STRATIFYING TRIANGULATED CATEGORIES

DAVE BENSON, SRIKANTH B. IYENGAR, AND HENNING KRAUSE

Abstract. A notion of stratification is introduced for any compactly generated triangulated category T endowed with an action of a graded commutative noetherian ring R. The utility of this notion is demonstrated by establishing diverse consequences which follow when T is stratified by R. Among them are a classification of the localizing subcategories of T in terms of subsets of the set of prime ideals in R; a classification of the thick subcategories of the subcategory of compact objects in T; and results concerning the support of the graded R-module of morphisms Hom_{T}^{∗}(C, D) leading to analogues of the tensor product theorem for support varieties of modular representation of groups.

Contents

1. Introduction 1 2. Local cohomology and support 4 3. A local-global principle 9 4. Stratification 13 5. Orthogonality 14 6. Classifying thick subcategories 18 7. Tensor triangulated categories 19 8. Formal differential graded algebras 21 References 23

1. Introduction

Over the last few decades, the theory of support varieties has played an increasingly important role in various aspects of representation theory. The original context was Carlson’s support varieties for modular representations of finite groups [12], but the method soon spread to restricted Lie algebras [14], complete intersections in commutative algebra [1, 2], Hochschild cohomological support for certain finite dimensional algebras [13], and finite group schemes [15, 16].

One of the themes in this development has been the classification of thick or localizing subcategories of various triangulated categories of representations. This story started with Hopkins’ classification [18] of thick subcategories of the perfect complexes over a commutative Noetherian ring R, followed by Neeman’s classification [25] of localizing subcategories of the full derived category of R; both involved a notion of support for complexes living in the prime ideal spectrum of R. Somewhat later came the classification by Benson, Carlson and Rickard [6] of the thick subcategories of the stable module category of finite dimensional representation of a finite group G in terms of the spectrum of its cohomology ring.

1991 Mathematics Subject Classification. 18G99 (primary); 13D45, 18E30, 20J06, 55P42.

Key words and phrases. localizing subcategory, thick subcategory, triangulated category, support, local cohomology.

The research of the first and second authors was undertaken during visits to the University of Paderborn, each supported by a research prize from the Humboldt Foundation. The research of the second author was partly supported by NSF grants, DMS 0602498 and DMS 0903493.

In [8] we established an analogous classification theorem for the localizing subcategories of the stable module category of all representations of G. The strategy of proof is a series of reductions and involves a passage through various other triangulated categories admitting a tensor structure. To execute this strategy, it was important to isolate a property which would permit one to classify localizing subcategories in tensor triangulated categories, and could be tracked easily under changes of categories. This is the notion of stratification introduced in [8] for tensor triangulated categories, inspired by work of Hovey, Palmieri, and Strickland [19]. For the stable module category of G, this condition yields a parameterization of localizing subcategories reminiscent of, and enhancing, Quillen stratification [30] of the cohomology algebra of G, whence the name.

In this work we present a notion of stratification for any compactly generated triangulated category T, and establish a number of consequences which follow when this property holds for T. The context is that we are given an action of a graded commutative ring R on T, namely a map from R to the graded center of T. We write Spec R for the set of homogeneous prime ideals of R. In [7] we developed a theory of support for objects in T, based on a construction of exact functors Γ_{𝔭}: T → T for each 𝔭 ∈ Spec R, which are analogous to local cohomology functors from commutative algebra. The support of any object X of T is the set

supp_{R} X = {𝔭 ∈ Spec R | Γ_{𝔭}X ≠ 0}.

In this paper, we investigate in detail what is needed in order to classify localizing subcategories in this general context, in terms of the set Spec R.

We separate out two essential ingredients of the process of classifying localizing subcategories. The first is the local-global principle: it states that for each object X of T, the localizing subcategory generated by X is the same as the localizing subcategory generated by the set of objects {Γ_{𝔭}X | 𝔭 ∈ Spec R}. We prove that T has this property when, for example, the dimension of Spec R is finite.

When the local-global principle holds for T the problem of classifying localizing subcategories of T can be tackled one prime at a time. This is the content of the following result, which is part of Proposition 3.6.

Theorem 1.1. When the local-global principle holds for T there is a one-to-one correspondence between localizing subcategories of T and functions assigning to each 𝔭 ∈ Spec R a localizing subcategory of Γ_{𝔭}T. The function corresponding to a localizing subcategory S sends 𝔭 to S ∩ Γ_{𝔭}T.

The second ingredient is that in good situations the subcategory Γ_{𝔭}T, which consists of objects supported at 𝔭, is either zero or contains no proper localizing subcategories. If this property holds for each 𝔭 and the local-global principle holds, then we say T is stratified by R. In this case, the map in Theorem 1.1 gives a one-to-one correspondence between localizing subcategories of T and subsets of supp_{R} T, which is the set of primes 𝔭 such that Γ_{𝔭}T ≠ 0; see Theorem 4.2.

We draw a number of further consequences of stratification. The best statements are available when T, in addition to be being stratified by R, is noetherian, meaning that the R-module End_{T}^{∗}(C) is finitely generated for each compact object C in T.

Theorem 1.2. If T is noetherian and stratified by R, then the map described in Theorem 1.1 gives a one-to-one correspondence between the thick subcategories of the compact objects in T and the specialization closed subsets of supp_{R} T.

This result is a rewording of Theorem 6.1 and can be deduced from the classification of localizing subcategories of T, using an argument due to Neeman [25]. We give a different proof based on the following result, which is Theorem 5.1.

Theorem 1.3. If T is noetherian and stratified by R, then for each pair of compact objects C, D in T there is an equality

supp_{R} Hom_{T}^{∗}(C, D) = supp_{R} C ∩ supp_{R} D .

When in addition R^{i} = 0 holds for i < 0, one has Hom_{T}^{n}(C, D) = 0 for n ≫ 0 if and only if Hom_{T}^{n}(D, C) = 0 for n ≫ 0.

The statement of this theorem is inspired by an analogous statement for modules over complete intersection local rings, due to Avramov and Buchweitz [2]. A stratification theorem is not yet available in this context; see however [21].

The stratification condition also implies that Ravenel’s ‘telescope conjecture’ [31], sometimes called the ‘smashing conjecture’, holds for T.

Theorem 1.4. If T is noetherian and stratified by R and L: T → T is a localization functor that preserves arbitrary coproducts, then the localizing subcategory Ker L is generated by objects that are compact in T.

This result is contained in Theorem 6.3, which establishes also a classification of localizing subcategories of T that are also closed under products. Another application, Corollary 5.7, addresses a question of Rickard. If S is a localizing subcategory of T, write ^{⊥}S for the full subcategory of objects X such that there are no nonzero morphisms from X to any object in S.

Theorem 1.5. Suppose that T is noetherian and stratified by R, and that S is a localizing subcategory of T. Then ^{⊥}S is the localizing subcategory corresponding to the set of primes {𝔭 ∈ Spec R | V(𝔭) ∩ supp_{R} S = ∅}.

In Section 7 we consider the case when T has a structure of a tensor triangulated category compatible with the R-action, and discuss a notion of stratification suitable for this context. A noteworthy feature is that the analogue of the local-global principle always holds, so stratification concerns only whether each Γ_{𝔭}T is minimal as tensor ideal localizing subcategory. When this property holds one has the following analogue of the tensor product theorem of modular representation theory as described in [5, Theorem 10.8]; cf. also Theorem 1.3.

