Affine weakly regular tensor triangulated categories
aa r X i v : . [ m a t h . C T ] M a y AFFINE WEAKLY REGULAR TENSOR TRIANGULATEDCATEGORIES
IVO DELL’AMBROGIO AND DONALD STANLEY
Abstract.
We prove that the Balmer spectrum of a tensor triangulated cat-egory is homeomorphic to the Zariski spectrum of its graded central ring, pro-vided the triangulated category is generated by its tensor unit and the gradedcentral ring is noetherian and regular in a weak sense. There follows a clas-sification of all thick subcategories, and the result extends to the compactlygenerated setting to yield a classification of all localizing subcategories as wellas the analog of the telescope conjecture. This generalizes results of Shamirfor commutative ring spectra. Introduction and results
Let K be an essentially small tensor triangulated category, with symmetric exacttensor product ⊗ and tensor unit object . Balmer [Bal05] defined a topologicalspace, the spectrum Spc K , that allows for the development of a geometric theoryof K , similarly to how the Zariski spectrum captures the intrinsic geometry ofcommutative rings; see the survey [Bal10b]. Among other uses, Balmer’s spectrumencodes the classification of the thick tensor ideals of K in terms of certain subsets.It is therefore of interest to find an explicit description of the spectrum in theexamples, but this is usually a difficult problem requiring some in-depth knowledgeof each example at hand.The goal of this note is to show that in some cases a concrete description of thespectrum can be obtained easily and completely formally. Let us denote by R := End ∗K ( ) = M i ∈ Z Hom K ( , Σ i )the graded endomorphism ring of the unit, where Σ : K → K is the suspensionfunctor. In the terminology of [Bal10a], this is the graded central ring of K . It isa graded commutative ring and therefore we can consider its spectrum of homo-geneous prime ideals, Spec R , equipped with the Zariski topology. As establishedin [Bal10a], there is always a canonical continuous map ρ : Spc K −→
Spec R comparing the two spectra. Under some mild hypotheses, e.g. when R is noetherian, ρ can be shown to be surjective, but it is less frequently injective and, when it is,the proof of injectivity is typically much harder.Here is our main result: Date : September 17, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Tensor triangulated category, thick subcategory, localizing subcate-gory, spectrum.First-named author partially supported by the Labex CEMPI (ANR-11-LABX-0007-01).
Theorem.
Assume that K satisfies the two following conditions: (a) K is classically generated by , i.e., as a thick subcategory: Thick( ) = K . (b) R is a (graded) noetherian ring concentrated in even degrees and, for everyhomogeneous prime ideal p of R , the maximal ideal of the local ring R p isgenerated by a (finite) regular sequence of homogeneous non-zero-divisors.Then the comparison map ρ : Spc K ∼ → Spec R is a homeomorphism. As in the title, we may refer to a tensor triangulated category K satisfyinghypotheses (a) and (b) as being affine and weakly regular , respectively. Note that R being noetherian implies that R = End K ( ) is a noetherian ring and that R afinitely generated R -algebra, by [GY83].The next result is an easy consequence of the theorem. Here Supp R H ∗ X denotes the (big) Zariski support of the cohomology graded R -module H ∗ X :=Hom ∗K ( , X ).1.2. Corollary. If K and R are as in the theorem, then there exists a canonicalinclusion-preserving bijection (cid:8) thick subcategories C of K (cid:9) ∼ / / (cid:8) specialization closed subsets V of Spec R (cid:9) o o mapping a thick subcategory C to V = S X ∈C Supp R H ∗ X and a specialization closedsubset V to C = { X ∈ K | Supp R H ∗ X ⊆ V } . In many natural examples, K occurs as the subcategory T c of compact objectsin a compactly generated tensor triangulated category T . By the latter we mean acompactly generated triangulated category T equipped with a symmetric monoidalstructure ⊗ which preserves coproducts and exact triangles in both variables, andsuch that the compact objects form a tensor subcategory T c (that is: is compactand the tensor product of two compact objects is again compact).In this case, the same hypotheses allow us to classify also the localizing subcat-egories of T , thanks to the stratification theory of compactly generated categoriesdue to Benson, Iyengar and Krause [BIK11]. The support supp R X ⊆ Spec R of anobject X ∈ T is defined in [BIK08], and can be described as the setsupp R X = { p ∈ Spec R | X ⊗ K ( p ) = 0 } where the residue field object K ( p ) of a prime ideal p is an object whose cohomologyis the graded residue field of R at p ; see § Theorem.
