , volume 117 of LMS L ect. Note , pages 73–96. Cambridge Univ. Press, 1987. [HPS97] Mark Ho vey , John H. Palmieri, and Neil P . Strickland. Axiomatic stable homotopy theory . Mem. Amer. Math. So c. , 128(610), 1997. [HS99] Mark Hovey and Neil P . Strickland. Morav a K -theories and lo calisation. Mem. Amer. Math. Soc. , 139(666):viii+100, 1999. [Kra23] Henning Krause. Completions of triangulated categories. A v ailable online at https: //arxiv.org/abs/2309.01260 , 2023. [Lin80] W en Hsiung Lin. On conjectures of Mahow ald, Segal and Sullivan. Math. Pr o c. Cambridge Philos. So c. , 87(3):449–458, 1980. [Lur09] Jacob Lurie. Higher top os the ory , volume 170 of Annals of Mathematics Studies . Princeton Universit y Press, Princeton, NJ, 2009. [Lur17] Jacob Lurie. Higher algebra. 1553 pages, av ailable from the author’s website, 2017. [MP12] J. P . Ma y and K. P onto. Mor e c oncise algebr aic top olo gy . Chicago Lectures in Mathematics. Univ ersity of Chicago Press, Chicago, IL, 2012. Localization, completion, and model categories. [Nee92a] Amnon Neeman. The chromatic tow er for D ( R ) . T op olo gy , 31(3):519–532, 1992. [Nee92b] Amnon Neeman. The connection b etw een the K -theory localization theorem of Thomason, Trobaugh and Yao and the smashing sub categories of Bousfield and Rav enel. Ann. Sci. Éc ole Norm. Sup. (4) , 25(5):547–566, 1992. [NP24] Niko Naumann and Luca Pol. Separable commutativ e algebras and Galois theory in stable homotopy theories. A dv. Math. , 449:Paper No. 109736, 67, 2024. [Ste18] Greg Stevenson. Filtrations via tensor actions. Int. Math. R es. Not. IMRN , (8):2535– 2558, 2018. [Tho97] R. W. Thomason. The classification of triangulated subcategories. Comp ositio Math. , 105(1):1–27, 1997. [TT90] R. W. Thomason and T. T robaugh. Higher algebraic K -theory of schemes and of derived categories. In The Gr othendie ck F estschrift, V ol. III , v olume 88 of Pr ogr. Math. , pages 247–435. Birkhäuser, Boston, MA, 1990. Drew Heard, Dep ar tment of Ma thema tical Sciences, Nor wegian University of Science and Technology, Tr ondheim Email address : drew.k.heard@ntnu.no URL : https://folk.ntnu.no/drewkh/ Marius Nielsen, Dep ar tment of Ma thema tical Sciences, Nor wegian University of Science and Technology, Tr ondheim Email address : marius.v.b.nielsen@ntnu.no URL : https://mariusnielsen.github.io/