Theorem 1.6. Let T be a tensor triangulated category with a canonical R-action. If R stratifies T, then for any objects X, Y in T there is an equality

supp_{R}(X ⊗ Y) = supp_{R} X ∩ supp_{R} Y .

This result reappears as Theorem 7.3. One can establish analogues of other results discussed above for tensor triangulated categories, but we do not do so; the arguments required are the same, and in any case, many of these results appear already in [8], at least for triangulated categories associated to modular representations of finite groups.

Most examples of stratified triangulated categories that appear in this work are imported from elsewhere in the literature. The one exception is the derived category of differential graded modules over any graded-commutative noetherian ring A. In Section 8 we verify that this triangulated category is stratified by the canonical Aaction, building on arguments from [8, §5] which dealt with the case A is a graded polynomial algebra over a field.

There are interesting classes of triangulated categories which cannot be stratified via a ring action, in the sense explained above; see Example 4.6. On the other hand, there are important contexts where it is reasonable to expect stratification, notably, modules over cocommutative Hopf algebras and modules over the Steenrod algebra, where analogues of Quillen stratification have been proved by Friedlander and Pevtsova [15] and Palmieri [29] respectively. One goal of [7] and the present work is to pave the way to such results.

Acknowledgments. It is our pleasure to thank Zhi-Wei Li for a critical reading of an earlier version of this manuscript.

2. Local cohomology and support

The foundation for this article is the work in [7] where we constructed analogues of local cohomology functors and support from commutative algebra for triangulated categories. In this section we further develop these ideas, as required, and along the way recall basic notions and constructions from op. cit.

Henceforth R denotes a graded-commutative noetherian ring and T a compactly generated R-linear triangulated category with arbitrary coproducts.

We begin by explaining what this means.

Compact generation. Let T be a triangulated category admitting arbitrary coproducts. A localizing subcategory of T is a full triangulated subcategory that is closed under taking coproducts. We write Loc_{T}(C) for the smallest localizing subcategory containing a given class of objects C in T, and call it the localizing subcategory generated by C.

An object C in T is compact if the functor Hom_{T}(C, −) commutes with all coproducts; we write T^{c} for the full subcategory of compact objects in T. The category T is compactly generated if it is generated by a set of compact objects.

We recall some facts concerning localization functors; see, for example, [7, §3].

Localization. A localization functor L: T → T is an exact functor that admits a natural transformation η: Id_{T} → L, called adjunction, such that L(ηX) is an isomorphism and L(ηX) = η(LX) for all objects X ∈ T. A localization functor L: T → T is essentially uniquely determined by the corresponding full subcategory

Ker L = {X ∈ T | LX = 0} .

This means that if L′ is a localization functor with Ker L ⊆ Ker L′ and η′ is its adjunction, then there is a unique morphism ι: L → L′ such that ιη = η′. Given such a localization functor L, the natural transformation Id_{T} → L induces for each object X in T a natural exact localization triangle

Γ X ⟶ X ⟶ LX ⟶

This exact triangle gives rise to an exact functor Γ: T → T with

Ker L = Im Γ and Ker Γ = Im L .

Here Im F, for any functor F: T → T, denotes the essential image: the full subcategory of T formed by objects {X ∈ T | X ≅ F Y for some Y in T}.

The next lemma provides the existence of localization functors with respect to a fixed localizing subcategory; see [26, Theorem 2.1] for the special case that the localizing subcategory is generated by compact objects.

Lemma 2.1. Let T be a compactly generated triangulated category. If a localizing subcategory S of T is generated by a set of objects, then there exists a localization functor L: T → T with Ker L = S.

Proof. In [28, Corollary 4.4.3] it is shown that the collection of morphisms between each pair of objects in the Verdier quotient T/S form a set. The quotient functor Q: T → T/S preserves coproducts, and a standard argument based on Brown’s representability theorem [23, 27] yields an exact right adjoint Q_{ρ}. Note that Q_{ρ} is fully faithful; see [17, Proposition I.1.3]. It follows that the composite L = Q_{ρ}Q is a localization functor satisfying Ker L = S; see [7, Lemma 3.1]. □

Central ring actions. Let R be a graded-commutative ring; thus R is ℤ-graded and satisfies r · s = (−1)^{|r||s|}s · r for each pair of homogeneous elements r, s in R. We say that the triangulated category T is R-linear, or that R acts on T, if there is a homomorphism R → Z^{∗}(T) of graded rings, where Z^{∗}(T) is the graded center of T. In this case, for all objects X, Y ∈ T the graded abelian group

Hom_{T}^{∗}(X, Y) = ⨁_{i∈ℤ} Hom_{T}(X, Σ^{i}Y)

carries the structure of a graded R-module.

Support. From now on, R denotes a graded-commutative noetherian ring and T a compactly generated R-linear triangulated category with arbitrary coproducts.

We write Spec R for the set of homogeneous prime ideals of R. Given a homogeneous ideal 𝔞 in R, we set

V(𝔞) = {𝔭 ∈ Spec R | 𝔭 ⊇ 𝔞} .

Let 𝔭 be a point in Spec R and M a graded R-module. We write M_{𝔭} for the homogeneous localization of M at 𝔭. When the natural map of R-modules M → M_{𝔭} is bijective M is said to be 𝔭-local. This condition is equivalent to: supp_{R} M ⊆ {𝔮 ∈ Spec R | 𝔮 ⊆ 𝔭}, where supp_{R} M is the support of M. The module M is 𝔭-torsion if each element of M is annihilated by a power of 𝔭; equivalently, if supp_{R} M ⊆ V(𝔭); see [7, §2] for proofs of these assertions.

The specialization closure of a subset U of Spec R is the set

cl U = {𝔭 ∈ Spec R | there exists 𝔮 ∈ U with 𝔮 ⊆ 𝔭} .

The subset U is specialization closed if cl U = U; equivalently, if U is a union of Zariski closed subsets of Spec R. For each specialization closed subset V of Spec R, we define the full subcategory of T of V-torsion objects as follows:

T_{V} = {X ∈ T | Hom_{T}^{∗}(C, X)_{𝔭} = 0 for all C ∈ T^{c}, 𝔭 ∈ Spec R \ V} .

This is a localizing subcategory and there exists a localization functor L_{V}: T → T such that Ker L_{V} = T_{V}; see [7, Lemma 4.3, Proposition 4.5]. For each object X in T the adjunction morphism X → L_{V}X induces the exact localization triangle

(2.2)  Γ_{V}X ⟶ X ⟶ L_{V}X ⟶ .

This exact triangle gives rise to an exact local cohomology functor Γ_{V}: T → T. In [7] we established a number of properties of these functors, to facilitate working with them. We single out one that is used frequently in this work: They commute with all coproducts in T; see [7, Corollary 6.5].

For each 𝔭 in Spec R set Z(𝔭) = {𝔮 ∈ Spec R | 𝔮 ⊈ 𝔭}, so V(𝔭) \ Z(𝔭) = {𝔭}, and

X_{𝔭} = L_{Z(𝔭)}X for each X ∈ T.