Let T be a compactly generated tensor triangulated category withcompact objects K := T c and graded central ring R satisfying conditions (a) and (b) .Then we have the following canonical inclusion-preserving bijection: (cid:8) localizing subcategories L ⊆ T (cid:9) ∼ / / (cid:8) subsets S ⊆ Spec R (cid:9) . o o The correspondence sends a localizing subcategory L to S = S X ∈L supp R X , andan arbitrary subset S to L = { X ∈ T | supp R X ⊆ S } . Moreover, the bijectionrestricts to localizing subcategories L = Loc( L ∩ K ) which are generated by compactobjects on the left and to specialization closed subsets S = S p ∈ S V ( p ) on the right. Note that here the affine condition (a) is equivalent to requiring that T is gener-ated by as a localizing subcategory. As Supp R H ∗ X = supp R X for all compact FFINE WEAKLY REGULAR TENSOR TRIANGULATED CATEGORIES 3 objects X ∈ K , one sees easily that in the compactly generated case Theorem 1.1and Corollary 1.2 are also a consequence of Theorem 1.3.The next corollary is another by-product of stratification. Recall that a localizingsubcategory L ⊆ T is smashing if the inclusion functor L ֒ → T admits a coproduct-preserving right adjoint.1.4. Corollary (The telescope conjecture in the affine weakly regular case) . In thesituation of Theorem 1.3, every smashing subcategory of T is generated by a set ofcompact objects of T . A few special cases of our formal results had already been observed, such aswhen R is even periodic and of global dimension at most one; see [DT12]. We nowconsider some more concrete examples. ∗ ∗ ∗ Example.
Let A be a commutative dg algebra and D ( A ) its derived categoryof dg modules. Then D ( A ) is an affine compactly generated tensor triangulatedcategory with respect to the standard tensor product ⊗ = ⊗ L A , and R = H ∗ A isthe cohomology algebra of A ; thus if the latter satisfies (b) all our results applyto D ( A ). Actually, in this example we can improve our results a little by eliminatingthe hypothesis that R is even and that the elements of the regular sequences arenon-zero-divisors:1.6. Theorem.
Let A be a commutative dg algebra such that its graded cohomologyring R = H ∗ A is noetherian and such that every local prime p R p is generated bya finite regular sequence. Then all the conclusions of Theorems 1.1 and 1.3 and ofCorollaries 1.2 and 1.4 hold for T = D ( A ) and K = D ( A ) c . We can apply this for instance to a graded polynomial algebra with any choiceof grading for the variables, seen as a strictly commutative formal dg algebra.1.7.
Example.
Let A be a commutative S -algebra (a.k.a. commutative highly struc-tured ring spectrum), and let D ( A ) be its derived category. (This covers Exam-ple 1.5, as commutative dga’s can be seen as commutative S -algebras.) Then D ( A )is an affine compactly generated tensor triangulated category, and R = π ∗ A is thestable homotopy algebra of A ; thus if the latter satisfies (b) all our results ap-ply to D ( A ). Shamir [Sha12] already treated this example under the additionalhypotheses that π ∗ A has finite Krull dimension. Working with ∞ -categories andE ∞ -rings, Mathew [Mat14b, Theorem 1.4] established the classification of thick sub-categories as in Corollary 1.2 for the case when π ∗ A is even periodic and π A regularnoetherian. Remarkably, in the special case of S -algebras defined over Q , Mathewwas also able to prove the classification of thick subcategories only assuming π ∗ A noetherian, i.e., without any regularity hypothesis; see [Mat14a, Theorem 1.4].The next two well-known examples show that neither hypothesis (a) nor (b) canbe weakened with impunity.1.8. Example.
The derived category T = D ( P k ) of the projective line over a field k is an example where R = End ∗ ( ) ≃ k certainly satisfies (b) but (a) does not hold.Indeed ρ can be identified with the structure map P k → Spec k and is therefore farfrom injective in this case; see [Bal10a, Remark 8.2]. IVO DELL’AMBROGIO AND DONALD STANLEY
Example. If T = D ( A ) is the derived category of a commutative (ungraded)ring A , Theorem 1.1 and the classification of thick subcategories always hold bya result of Thomason [Tho97] (see [Bal10a, Proposition 8.1]); the classificationof localizing subcategories and the telescope conjecture hold if A is noetherianby Neeman [Nee92a]. On the other hand, Keller [Kel94] found examples of non-noetherian rings A for which the two latter results fail.In view of these examples, it would be interesting to know how far our weakregularity hypothesis (b) can be weakened in general. Would noetherian suffice?2. Recollections
Let K be an essentially small tensor triangulated category.For any two objects X, Y ∈ K , consider the Z -graded group Hom ∗K ( X, Y ) = L i ∈ Z Hom K ( X, Σ i Y ). Recall that the symmetric tensor product of K canonicallyinduces on R := Hom ∗K ( , ) the structure of a graded commutative ring, andon each Hom ∗K ( X, Y ) the structure of a (left and right) graded R -module. Thecomposition of maps in K and the tensor functor − ⊗ − are (graded) bilinear forthis action. See [Bal10a, §
3] for details.Since we are using cohomological gradings, we write H ∗ X for the R -moduleHom ∗K ( , X ) and call it the cohomology of X . Supports for graded modules.
We denote by Spec R the Zariski spectrum ofall homogeneous prime ideals in R . If M is an R -module (always understood tobe graded) and p ∈ Spec R , the graded localization of M at p is the R -module M p obtained by inverting the action of all the homogeneous elements in R r p . The bigsupport of M is the following subset of the spectrum:Supp R M = { p ∈ Spec R | M p = 0 } . Since our graded ring R is noetherian we also dispose of the small support , definedin terms of the indecomposable injective R -modules E ( R/ p ):supp R M = { p | E ( R/ p ) occurs in the minimal injective resolution of M } . We recall from [BIK08, §
2] some well-known properties of supports. In general wehave supp R M ⊆ Supp R M . If M is finitely generated, these two sets are equal andalso coincide with the Zariski closed set V (Ann R M ). For a general M , Supp R M is always specialization closed : if it contains any point p then it must contain itsclosure V ( p ) = { q | p ⊆ q } . In fact Supp R M is equal to the specialization closureof supp R M : Supp R M = S p ∈ supp R M V ( p ). The small support plays a fundamentalrole in the Benson-Iyengar-Krause stratification theory, but in this note it will onlyappear implicitly.The next lemma follows by a standard induction on the length of the objects.2.1. Lemma. If K = Thick( ) is affine and R is noetherian, the graded R -module Hom ∗K ( X, Y ) is finitely generated for all X, Y ∈ K . (cid:3) To be precise, graded commutativity means here that fg = ǫ | f || g | gf for any two homogeneouselements f ∈ Hom K ( , Σ | f | ) and g ∈ Hom K ( , Σ | g | ), where ǫ ∈ R is a constant with ǫ = 1induced by the symmetry isomorphism Σ ⊗ Σ ∼ → Σ ⊗ Σ . In most cases we have ǫ = −
1, e.g.if K admits a symmetric monoidal model, but usually no extra difficulty arises by allowing thegeneral case. Of course, this is immaterial for R even. FFINE WEAKLY REGULAR TENSOR TRIANGULATED CATEGORIES 5
The comparison map of spectra.
Recall from [Bal05] that, as a set, the spec-trum Spc K is defined to be the collection of all proper thick subcategories P ( K which are prime tensor ideals : X ⊗ Y ∈ P ⇔ X ∈ P or Y ∈ P . For every P ∈
Spc K , let ρ K ( P ) denote the ideal of Spec R generated by the set of homo-geneous elements { f : → Σ | f | | cone( f )
6∈ P} . By [Bal10a, Theorem 5.3], theassignment
P 7→ ρ K ( P ) defines a continuous map ρ K : Spc K →
Spec R , naturalin K . Moreover, the two spaces Spc K and Spec R are spectral in the sense ofHochster [Hoc69], and ρ K is a spectral map in that the preimage of a compact openset is again compact.2.2. Lemma. If ρ K is bijective then it is a homeomorphism.Proof. This is an immediate consequence of [Hoc69, Proposition 15], which saysthat for a spectral map of spectral topological spaces to be a homeomorphism itsuffices that it is an order isomorphism for the specialization order of the two spaces.Recall that the specialization order is defined for the points of any topological spaceby x ≥ y ⇔ x ∈ { y } . Indeed ρ := ρ K is inclusion reversing, Q ⊆ P ⇔ ρ ( Q ) ⊇ ρ ( P ),hence it maps the closure {P} = {Q | Q ⊆ P} in Spc K of any point P to theZariski closure V ( ρ ( P )) = { q | q ⊇ ρ ( P ) } in Spec R of the corresponding point. (cid:3) Central localization.
For every prime ideal p of the graded central ring R of K ,there exists by [Bal10a, Theorem 3.6] a tensor triangulated category K p having thesame objects as K and such that its graded Hom modules are the localizationsHom ∗K p ( X, Y ) = Hom ∗K ( X, Y ) p . In particular the graded central ring of K p is the local ring R p . There is a canonicalexact functor q p : K → K p , which is in fact the Verdier quotient by the thick tensorideal generated by { cone( f ) ∈ K | f ∈ R r p homogeneous } . For emphasis, we willsometimes write X p for X = q p X when considered as an object of K p .Clearly if K is generated by then K p is generated by p . Later we will use thefact that if a tensor triangulated category is generated by its unit then every thicksubcategory is automatically a tensor ideal.Let ℓ p : R → R p denote the localization map between the graded central rings ofthe two categories. By [Bal10a, Theorem 5.4], we have a pullback square of spaces(2.3) Spc( K p ) ρ K p (cid:15) (cid:15) (cid:31) (cid:127) Spc( q p ) / / Spc( K ) ρ K (cid:15) (cid:15) Spec( R p ) (cid:31) (cid:127) Spec( ℓ p ) / / Spec( R )where the horizontal maps are injective. Koszul objects.