The notation is justified by the next result which enhances [7, Theorem 4.7].

Proposition 2.3. Let 𝔭 be a point in Spec R and X, Y objects in T. The R-modules Hom_{T}^{∗}(X, Y_{𝔭}) and Hom_{T}^{∗}(X_{𝔭}, Y) are 𝔭-local, so the adjunction morphism Y → Y_{𝔭} induces a unique homomorphism of R-modules

Hom_{T}^{∗}(X, Y)_{𝔭} ⟶ Hom_{T}^{∗}(X, Y_{𝔭}) .

This map is an isomorphism if X is compact.

Proof. The last assertion in the statement is [7, Theorem 4.7]. It implies that the Rmodule Hom_{T}^{∗}(C, Y_{𝔭}) is 𝔭-local for each compact object C in T. It then follows that Hom_{T}^{∗}(X, Y_{𝔭}) is 𝔭-local for each object X, since X is in the localizing subcategory generated by the compact objects, and the subcategory of 𝔭-local modules is closed under taking products, kernels, cokernels and extensions; see [7, Lemma 2.5].

At this point we know that End_{T}^{∗}(X_{𝔭}) is 𝔭-local, and hence so is Hom_{T}^{∗}(X_{𝔭}, Y), since the R-action on it factors through the homomorphism R → End_{T}^{∗}(X_{𝔭}). □

Consider the exact functor Γ_{𝔭}: T → T obtained by setting

Γ_{𝔭}X = Γ_{V(𝔭)}(X_{𝔭}) .

for each object X in T. The essential image of the functor Γ_{𝔭} is denoted by Γ_{𝔭}T, and an object X in T belongs to Γ_{𝔭}T if and only if Hom_{T}^{∗}(C, X) is 𝔭-local and 𝔭-torsion for every compact object C; see [7, Corollary 4.10]. From this it follows that Γ_{𝔭}T is a localizing subcategory.

The support of an object X in T is a subset of Spec R defined as follows:

supp_{R} X = {𝔭 ∈ Spec R | Γ_{𝔭}X ≠ 0} .

In addition to properties of the functors Γ_{V} and L_{V}, and support, given in [7], we require also the following ones.

Lemma 2.4. Let V ⊆ Spec R be a specialization closed subset and 𝔭 ∈ Spec R. Then for each object X in T one has

Γ_{𝔭}(Γ_{V}X) ≅ \begin{cases}Γ_{𝔭}X & when 𝔭 ∈ V, \\ 0 & otherwise, \end{cases}  and  Γ_{𝔭}(L_{V}X) ≅ \begin{cases}Γ_{𝔭}X & when 𝔭 ∉ V, \\ 0 & otherwise. \end{cases}

Proof. Apply the exact functor Γ_{𝔭} to the exact triangle Γ_{V}X → X → L_{V}X →. The assertion then follows from the fact that either Γ_{𝔭}(L_{V}X) = 0 (and this happens precisely when 𝔭 ∈ V) or Γ_{𝔭}(Γ_{V}X) = 0; see [7, Theorem 5.6]. □

Further results involve a useful construction from [7, 5.10].

Koszul objects. Let r ∈ R be a homogeneous element of degree d and X an object in T. We denote X//r any object that appears in an exact triangle

(2.5)  X ⟶^{r} Σ^{d}X ⟶ X//r ⟶

and call it a Koszul object of r on X; it is well defined up to (nonunique) isomorphism. Given a homogeneous ideal 𝔞 in R we write X//𝔞 for any Koszul object obtained by iterating the construction above with respect to some finite sequence of generators for 𝔞. This object may depend on the choice of the generating sequence for 𝔞, but one has the following uniqueness statement; see also Proposition 2.11(2).

Lemma 2.6. Let 𝔞 be a homogenous ideal in R. Each object X in T satisfies

supp_{R}(X//𝔞) = V(𝔞) ∩ supp_{R} X .

Proof. We verify the claim for 𝔞 = (r); an obvious iteration gives the general result.

Fix a point 𝔭 in Spec R and a compact object C in T. Applying the exact functor Γ_{𝔭} to the exact triangle (2.5), and then the functor Hom_{T}^{∗}(C, −) results in an exact sequence of R-modules

Hom_{T}^{∗}(C, Γ_{𝔭}X) ⟶^{∓r} Hom_{T}^{∗}(C, Γ_{𝔭}X)[d] ⟶

⟶ Hom_{T}^{∗}(C, Γ_{𝔭}(X//r)) ⟶ Hom_{T}^{∗}(C, Γ_{𝔭}X)[1] ⟶^{±r} Hom_{T}^{∗}(C, Γ_{𝔭}X)[d + 1] .

Set H = Hom_{T}^{∗}(C, Γ_{𝔭}X). The R-module H is 𝔭-local and 𝔭-torsion, see [7, Corollary 4.10], and this is used as follows. If Hom_{T}^{∗}(C, Γ_{𝔭}(X//r)) ≠ 0 holds, then H ≠ 0 and r ∈ 𝔭 since H is 𝔭-local. On the other hand, H ≠ 0 and r ∈ 𝔭 implies that Hom_{T}^{∗}(C, Γ_{𝔭}(X//r)) ≠ 0 since H is 𝔭-torsion. This implies the desired equality. □

The result below is [8, Proposition 3.5], except that there G is assumed to consist of a single object. The argument is however the same, so we omit the proof.

Proposition 2.7. Let G be a set of compact objects which generate T, and let V be a specialization closed subset of Spec R. For any decomposition V = ⋃_{i∈I} V(𝔞_{i}) where each 𝔞_{i} is an ideal in R, there are equalities

T_{V} = Loc_{T}({C//𝔞_{i} | C ∈ G, i ∈ I}) = Loc_{T}({Γ_{V(𝔞_{i})}C | C ∈ G, i ∈ I}) .  □

An element r ∈ R^{d} is invertible on an R-module M if the map M ⟶^{r} M[d] is an isomorphism. In the same vein, we say r is invertible on an object X in T if the natural morphism X ⟶^{r} Σ^{d}X is an isomorphism; equivalently, if X//r is zero.

Lemma 2.8. Let X be an object in T and V ⊆ Spec R a specialization closed subset. Each element r ∈ R with V(r) ⊆ V is invertible on L_{V}X, and hence on the R-modules Hom_{T}^{∗}(L_{V}X, Y) and Hom_{T}^{∗}(Y, L_{V}X), for any object Y in T.

Proof. From [7, Theorem 5.6] and Lemma 2.6 one gets equalities

supp_{R} L_{V}(X//r) = V(r) ∩ supp_{R} X ∩ (Spec R \ V(r)) = ∅ .

Therefore L_{V}(X//r) = 0, by [7, Theorem 5.2]. Applying L_{V} to the exact triangle (2.5) yields an isomorphism L_{V}X ⟶^{r} Σ^{|r|}L_{V}X, which is the first part of the statement. Applying Hom_{T}^{∗}(−, Y) and Hom_{T}^{∗}(Y, −) to it gives the second part. □

Homotopy colimits. Let X_{1} ⟶^{f_{1}} X_{2} ⟶^{f_{2}} X_{3} ⟶^{f_{3}} ⋯ be a sequence of morphisms in T. Its homotopy colimit, denoted hocolim X_{n}, is defined by an exact triangle

⨁_{n⩾1} X_{n} ⟶^{θ} ⨁_{n⩾1} X_{n} ⟶ hocolim X_{n} ⟶

where θ is the map (id −f_{n}); see [11].

Now fix a homogeneous element r ∈ R of degree d. For each X in T and each integer n set X_{n} = Σ^{nd}X and consider the commuting diagram

[figure: X; r; X; r^{2}; X; r^{3}; ⋯; X_{1}; r; X_{2}; r; X_{3}; r; ⋯; X//r; X//r^{2}; X//r^{3}; ⋯]

where each vertical sequence is given by the exact triangle defining X//r^{n}, and the morphisms in the last row are the (non-canonical) ones induced by the commutativity of the upper squares. The gist of the next result is that the homotopy colimits of the horizontal sequences in the diagram compute L_{V(r)}X and Γ_{V(r)}X.

Proposition 2.9. Let r ∈ R be a homogeneous element of degree d. For each X in T the adjunction morphisms X → L_{V(r)}X and Γ_{V(r)}X → X induce isomorphisms

hocolim X_{n} ⟶^{∼} L_{V(r)}X and hocolim Σ^{−1}(X//r^{n}) ⟶^{∼} Γ_{V(r)}X .

Proof. Applying the functor Γ_{V(r)} to the middle row of the diagram above yields a sequence of morphisms Γ_{V(r)}X_{1} → Γ_{V(r)}X_{2} → · · ·. For each compact object C in T, this induces a sequence of morphisms of R-modules

Hom_{T}^{∗}(C, Γ_{V(r)}X_{1}) ⟶^{g_{1}} Hom_{T}^{∗}(C, Γ_{V(r)}X_{2}) ⟶^{g_{2}} ⋯

Each R-module Hom_{T}^{∗}(C, Γ_{V(r)}X_{n}) is (r)-torsion and, identifying this module with Hom_{T}^{∗}(C, Γ_{V(r)}X)[nd], the map g_{n} is given by multiplication with r. Thus the colimit of the sequence above, in the category of R-modules, satisfies:

(2.10)  colim Hom_{T}^{∗}(C, Γ_{V(r)}X_{n}) = 0

Applying the functor L_{V(r)} to the canonical morphism ϕ: X → hocolim X_{n} yields the following commutative square.

\xymatrix{X \ar[r]^{ϕ} \ar[d]_{ηX} & hocolim X_{n} \ar[d]^{η hocolim X_{n}} \\ L_{V(r)}X \ar[r]^{L_{V(r)}ϕ} & L_{V(r)} hocolim X_{n}}

The morphism η hocolim X_{n} is an isomorphism since Γ_{V(r)} hocolim X_{n} = 0. The equality holds because, for each compact object C, there is a chain of isomorphisms

Hom_{T}^{∗}(C, Γ_{V(r)} hocolim X_{n}) ≅ Hom_{T}^{∗}(C, hocolim Γ_{V(r)}X_{n})
≅ colim Hom_{T}^{∗}(C, Γ_{V(r)}X_{n})
≅ 0

where the second one holds because C is compact and the last one is by (2.10). On the other hand, L_{V(r)}ϕ is an isomorphism, since

L_{V(r)} hocolim X_{n} ≅ hocolim L_{V(r)}X_{n}

and r is invertible on L_{V(r)}X, by Lemma 2.8. Thus hocolim X_{n} ≅ L_{V(r)}X.

Now consider the canonical morphism ψ: hocolim Σ^{−1}(X//r^{n}) → X. Applying the functor Γ_{V(r)} to it yields a commutative square:

\xymatrix{Γ_{V(r)} hocolim Σ^{−1}(X//r^{n}) \ar[d]_{θ hocolim Σ^{−1}(X//r^{n})} \ar[r]^{Γ_{V(r)}ψ} & Γ_{V(r)}X \ar[d]^{θX} \\ hocolim Σ^{−1}(X//r^{n}) \ar[r]^{ψ} & X}

By [7, Lemma 5.11], each X//r^{n} is in T_{V(r)} and hence so is hocolim Σ^{−1}(X//r^{n}). Thus the morphism θ hocolim Σ^{−1}(X//r^{n}) is an isomorphism. It remains to show that Γ_{V(r)}ψ is an isomorphism; equivalently, that the map Hom_{T}^{∗}(C, Γ_{V(r)}ψ) is an isomorphism for each compact object C.

The exact triangle X → X_{n} → X//r^{n} → induces an exact sequence of R-modules:

Hom_{T}^{∗}(C, Σ^{−1}Γ_{V(r)}X_{n}) ⟶ Hom_{T}^{∗}(C, Σ^{−1}(Γ_{V(r)}X//r^{n})) ⟶

⟶ Hom_{T}^{∗}(C, Γ_{V(r)}X) ⟶ Hom_{T}^{∗}(C, Γ_{V(r)}X_{n})

In view of (2.10), passing to their colimits yields that Hom_{T}^{∗}(C, Γ_{V(r)}ψ) is an isomorphism, as desired. □

Proposition 2.11. Let 𝔞 be an ideal in R. For each object X in T the following statements hold:

(1) X//𝔞 is in Thick_{T}(Γ_{V(𝔞)}X) and Γ_{V(𝔞)}X is in Loc_{T}(X//𝔞);

(2) Loc_{T}(X//𝔞) = Loc_{T}(Γ_{V(𝔞)}X);

(3) Γ_{V(𝔞)}X and L_{V(𝔞)}X are in Loc_{T}(X).

Proof. (1) By construction X//𝔞 is in Thick_{T}(X). As Γ_{V(𝔞)} is an exact functor, one obtains that Γ_{V(𝔞)}(X//𝔞) is in Thick_{T}(Γ_{V(𝔞)}X). This justifies the first claim in (1), since X//𝔞 is in T_{V(𝔞)} by [7, Lemma 5.11].

Now we verify that Γ_{V(𝔞)}X is in the localizing subcategory generated by X//𝔞. Consider the case where 𝔞 is generated by a single element, say a. Claim: X//a^{n} is in Thick_{T}(X//a), for each n ≥ 1. Indeed, this is clear for n = 1. For any n ≥ 1, the composition of maps

X ⟶^{a^{n}} Σ^{n|a|}X ⟶^{a} Σ^{(n+1)|a|}X

yields, by the octahedral axiom, an exact triangle

X//a^{n} ⟶ X//a^{n+1} ⟶ Σ^{n|a|}X//a ⟶ .

Thus, when X//a^{n} is in Thick_{T}(X//a), so is X//a^{n+1}. This justifies the claim.

It follows from Proposition 2.9 that Γ_{V(a)}X is the homotopy colimit of objects Σ^{−1}X//a^{n}, and hence in Loc_{T}(X//a), by the claim above.