We adapt some convenient notation from [BIK08]. For any object X ∈ K and homogeneous element f ∈ R , let X//f := cone( f · X ) be any choice ofmapping cone for the map f · X : Σ −| f | X → X given by the R -action. If f , . . . , f n is a finite sequence of homogeneous elements, define recursively X := X and X i := X// ( f , . . . , f i ) := ( X// ( f , . . . , f i − )) //f i for i ∈ { , . . . , n } . Thus by constructionwe have exact triangles(2.4) Σ −| f i | X i − f i · X i − / / X i − / / X i / / Σ −| f i | +1 X i − , IVO DELL’AMBROGIO AND DONALD STANLEY and moreover, since the tensor product is exact, we have isomorphisms
X// ( f , . . . , f i ) ≃ X ⊗ //f ⊗ . . . ⊗ //f i for all i ∈ { , . . . , n } . In the following, we will perform this construction inside the p -local category K p .We need the following triangular version of the Nakayama lemma, for K affine.2.5. Lemma. If X ∈ K p is any object and f , . . . , f n is a set of homogenous gen-erators for p R p , then in K p we have X = 0 if and only if X// ( f , . . . , f n ) = 0 .Proof. Since K and thus K p are generated by their tensor unit, is suffices to showthat H ∗ X p = 0 if and only if H ∗ ( X// ( f , . . . , f n )) p = 0, and the latter can be provedas in [BIK08, Lemma 5.11 (3)]. We give the easy argument for completeness.With the above notation, by taking cohomology H ∗ = Hom ∗K p ( p , − ) of thetriangle (2.4) of K p we obtain the long exact sequence of R p -modules . . . / / H ∗−| f i | X i − f i / / H ∗ X i − / / H ∗ X i / / H ∗−| f i | +1 X i − / / . . . where each module is finitely generated by Lemma 2.1. Since f i ∈ p , if H ∗ X i − = 0the first map in the sequence is not invertible by the Nakayama lemma, hence H ∗ X i = 0. The evident recursion shows that H ∗ X = 0 implies H ∗ X n = 0. Thevery same exact sequences also show that if H ∗ X = 0 then H ∗ X n = 0. (cid:3) Thick subcategories
Assume from now on that K satisfies the conditions (a) and (b) of Theorem 1.1. Residue field objects.
By hypothesis, for every prime ideal p ∈ Spec R thereexists a regular sequence f , . . . , f n of homogeneous non-zero-divisors of R p whichgenerate the ideal p R p . Choose one such sequence once and for all, and constructthe associated Koszul object K ( p ) := p // ( f , . . . , f n ) ≃ p //f ⊗ . . . ⊗ p //f n in the p -local tensor triangulated category K p .3.1. Lemma.
For every object X ∈ K p and every i ∈ { , . . . , n } , each element f ofthe ideal ( f , . . . , f i ) ⊂ R p acts as zero on X// ( f , . . . , f i ) , i.e., f · X// ( f , . . . , f i ) = 0 .Proof. Recall that, as an immediate consequence of the R p -bilinearity of the com-position in K p , the elements of R p acting as zero on an object Y form an ideal(coinciding with the annihilator of the R p -module Hom ∗K p ( Y, Y )). Thanks to theisomorphism
X// ( f , . . . , f i ) ≃ X ⊗ p //f ⊗ . . . ⊗ p //f i and the R p -linearity ofthe tensor product, it will therefore suffice to prove that f i acts as zero on p //f i .Consider the following commutative diagramΣ −| f i | p f i / / p g / / f i (cid:15) (cid:15) % % ❑❑❑❑❑❑❑❑❑❑❑ p //f if i (cid:15) (cid:15) / / Σ −| f i | +1 p h x x Σ | f i | p g / / Σ | f i | p //f i where the top row is the exact triangle defining p //f i . Being the composite of twoconsecutive maps in a triangle, gf i is zero. Up to a suspension, this is also the FFINE WEAKLY REGULAR TENSOR TRIANGULATED CATEGORIES 7 diagonal map in the square. Hence f i · p //f i factors through a map h as pictured.Since R is even by hypothesis, then R p is even and we claim that also(3.2) H n ( p //f i ) = 0 for all odd n . This implies h = 0 and therefore f i · p //f i = 0, as required. To prove the claim,note that the defining triangle of p //f i induces the exact sequence R n −| f i | p f i / / R n p / / H n ( p //f i ) / / R n −| f i | +1 p f i / / R n +1 p where the first and last maps are injective by the hypothesis that f i is a non-zero-divisor in R p . Thus (3.2), and even H ∗ ( p //f i ) ≃ R p / ( f i ), follow immediately. (cid:3) Corollary. H ∗ ( X ⊗ K ( p )) is a graded k ( p ) -vector space for every X ∈ K p .Proof. By Lemma 3.1 together with the R -linearity of the tensor product, each f ∈ p R p acts as zero on X ⊗ K ( p ) ≃ X ⊗ p // ( f , . . . , f n ). Therefore all such f alsoact as zero on H ∗ ( X ⊗ K ( p )) by the R -linearity of composition. (cid:3) Lemma.
There is an isomorphism H ∗ ( p // ( f , . . . , f i )) ≃ R p / ( f , . . . , f i ) of R -modules for all i ∈ { , . . . , n } . In particular H ∗ K ( p ) is isomorphic to the residuefield k ( p ) := R p / p R p .Proof. Write C = p and C i := p // ( f , . . . , f i ) for short. Then K ( p ) = C n , andfor all i ∈ { , . . . , n } we have exact trianglesΣ −| f i | C i − f i · C i − / / C i − / / C i / / Σ −| f i | +1 C i − . The claim follows by recursion on i . Indeed H ∗ C = R p , and assume that H ∗ C i − ≃ R p / ( f , . . . , f i − ). Then the above triangle induces an exact sequence H ∗−| f i | C i − f i / / H ∗ C i − / / H ∗ C i / / H ∗−| f i | +1 C i − f i / / H ∗ +1 C i − where the first and last maps are injective because by hypothesis f i is a nonzerodivisor in the ring R p / ( f , . . . , f i − ). We thus obtain a short exact sequence 0 → f i R p / ( f , . . . , f i − ) → R p / ( f , . . . , f i − ) → H ∗ C i →
0, proving the claim for i . (cid:3) Remark.