Now suppose 𝔞 = (a_{1}, …, a_{n}), and set 𝔞′ = (a_{1}, …, a_{n−1}). Then the equality V(𝔞) = V(a_{1}) ∩ V(𝔞′) yields Γ_{V(𝔞)} = Γ_{V(a_{n})}Γ_{V(𝔞′)} by [7, Proposition 6.1]. By induction on n one may assume that Γ_{V(𝔞′)}X is in Loc_{T}(X//𝔞′). Therefore Γ_{V(𝔞)}X is in Loc_{T}(Γ_{V(a_{n})}(X//𝔞′)). The basis of the induction implies that Γ_{V(a_{n})}(X//𝔞′) is in the localizing subcategory generated by (X//𝔞′)//a_{n}, that is to say, by X//𝔞. Therefore, Γ_{V(𝔞)}X is in Loc_{T}(X//𝔞), as claimed.

(2) is an immediate consequence of (1).

(3) Since X//𝔞 is in Thick_{T}(X), it follows from (1) that Γ_{V(𝔞)}X is in Loc_{T}(X). The localization triangle (2.2) then yields that L_{V(𝔞)}X is also in Loc_{T}(X). □

3. A local-global principle

We introduce a local-global principle for T and explain how, when it holds, the problem of classifying the localizing subcategories can be reduced to one of classifying localizing subcategories supported at a single point in Spec R.

Recall that T is a compactly generated R-linear triangulated category. If for each object X in T there is an equality

Loc_{T}(X) = Loc_{T}({Γ_{𝔭}X | 𝔭 ∈ Spec R})

we say that the local-global principle holds for T.

Theorem 3.1. Let T be a compactly generated R-linear triangulated category. The local-global principle is equivalent to each of the following statements.

(1) For any X ∈ T and any localizing subcategory S of T, one has

X ∈ S ⟺ Γ_{𝔭}X ∈ S for each 𝔭 ∈ Spec R .

(2) For any X ∈ T, one has X ∈ Loc_{T}({Γ_{𝔭}X | 𝔭 ∈ Spec R}).

(3) For any X ∈ T and any specialization closed subset V of Spec R, one has

Γ_{V}X ∈ Loc_{T}({Γ_{𝔭}X | 𝔭 ∈ V}) .

(4) For any X ∈ T, one has Loc_{T}(X) = Loc_{T}({X_{𝔭} | 𝔭 ∈ Spec R}).

(5) For any X ∈ T and any localizing subcategory S of T, one has

X ∈ S ⟺ X_{𝔭} ∈ S for each 𝔭 ∈ Spec R .

(6) For any X ∈ T, one has X ∈ Loc_{T}({X_{𝔭} | 𝔭 ∈ Spec R}).

The proof uses some results which may also be useful elsewhere.

Lemma 3.2. Let X be an object in T. Suppose that for any specialization closed subset V of Spec R, one has

Γ_{V}X ∈ Loc_{T}({Γ_{𝔭}X | 𝔭 ∈ V}) .

Then Γ_{V}X and L_{V}X belong to Loc_{T}(X) for every specialization closed V ⊆ Spec R.

Proof. It suffices to prove that Γ_{𝔭}X is in Loc_{T}(X) for each 𝔭, that is to say, that the set U = {𝔭 ∈ Spec R | Γ_{𝔭}X ∉ Loc_{T}(X)} is empty. Assume U is not empty and choose a maximal element, say 𝔭, with respect to inclusion. This is possible since R is noetherian. Set W = V(𝔭) \ {𝔭}, and consider the localization triangle

Γ_{W}X ⟶ Γ_{V(𝔭)}X ⟶ Γ_{𝔭}X ⟶

of Γ_{V(𝔭)}X with respect to W. The hypothesis implies the first inclusion below

Γ_{W}X ∈ Loc_{T}({Γ_{𝔮}X | 𝔮 ∈ W}) ⊆ Loc_{T}(X) ,

and the second one follows from the choice of 𝔭. The object Γ_{V(𝔭)}X is in Loc_{T}(X), by Proposition 2.11, so the exact triangle above yields that Γ_{𝔭}X is in Loc_{T}(X). This contradicts the choice of 𝔭, and hence U = ∅, as desired. □

Finite dimension. The dimension of a subset U of Spec R, denoted dim U, is the supremum of all integers n such that there exists a chain 𝔭_{0} ⊊ 𝔭_{1} ⊊ ⋯ ⊊ 𝔭_{n} in U. The set U is called discrete if dim U = 0.

Proposition 3.3. Let X be an object of T and set U = supp_{R} X. If U is discrete, then there are natural isomorphisms

X ⟵^{∼} ∐_{𝔭∈U} Γ_{V(𝔭)}X ⟶^{∼} ∐_{𝔭∈U} Γ_{𝔭}X .

Proof. Arguing as in the proof of [7, Theorem 7.1] one gets that the morphisms Γ_{V(𝔭)}X → X induce the isomorphism on the left, in the statement above. The isomorphism on the right holds since for each 𝔭 ∈ U the morphism Γ_{V(𝔭)}X → Γ_{𝔭}X is an isomorphism by Lemma 2.4. □

Theorem 3.4. Let T be a compactly generated R-linear triangulated category and X an object of T. If dim supp_{R} X < ∞, then X is in Loc_{T}({Γ_{𝔭}X | 𝔭 ∈ supp_{R} X}).

Proof. Set U = supp_{R} X and S = Loc_{T}({Γ_{𝔭}X | 𝔭 ∈ U}). The proof is an induction on n = dim U. The case n = 0 is covered by Proposition 3.3. For n > 0 set U′ = U \min U, where min U is the set of minimal elements with respect to inclusion in U, and set V = cl U′. It follows from Lemma 2.4 that supp_{R} Γ_{V}X = U′. Since dim U′ = dim U − 1, the induction hypothesis yields that Γ_{V}X is in S. On the other hand, supp_{R} L_{V}X = min U is discrete and therefore L_{V}X belongs to S by Proposition 3.3 and Lemma 2.4. Thus X is in S, in view of the localization triangle (2.2). □

Proof of Theorem 3.1. It is easy to check that the local-global principle is equivalent to (1). Also, the implications (1) ⟹ (2) and (4) ⟺ (5) ⟹ (6) are obvious.

(2) ⟹ (3): Fix X ∈ T and a specialization closed subset V of Spec R. Then

Γ_{V}X ∈ Loc_{T}({Γ_{𝔭}Γ_{V}X | 𝔭 ∈ Spec R}) = Loc_{T}({Γ_{𝔭}X | 𝔭 ∈ V})

hold, where the last equality follows from Lemma 2.4.

(3) ⟹ (1): Since Γ_{𝔭} = Γ_{V(𝔭)}L_{Z(𝔭)}, it follows from condition (3) and Lemma 3.2 that Γ_{𝔭}X is in Loc_{T}(X). This implies Loc_{T}(X) ⊇ Loc_{T}({Γ_{𝔭}X | 𝔭 ∈ Spec R}) and the reverse inclusion holds by condition (3) for V = Spec R. Thus the local-global principle, which is equivalent to condition (1), holds.

(3) ⟹ (4): We have Γ_{𝔭}X = Γ_{V(𝔭)}X_{𝔭} ∈ Loc_{T}(X_{𝔭}) for each prime ideal 𝔭 by Proposition 2.11 and hence the hypothesis implies X ∈ Loc_{T}({X_{𝔭} | 𝔭 ∈ Spec R}). On the other hand, X_{𝔭} ∈ Loc_{T}(X) for each prime ideal 𝔭 by Lemma 3.2.