Of the weak regularity hypothesis (b), the proof of Lemma 3.4 onlyuses that f , . . . , f n is a regular sequence, while the proof of Lemma 3.1 only usesthat the f i are non-zero-divisors in R p and that the ring R is even. These are theonly places where we make use of these assumptions (the noetherian hypothesis,on the other hand, will be needed on several occasions). Note that, although wealready know by Corollary 3.3 that H ∗ K ( p ) is a k ( p )-vector space, for the nextproposition we also need it to be one-dimensional as per Lemma 3.4.3.6. Proposition.
For all p ∈ Spec R and X ∈ K p , the tensor product X ⊗ K ( p ) decomposes into a coproduct of shifted copies of the residue field object: a α Σ n α K ( p ) ∼ −→ X ⊗ K ( p ) . Proof.
By Corollary 3.3 we know that H ∗ ( X ⊗ K ( p )) is a graded k ( p )-vector space.Choose a graded basis { x α } α , corresponding to a morphism ` α Σ n α p → X ⊗ K ( p ).We will show that this map extends nontrivially to the Koszul object( a α Σ n α p ) // ( f , . . . , f n ) = a α (Σ n α p // ( f , . . . , f n )) . IVO DELL’AMBROGIO AND DONALD STANLEY
For this, it will suffice to extend each individual map x α : Σ n α p → X ⊗ K ( p ). Asbefore, we proceed recursively along the regular sequence f , . . . , f n . Consider thefollowing commutative diagramΣ n α −| f | p Σ −| f | x α (cid:15) (cid:15) f / / ' ' PPPPPPPPPPPP Σ n α p / / x α (cid:15) (cid:15) Σ n α p //f / / x α x x Σ −| f | X ⊗ K ( p ) f =0 / / X ⊗ K ( p )where the top row is a rotation of the defining triangle for p //f . The left-bottomcomposite vanishes because f acts trivially on X ⊗ K ( p ) by Lemma 3.1. Hence weobtain the map x α on the right. Note that x α = 0 because x α = 0. Now we repeatthe procedure for i = 2 , . . . , n , using the triangleΣ −| f i | p // ( f , . . . , f i − ) f i / / p // ( f , . . . , f i − ) / / p // ( f , . . . , f i ) / / in order to extend x i − α to a nonzero map x iα : Σ n α p // ( f , . . . , f i ) → X ⊗ K ( p )hitting the same element in cohomology. In particular we obtain the announcedextension x nα : Σ n α K ( p ) → X ⊗ K ( p ). As a nonzero map on a one-dimensional k ( p )-vector space (Lemma 3.4), the induced map H ∗ ( x nα ) must be injective. Hence,collectively, the maps { x nα } α yield an isomorphism as required. (cid:3) Proposition.
For every p , the thick subcategory Thick( K ( p )) of K p is minimal ,meaning that it contains no proper nonzero thick subcategories.Proof. Note that for every nonzero object X of K p we have X ⊗ K ( p ) = 0. Indeedif X ⊗ K ( p ) = X// ( f , . . . , f n ) = 0 then X p = 0 by Lemma 2.5.Let C be a thick subcategory of Thick( K ( p )). Because C is a tensor ideal, if itcontains a nonzero object X then it also contains X ⊗ K ( p ), which is again nonzeroby the above observation. Therefore C must contain a shifted copy of K ( p ) byProposition 3.6, hence C = Thick( K ( p )). This proves the claim. (cid:3) Proof of Theorem 1.1.
Now we show how to deduce our main result from theminimality of the thick subcategories Thick( K ( p )). By Lemma 2.2 it will suffice toshow that the map ρ K : Spc K →
Spec R is bijective. Since R is graded noetherian, ρ K is surjective by [Bal10a, Theorem 7.3]. It remains to prove it is injective.Let p ∈ Spec R be any homogeneous prime. We must show that the fiber ofthe comparison map ρ K : Spc K →
Spec R over p consists of a single prime tensorideal. By the pullback square (2.3), every point of Spc K lying over p must belongto Spc K p . Hence it will suffice to show that the fiber of ρ := ρ K p over the maximalideal m := p R p of R p consists of a single point. In fact a stronger statement is true:if P ∈
Spc K p is such that ρ ( P ) = m , then P = { } . Let us prove this.By definition of the comparison map we have ρ ( P ) = h{ f ∈ R p | f is homogeneous and p //f
6∈ P}i , and as ρ ( P ) ⊆ m always holds by the maximality of m , the hypothesis ρ ( P ) = m precisely means that p //f
6∈ P for all homogeneous elements f ∈ m . In particular p //f i
6∈ P for the elements f i in the chosen regular sequence for m . As P is atensor prime, we deduce further that(3.8) K ( p ) ≃ p //f ⊗ . . . ⊗ p //f n
6∈ P . FFINE WEAKLY REGULAR TENSOR TRIANGULATED CATEGORIES 9
Now let X ∈ P and assume that X = 0. Then X ⊗ K ( p ) = 0 by Lemma 2.5, hence(3.9) Thick( X ⊗ K ( p )) = Thick( K ( p ))by the minimality of Thick( K ( p )), Proposition 3.7. As P is a thick tensor ideal wealso have X ⊗ K ( p ) ∈ P and therefore K ( p ) ∈ P by (3.9), but this contradicts (3.8).Therefore X = 0 and we conclude that P = { } , proving the claim.This concludes the proof of Theorem 1.1. Proof of Corollary 1.2.