(6) ⟹ (2): Fix X ∈ T. For every prime ideal 𝔭, one has, for example from Lemma 2.4, that supp_{R} X_{𝔭} is a subset of {𝔮 ∈ Spec R | 𝔮 ⊆ 𝔭}. In particular, it is finite dimensional, since R is noetherian, so X_{𝔭} ∈ Loc_{T}({Γ_{𝔮}X_{𝔭} | 𝔮 ∈ Spec R}) holds, by Theorem 3.4. Thus

X ∈ Loc_{T}({X_{𝔭} | 𝔭 ∈ Spec R}) ⊆ Loc_{T}({Γ_{𝔮}X_{𝔭} | 𝔭, 𝔮 ∈ Spec R})
= Loc_{T}({Γ_{𝔮}X | 𝔮 ∈ Spec R}),

where the last equality follows from Lemma 2.4. □

The result below is an immediate consequence of Theorems 3.4 and 3.1.

Corollary 3.5. When dim Spec R is finite the local-global principle holds for T. □

Classifying localizing subcategories. Localizing subcategories of T are related to subsets of V = supp_{R} T via the following maps

\{ \begin{matrix}Localizing \\ subcategories of T \end{matrix} \} [figure: σ; τ]\{ \begin{matrix}Families (S(𝔭))_{𝔭∈V} with S(𝔭) a \\ localizing subcategory of Γ_{𝔭}T \end{matrix} \}

which are defined by σ(S) = (S ∩ Γ_{𝔭}T)_{𝔭∈V} and τ(S(𝔭))_{𝔭∈V} = Loc_{T} (S(𝔭) | 𝔭 ∈ V). The next result expresses the local-global principle in terms of these maps.

Proposition 3.6. The following conditions are equivalent.

(1) The local-global principle holds for T.

(2) The map σ is bijective, with inverse τ.

(3) The map σ is one-to-one.

Proof. We repeatedly use the fact that Γ_{𝔭} is an exact functor preserving coproducts. For each localizing subcategory S of T and each 𝔭 in Spec R there is an inclusion

(3.7)  S ∩ Γ_{𝔭}T ⊆ Γ_{𝔭}S .

We claim that στ is the identity, that is to say, that for any family (S(𝔭))_{𝔭∈V} of localizing subcategories with S(𝔭) ⊆ Γ_{𝔭}T the localizing subcategory generated by all the S(𝔭), call it S, satisfies

S ∩ Γ_{𝔭}T = S(𝔭) , for each 𝔭 ∈ V.

To see this, note that Γ_{𝔭}S = S(𝔭) holds, since Γ_{𝔭}Γ_{𝔮} = 0 when 𝔭 ≠ 𝔮. Hence (3.7) yields an inclusion S ∩ Γ_{𝔭}T ⊆ S(𝔭). The reverse inclusion is obvious.

(1) ⟹ (2): It suffices to show that τσ equals the identity, since στ = id holds. Fix a localizing subcategory S of T. It is clear that τσ(S) ⊆ S. As to the reverse inclusion: Fixing X in S, it follows from Theorem 3.1(1) that Γ_{𝔭}X is in S∩Γ_{𝔭}T and hence in τσ(S), for each 𝔭 ∈ Spec R. Thus, X is in τσ(S), again by Theorem 3.1(1).

(2) ⟹ (3): Clear.

(3) ⟹ (1): Since στ = id and σ is one-to-one, one gets τσ = id. For each object X in T there is thus an equality:

Loc_{T}(X) = Loc_{T}({Loc_{T}(X) ∩ Γ_{𝔭}T | 𝔭 ∈ Spec R})
⊆ Loc_{T}({Γ_{𝔭}X | 𝔭 ∈ Spec R})

The inclusion follows from (3.7). Now apply Theorem 3.1. □

The local-global principle focuses attention on the subcategory Γ_{𝔭}T. Next we describe some of its properties, even though these are not needed in the sequel.

Local structure. Let 𝔭 be a point in Spec R. In analogy with the case of Rmodules, we say that an object X in T is 𝔭-local if

supp_{R} X ⊆ {𝔮 ∈ Spec R | 𝔮 ⊆ 𝔭}

and that X is 𝔭-torsion if

supp_{R} X ⊆ {𝔮 ∈ Spec R | 𝔮 ⊇ 𝔭}.

The objects of Γ_{𝔭}T are precisely those that are both 𝔭-local and 𝔭-torsion; see [7, Corollary 5.9] for alternative descriptions. Set

X(𝔭) = (X//𝔭)_{𝔭} .

The subcategory Loc_{T}(X(𝔭)) is independent of the choice of a generating set for the ideal 𝔭 used to construct X//𝔭; this follows from the result below.

Lemma 3.8. The following statements hold for each X ∈ T and 𝔭 ∈ Spec R. (1) X(𝔭) is 𝔭-local and 𝔭-torsion.

(2) Loc_{T}(X(𝔭)) = Loc_{T}(Γ_{𝔭}X).

(3) Hom_{T}(W, X(𝔭)) = 0 for any object W that is 𝔮-local and 𝔮-torsion with 𝔮 ≠ 𝔭.

Proof. The argument is based on the fact that the localization functor that takes an object X to X_{𝔭} is exact and preserves coproducts.

(1) Exactness of localization implies (X//𝔭)_{𝔭} can be realized as X_{𝔭}//𝔭. Hence X(𝔭) belongs to Thick_{T}(X_{𝔭}), so that it is 𝔭-local; it is 𝔭-torsion by [7, Lemma 5.11].

(2) Applying the localization functor to the equality Loc_{T}(X//𝔭) = Loc_{T}(Γ_{V(𝔭)}X) in Proposition 2.11 yields (2).

(3) If 𝔮 ⊈ 𝔭 holds, then Γ_{V(𝔮)}(X(𝔭)) = 0 and hence the desired claim follows from the adjunction isomorphism Hom_{T}(W, X(𝔭)) ≅ Hom_{T}(W, Γ_{V(𝔮)}X(𝔭)). If 𝔮 ⊆ 𝔭, then the R-module Hom_{T}^{∗}(W, X(𝔭)) is 𝔮-local, by Proposition 2.3, and 𝔭-torsion, by [7, Lemma 5.11], and hence zero since 𝔮 ≠ 𝔭. □

Proposition 3.9. For each 𝔭 in Spec R and each compact object C in T, the object C(𝔭) is compact in Γ_{𝔭}T, and both {C(𝔭) | C ∈ T^{c}} and {Γ_{𝔭}C | C ∈ T^{c}} generate the triangulated category Γ_{𝔭}T. Furthermore, the R-linear structure on T induces a natural structure of an R_{𝔭}-linear triangulated category on Γ_{𝔭}T.

Proof. Recall that Γ_{𝔭}T is a localizing subcategory of T, so the coproduct in it is the same as the one in T. Each object X in Γ_{𝔭}T is 𝔭-local, so there is an isomorphism

Hom_{T}(C(𝔭), X) ≅ Hom_{T}(C//𝔭, X) .

When C is compact in T, so is C//𝔭. Thus the isomorphism above implies that C(𝔭) is compact in Γ_{𝔭}T. Furthermore, the collection of objects C//𝔭 with C compact in T generates T_{V(𝔭)} by Proposition 2.7, and hence the C(𝔭) generate Γ_{𝔭}T.