In order to deduce Corollary 1.2 from the theorem, wemust verify that the homeomorphism ρ K identifies Supp R H ∗ X ⊆ Spec R , the ring-theoretic support of an object X ∈ K , with supp X := {P ∈ Spc
K | X
6∈ P} , theuniversal support datum of X :3.10. Lemma.
We have
Supp R H ∗ X = ρ K (supp X ) for all X ∈ K .Proof. Let p = ρ K ( P ). It follows from (2.3) that X ∈ P iff X p ∈ P p , where P p denotes P seen as an element of Spc K p . We have just proved that ρ K p : Spc K p ∼ → Spec R p is a bijection sending { } to p R p , so we must have P p = { } . Therefore p ∈ Supp R H ∗ X ⇔ H ∗ X p = 0 ⇔ X p = 0 ⇔ X p
6∈ P p ⇔ P ∈ supp X . (cid:3) Now it suffices to appeal to the abstract classification theorem [Bal05, Theo-rem 4.10]. Indeed, since R is noetherian, the space Spec R is noetherian and there-fore its specialization closed subsets and its Thomason subsets coincide (cf. [Bal05,Remark 4.11]). Moreover, since K is generated by its tensor unit, all its objects aredualizable (because dualizable objects form a thick subcategory and is dualizable)and therefore all its thick tensor ideals are radical (see [Bal07, Proposition 2.4]).Hence by Theorem 1.1 and Lemma 3.10 the classification of [Bal05, Theorem 4.10]immediately translates into the classification described in Corollary 1.2, as wished.4. Localizing subcategories
Assume from now on that T is a compactly generated tensor triangulated cat-egory such that its subcategory K := T c of compact objects satisfies hypotheses(a) and (b) from the statement of Theorem 1.1. Thus in particular T is generatedas a localizing subcategory by the tensor unit: Loc( ) = T . If follows that everylocalizing subcategory of T is automatically a tensor ideal.Since T is compactly generated, the (Verdier) p -localization functor q p : K → K p we used so far can be extended to a finite (Bousfield) localization functor( − ) p : T −→ T . We briefly recall its properties, referring to [BIK11, §
2] or [Del10, §
2] for allproofs. Let L = Loc( { cone( f ) | f ∈ R r p homogeneous } ). Then the Verdierquotient Q : T → T / L =: T p has a fully faithful right adjoint, I : T p ֒ → T , andthe functor ( − ) p can be defined to be the composite ( − ) p := I ◦ Q . As L isgenerated by a tensor ideal of dualizable objects, we have X p ∼ = X ⊗ p for all X ∈ T . Moreover, the unit X → X p of the ( Q, I )-adjunction induces a naturalmap Hom ∗T ( Y, X ) p → Hom ∗T ( Y, X p ) which is an isomorphism whenever Y ∈ K (see[BIK11, Proposition 2.3] or [Del10, Theorem 2.33 (h)]). In particular we have theidentification ( H ∗ X ) p ∼ → H ∗ ( X p ) for all X ∈ T . It follows also that the restriction of Q to compact objects X, Y ∈ K agrees with q p , so that we may identify K p with the full subcategory I ( K p ) of T (and thereby eliminate the slight ambiguity of the notation “ X p ”).Recall the residue field objects K ( p ) defined in the previous section: K ( p ) := p // ( f , . . . , f n ) ≃ p //f ⊗ . . . ⊗ p //f n ∈ T (as before, f , . . . , f n denotes the chosen regular sequence of non-zero-divisors gen-erating the prime p ).The main point of this section is that the crucial minimality result of Proposi-tion 3.7 can be extended to localizing subcategories of T , as we verify next.4.1. Lemma.
For every object X ∈ T and every i ∈ { , . . . , n } , each element f of ( f , . . . , f i ) ⊂ R acts as zero on X p // ( f , . . . , f i ) , i.e., f · X p // ( f , . . . , f i ) = 0 . Inparticular, the R -module H ∗ ( X ⊗ K ( p )) is a graded k ( p ) -vector space.Proof. Exactly the same proof as for Lemma 3.1 and Corollary 3.3. (Use that X ⊗ K ( p ) = X p ⊗ K ( p ) to work inside the big p -local category T p .) (cid:3) Proposition.
For all p ∈ Spec R and X ∈ T , the tensor product X ⊗ K ( p ) decomposes into a coproduct of shifted copies of the residue field object: a α Σ n α K ( p ) ∼ −→ X ⊗ K ( p ) . Proof.
Exactly the same as for Proposition 3.6, using Lemma 4.1. (cid:3)
Proposition.
For every p , the localizing subcategory Loc( K ( p )) of T is mini-mal, meaning that it contains no proper nonzero localizing subcategories.Proof. This follows from Proposition 4.2 precisely as in the proof of Proposition 3.7,except that we cannot use Lemma 2.5 to show that X ⊗ K ( p ) = 0 for every nonzeroobject X ∈ Loc( K ( p )). Instead, we may use the following argument.First note that X ⊗ K ( q ) = 0 for all q ∈ Spec R r { p } . Indeed, this property holdsfor X = K ( p ) by Lemma 4.1 (because if p = q then some homogeneous elementof R must act on K ( p ) ⊗ K ( q ) both as zero and invertibly) and is stable undertaking coproducts and mapping cones (as the latter are preserved by − ⊗ K ( p ));hence it must hold for all objects of Loc( K ( p )), as wished. Now combine this withProposition 4.5 below. (cid:3) Lemma.
Let M be any nonzero module, possibly infinitely generated, over anoetherian Z -graded commutative ring S . Then there exists a minimal prime in Supp S M := { p ∈ Spec S | M p = 0 } , the big Zariski support of M .Proof. If M = 0 then M p = 0 for some prime p , so the support is not empty.Moreover, it suffices to prove the claim for the nonzero module M p over S p , becausea minimal prime of Supp S p M p yields a minimal prime in Supp S M ; hence we mayassume that S is local. By Zorn’s lemma it suffices to show that in Supp S M everychain of primes admits a minimum. Indeed, each such chain must stabilize, becausea local commutative noetherian ring has finite Krull dimension. In the ungradedcase, the latter is a well-known corollary of Krull’s principal ideal theorem. A proofof the analogous result for graded rings can be found in [BH93, Theorem 1.5.8]or [PP11, Theorem 3.5]. (cid:3) FFINE WEAKLY REGULAR TENSOR TRIANGULATED CATEGORIES 11
Proposition.
If an object X ∈ T is such that X ⊗ K ( p ) = 0 for all p ∈ Spec R then X = 0 .Proof. We prove the contrapositive. Assume that X = 0. Then H ∗ X = 0, hencefor some p ∈ Spec R we must have H ∗ ( X p ) = ( H ∗ X ) p = 0 and therefore X p = 0.By Lemma 4.4, we may choose a prime p which is minimal among the primes withthis property. Thus the big support of the R -module H ∗ X p consists precisely ofthe prime p . We are going to recursively show that X i := X p // ( f , . . . , f i ) satisfiesSupp R H ∗ X i = { p } for all i ∈ { , . . . , n } . Thus in particular X ⊗ K ( p ) = X n = 0,which proves the proposition. We already know that Supp R H ∗ X = { p } for X := X p , and suppose we have shown that Supp R H ∗ X i − = { p } . The exact triangleΣ −| f i | X i − f i / / X i − / / X i / / Σ −| f i | +1 X i − implies that Supp R H ∗ X i ⊆ { p } . Hence X i = 0 is equivalent to Supp R H ∗ X i = { p } .By the triangle again, if X i = 0 were the case f i would act invertibly on X i − andthus on H ∗ X i − . This implies H ∗ X i − = ( H ∗ X i − )[ f − i ], and since f i ∈ p we wouldconclude that p Supp R H ∗ X i − , in contradiction with the induction hypothesis.Therefore X i = 0, as claimed. (cid:3) Proof of Theorem 1.3.