The class of compact objects C generates T hence the objects Γ_{𝔭}C generate Γ_{𝔭}T. Proposition 2.3 implies that for each pair of objects X, Y in Γ_{𝔭}T the R-module

Hom_{T}^{∗}(X, Y) is 𝔭-local, so that they admit a natural R_{𝔭}-module structure. This translates to an action of R_{𝔭} on Γ_{𝔭}T. □

4. Stratification

In this section we introduce a notion of stratification for triangulated categories with ring actions. It is based on the concept of a minimal subcategory introduced by Hovey, Palmieri, and Strickland [19, §6].

As before T is a compactly generated R-linear triangulated category.

Minimal subcategories. A localizing subcategory of T is said to be minimal if it is nonzero and has no proper nonzero localizing subcategories.

Lemma 4.1. A nonzero localizing subcategory S of T is minimal if and only if for all nonzero objects X, Y in S one has Hom_{T}^{∗}(X, Y) ≠ 0.

Proof. When S is minimal and X a nonzero object in it Loc_{T}(X) = S, by minimality, so if Hom_{T}^{∗}(X, Y) = 0 for some Y in S, then Hom_{T}^{∗}(Y, Y) = 0, that is to say, Y = 0.

Suppose S contains a nonzero proper localizing subcategory S′; we may assume S′ = Loc_{T}(X) for some nonzero object X. For each object W in T there is then an exact triangle W′ ⟶^{θ} W ⟶^{η} W′′ → with W′ ∈ S′, Hom_{T}^{∗}(X, W′′) = 0, and θ invertible if and only if and W is in S′; see Lemma 2.1. It remains to pick an object W in S \ S′, set Y = W′′, and note that Y is in S and nonzero. □

Stratification. We say that T is stratified by R if the following conditions hold: (S1) The local-global principle, discussed in Section 3, holds for T. (S2) For each 𝔭 ∈ Spec R the localizing subcategory Γ_{𝔭}T is either zero or minimal.

The crucial condition here is (S2); for example, (S1) holds when the dimension of Spec R is finite, by Corollary 3.5. Since the objects in Γ_{𝔭}T are precisely the 𝔭-local and 𝔭-torsion ones in T, condition (S2) is that each nonzero 𝔭-local 𝔭-torsion object builds every other such object.

Given a localizing subcategory S of T and a subset U of Spec R set

supp_{R} S = ⋃_{X∈S} supp_{R} X and supp_{R}^{−1} U = {X ∈ T | supp_{R} X ⊆ U}.

Observe that supp_{R} and supp_{R}^{−1} both preserve inclusions.

Theorem 4.2. Let T be a compactly generated R-linear triangulated category. If T is stratified by R, then there are inclusion preserving inverse bijections:

\{ \begin{matrix}Localizing \\ subcategories of T \end{matrix} \} [figure: ^{supp^{R}}; _{supp{−1 \atop R}}]\{ Subsets of supp_{R} T \}

Conversely, if the map supp_{R} is injective, then T must be stratified by R.

Proof. For each 𝔭 ∈ Spec R the subcategory Ker Γ_{𝔭} is localizing. This implies that for any subset U of Spec R the subcategory supp_{R}^{−1} U is localizing, for

supp_{R}^{−1} U = ⋂_{𝔭∉U} Ker Γ_{𝔭} .

Moreover, it is clear that supp_{R}(supp_{R}^{−1} U) = U for each subset U of supp_{R} T, and that S ⊆ supp_{R}^{−1}(supp_{R} S) holds for any localizing subcategory S. The moot point is whether S contains supp_{R}^{−1}(supp_{R} S); equivalently, whether supp_{R} is one-to-one.

The map supp_{R} factors as σ′σ with σ as in Proposition 3.6 and σ′ the map

\{ \begin{matrix}Families (S(𝔭))_{𝔭∈supp_{R} T} with S(𝔭) \\ a localizing subcategory of Γ_{𝔭}T \end{matrix} \} ⟶ \{ Subsets of supp_{R} T \}

where σ′(S(𝔭)) = {𝔭 ∈ Spec R | S(𝔭) ≠ {0}}. Evidently σ′ is one-to-one if and only if it is bijective, if and only if the minimality condition (S2) holds. The map σ is also one-to-one if and only if it is bijective; moreover this holds precisely when the local-global principle holds for T, by Proposition 3.6. The desired result follows. □

Corollary 4.3. If R stratifies T and G is a set of generators for T, then each localizing subcategory S of T is generated by the set S ∩ {Γ_{𝔭}X | X ∈ G, 𝔭 ∈ Spec R}. In particular, there exists a localization functor L: T → T such that S = Ker L.

Proof. The first assertion is an immediate consequence of Theorem 4.2, since S and the localizing subcategory generated by the given set of objects have the same support. Given this, the second one follows from Lemma 2.1. □

Other consequences of stratification are given in Sections 5 and 6. Now we provide examples of triangulated categories that are stratified; see also Example 7.4.

Example 4.4. Let A be a commutative noetherian ring and D(A) the derived category of the category of A-modules. The category D(A) is compactly generated, A-linear, and triangulated. This example is discussed in [7, §8], where it is proved that the notion of support introduced in [7] coincides with the usual one, due to Foxby and Neeman; see [7, Theorem 9.1]. In view of Theorem 4.2, one can reformulate [25, Theorem 2.8] as: The A-linear triangulated category D(A) is stratified by A. This example will be subsumed in Theorem 8.1.

Example 4.5. Let k be a field and Λ an exterior algebra over k in finitely many indeterminates of negative odd degree; the grading is upper. We view Λ as a dg algebra, with differential zero. In [8, §6] we introduced the homotopy category of graded-injective dg Λ-modules and proved that it is stratified by a natural action of its cohomology algebra, Ext_{Λ}^{∗}(k, k).

The next example shows that there are triangulated categories which cannot be stratified by any ring action.

Example 4.6. Let k be a field and Q a quiver of Dynkin type; see, for example, [4, Chapter 4]. The path algebra kQ is a finite dimensional hereditary algebra of finite representation type. It is easily checked that the graded center of the derived category D(kQ) is isomorphic to k. In fact, each object in D(kQ) is a direct sum of indecomposable objects, and End_{D(kQ)}^{∗}(X) ≅ k for each indecomposable object X. The localizing subcategories of D(kQ) are parameterized by the noncrossing partitions associated to Q; this can be deduced from work of Ingalls and Thomas [20]. Thus the triangulated category D(kQ) is stratified by some ring acting on it if and only if the quiver consists of one vertex and has no arrows.

5. Orthogonality

Let X and Y be objects in T. The discussion below is motivated by the question: when is Hom_{T}^{∗}(X, Y) = 0? The orthogonality property [7, Corollary 5.8] says that if cl(supp_{R} X) and supp_{R} Y are disjoint, then one has the vanishing. What we seek are converses to this statement, ideally in terms of the supports of X and Y. Lemma 4.1 suggests that one can expect satisfactory answers only when T is stratified. In this section we establish some results addressing this question and give examples which indicate that these may be the best possible.