The result now follows easily from the machinery devel-oped by Benson, Iyengar and Krause in [BIK08] and [BIK11]. Indeed, by [BIK11,Theorem 4.2] in order to obtain the claimed classification of localizing subcategoriesit suffices to verify that the action of R stratifies T . By definition, this means thatthe following two axioms are satisfied: • The local-global principle:
For every object X ∈ T we have the equalityLoc( X ) = Loc( { Γ p X | p ∈ Spec R } )of localizing subcategories of T . • Minimality:
For every p ∈ Spec R the localizing subcategory Γ p T of T isminimal or zero.The functors Γ p : T → T are introduced in [BIK08], but we don’t need to know howthey are defined. In our context, i.e. where T is a tensor category and the action of R is the canonical one of the central ring, the local-global principle always holds by[BIK11, Theorem 7.2] (see also [Ste13, Theorem 6.8]). Moreover Γ p X = X ⊗ Γ p for all X ∈ T , which implies that Γ p T = Loc( Γ p ) since T is generated by .Therefore the remaining minimality condition follows from Proposition 4.3, becauseLoc( K ( p )) = Loc( Γ p ) by [BIK11, Lemma 3.8 (2)] (indeed, by construction K ( p )is a particular instance of the objects collectively denoted by ( p ) in loc. cit. ).This establishes the first bijection in Theorem 1.3.The claimed identification of the Benson-Iyengar-Krause support, supp R X = { p ∈ Spec R | X ⊗ Γ p = 0 } , with the set { p ∈ Spec R | X ⊗ K ( p ) = 0 } is an easyconsequence of the equality Loc( K ( p )) = Loc( Γ p ) mentioned above.It remains to verify the moreover part of Theorem 1.3. Let us begin by notingthat, if X ∈ K is a compact object, we have(4.6) supp R X = supp R H ∗ X = Supp R H ∗ X by [BIK08, Theorem 5.5 (1)] and Lemma 2.1. Now let
L ⊆ T be such that L = Loc( L ∩ K ). Then [ X ∈L supp R X = [ X ∈L∩K supp R X = [ X ∈L∩K Supp R H ∗ X by (4.6), and the latter is a specialization closed subset of the spectrum. Conversely,if S ⊆ Spec R is specialization closed the corresponding localizing subcategory { X ∈ T | supp R X ⊆ S } is generated by compact objects by [BIK08, Theorem 6.4],hence L = Loc( L ∩ K ). This concludes the proof of the theorem.It is well-known that the assignments
C 7→
Loc( C ) and L 7→ L ∩ K are mutuallyinverse bijections between thick subcategories
C ⊆ K and localizing subcategories
L ⊆ T which are generated by compact objects of T (see [Nee92b]). Togetherwith (4.6), this shows how to deduce the classification of thick subcategories ofCorollary 1.2 from Theorem 1.3.Finally, there are several ways to derive the telescope conjecture of Corollary 1.4from the previous results. For instance, we may proceed as in [BIK11, § Remark.
Using the theory of coherent functors, Benson, Iyengar and Krausehave recently developed in [BIK15] an analogue of their stratification theory ofcompactly generated categories that can be applied to general essentially smalltriangulated categories. Their theory, and more specifically [BIK15, Theorem 7.4],provides an alternative way to derive Theorem 1.1 from Proposition 3.7.
The case of commutative dg algebras.
We still owe our reader a proof ofTheorem 1.6. Let A be a commutative dg algebra and let D ( A ) be the derivedcategory of (left, say) dg- A -modules. The following elementary fact was pointedout to us by the referee.4.8. Lemma.
Every f ∈ H ∗ A acts as zero on its own mapping cone C ( f ) .Proof. A (homogeneous) element f ∈ H ∗ A of degree | f | = − n is (represented by) amorphism f : Σ n A → A of left dg- A -modules. Let us write s a ( a ∈ A ) for a genericelement of degree | a | − A := Σ A ; here we use the Koszul signconvention and treat s as a symbol of degree −
1. The cone C ( f ) has elements( a, s n +1 b ) (for a, b ∈ A ). Then f acts on C ( f ) by a morphism s n C ( f ) → C ( f )which, under the isomorphism s n C ( f ) ∼ = C (s n f ), is written as follows: g : C (s n f ) → C ( f ) g (s n a, s n +1 b ) = ( f (s n a ) , s n +1 f (s n b ))(recall that the suspension s h : s B → s C of a morphism h : B → C is given by(s h )(s b ) = s( h ( b ))). With these notations, the map H : C (s n f ) → C ( f ) definedby H (s n a, s n +1 b ) := (0 , s n +1 a ) is easily seen to satisfy H ( tx ) = ( − | t | tH ( x ) (for t ∈ A, x ∈ C (s n f )) and dH + Hd = − g ; in other words, H is a homotopy g ∼ A . (cid:3) As noted in Remark 3.5, Lemma 3.1 was the only place in all of our argumentswhere we made use of the hypothesis that R is concentrated in even degrees andthat in the regular sequences we may choose the elements to be non-zero-divisors.But if we consider the example K := D ( A ) c , T := D ( A ) and R := H ∗ A , wesee immediately that the conclusion of the lemma also follows from the aboveLemma 4.8. Hence in this case we can get rid of the extra hypotheses, whilethe rest of our arguments go through unchanged. This proves Theorem 1.6. FFINE WEAKLY REGULAR TENSOR TRIANGULATED CATEGORIES 13
Indeed, in general in Theorem 1.1 (b) we could similarly renounce the evennessof R if we substitute the requirement that all elements f i of the regular sequencesbe non-zero-divisors with the requirement that f i · //f i = 0. Acknowledgements.
We are very grateful to Paul Balmer and an anonymousreferee for their useful comments, and to the referee for suggesting Theorem 1.6.We also thank Joseph Chuang for the reference [Mat14a].
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Ivo Dell’Ambrogio, Laboratoire de Math´ematiques Paul Painlev´e, Universit´e de Lille 1,Cit´e Scientifique – Bˆat. M2, 59665 Villeneuve-d’Ascq Cedex, France.
E-mail address : [email protected] URL : http://math.univ-lille1.fr/ ∼ dellambr Donald Stanley, Department of Mathematics and Statistics, University of Regina,3737 Wascana Parkway, Regina, Saskatchewan, S4S 0A2 Canada.
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