For any graded R-module M set Supp_{R} M = {𝔭 ∈ Spec R | M_{𝔭} ≠ 0}. This subset is sometimes referred to as the ‘big support’ of M to distinguish it from its ‘homological’ support, supp_{R} M. Analogously, for any object X in T, we set

Supp_{R} X = ⋃_{C∈T^{c}} Supp_{R} Hom_{T}^{∗}(C, X) .

It follows from [7, Theorem 5.15(1) and Lemma 2.2(1)] that there is an equality:

Supp_{R} X = cl(supp_{R} X) .

We use this equality without further comment.

Theorem 5.1. Let T be a compactly generated R-linear triangulated category. If R stratifies T, then for each compact object C and each object Y, there is an equality

Supp_{R} Hom_{T}^{∗}(C, Y) = Supp_{R} C ∩ Supp_{R} Y .

The proof requires only stratification condition (S2), never (S1).

Proof. Fix a prime ideal 𝔭 ∈ Spec R. Suppose Hom_{T}^{∗}(C, Y)_{𝔭} ≠ 0; by definition, one then has 𝔭 ∈ Supp_{R} Y. Moreover End_{T}^{∗}(C)_{𝔭} ≠ 0 since the R-action on Hom_{T}^{∗}(C, Y)_{𝔭} factors through it, hence 𝔭 is also in Supp_{R} C. Thus there is an inclusion

Supp_{R} Hom_{T}^{∗}(C, Y) ⊆ Supp_{R} C ∩ Supp_{R} Y .

Now suppose Hom_{T}^{∗}(C, Y)_{𝔭} = 0. One has to verify that that for any prime ideal 𝔮 ⊆ 𝔭 either Γ_{𝔮}C = 0 or Γ_{𝔮}Y = 0. By [7, Theorem 4.7], see also Proposition 2.3, since C is compact the adjunction morphism Y → Y_{𝔮} induces an isomorphism

0 = Hom_{T}^{∗}(C, Y)_{𝔮} ≅ Hom_{T}^{∗}(C, Y_{𝔮}) .

As Γ_{V(𝔮)}Y is in Loc_{T}(Y), by Proposition 2.11, one obtains that Γ_{𝔮}Y is in Loc_{T}(Y_{𝔮}), hence the calculation above yields Hom_{T}^{∗}(C, Γ_{𝔮}Y) = 0. As Γ_{𝔮}Y is 𝔮-local the adjunction morphism C → C_{𝔮} induces the isomorphism below

Hom_{T}^{∗}(C_{𝔮}, Γ_{𝔮}Y) ≅ Hom_{T}^{∗}(C, Γ_{𝔮}Y) = 0 .

Using now the fact that Γ_{𝔮}C is in Loc_{T}(C_{𝔮}) one gets Hom_{T}^{∗}(Γ_{𝔮}C, Γ_{𝔮}Y) = 0. Our hypothesis was that R stratifies T. Thus one of Γ_{𝔮}C or Γ_{𝔮}Y is zero. □

The example below shows that the conclusion of the preceding theorem need not hold when C is not compact. See also Example 5.9

Example 5.2. Let A be a commutative noetherian ring with Krull dimension at least one and 𝔪 a maximal ideal of A that is not also a minimal prime. For example, take A = ℤ and 𝔪 = (p), where p is a prime number.

Let T be the derived category of A-modules, viewed as an A-linear category; see Example 4.4. Let E be the injective hull of A/𝔪. The A-module Hom_{T}^{∗}(E, E) is then the 𝔪-adic completion of A, so it follows that

Supp_{A} Hom_{T}^{∗}(E, E) = {𝔭 ⊆ 𝔪 | 𝔭 ∈ Spec R} ⊋ {𝔪} = Supp_{A} E .

Observe that supp_{A} Hom_{T}^{∗}(E, E) = Supp_{A} Hom_{T}^{∗}(E, E) and supp_{A} E = Supp_{A} E.

One drawback of Theorem 5.1 is that it involves the big support Supp_{R}, while one is mainly interested in supp_{R}. Next we identify a rather natural condition on T under which one can obtain results in the desired form.

Noetherian categories. We call a compactly generated R-linear triangulated category noetherian if for any compact object C in T the R-module End_{T}^{∗}(C) is finitely generated. This is equivalent to the condition that for all compact objects C, D the R-module Hom_{T}^{∗}(C, D) is finitely generated: consider End_{T}^{∗}(C ⊕ D). If C generates T, then T is noetherian if and only if the R-module End_{T}^{∗}(C) is noetherian.

As a consequence of Theorem 5.1 one gets:

Corollary 5.3. If T is noetherian and stratified by R, then for each pair of compact objects C, D in T there is an equality

supp_{R} Hom_{T}^{∗}(C, D) = supp_{R} C ∩ supp_{R} D .

When in addition R^{i} = 0 holds for i < 0, one has Hom_{T}^{n}(C, D) = 0 for n ≫ 0 if and only if Hom_{T}^{n}(D, C) = 0 for n ≫ 0.

Proof. In view of the noetherian hypothesis and [7, Lemma 2.2(1), Theorem 5.5(2)], the desired equality follows from Theorem 5.1. It implies in particular that

supp_{R} Hom_{T}^{∗}(C, D) = supp_{R} Hom_{T}^{∗}(D, C) .

When R^{i} = 0 holds for i < 0 and M is a noetherian R-module one has M^{n} = 0 for n ≫ 0 if and only if supp_{R} M ⊆ {𝔭 ∈ Spec R | 𝔭 ⊇ R^{⩾1}}; see [10, Proposition 2.4]. The last part of the corollary now follows from the equality above. □

There is a version of the preceding result where the objects C and D need not be compact. This is the topic of the next theorem. As preparation for its proof, and for later applications, we further develop the material in [7, Definition 4.8]. Let C be a compact object in T. For each injective R-module I, the Brown representability theorem [23, 27] yields an object T_{C}(I) in T such that there is a natural isomorphism:

Hom_{T}(−, T_{C}(I)) ≅ Hom_{R}(Hom_{T}^{∗}(C, −), I) .

Moreover, the assignment I ↦ T_{C}(I) defines a functor T_{C}: Inj R → T from the category of injective R-modules to T.

Proposition 5.4. Let C be a compact object in T. The functor T_{C}: Inj R → T preserves products. If the R-linear category T is noetherian, each I ∈ Inj R satisfies:

supp_{R} T_{C}(I) = supp_{R} C ∩ supp_{R} I = supp_{R} End_{T}^{∗}(C) ∩ supp_{R} I .

In particular, for each 𝔭 ∈ Spec R the object T_{C}(E(R/𝔭)) is in Γ_{𝔭}T.

Proof. It follows by construction that T_{C} preserves products. For each compact object D in T, there is an isomorphism of R-modules

Hom_{T}^{∗}(D, T_{C}(I)) ≅ Hom_{R}^{∗}(Hom_{T}^{∗}(C, D), I) .

When T is noetherian, so that the R-module Hom_{T}^{∗}(C, D) is finitely generated, the isomorphism above gives the first equality below:

supp_{R} Hom_{T}^{∗}(D, T_{C}(I)) = supp_{R} Hom_{T}^{∗}(C, D) ∩ supp_{R} I
= supp_{R} C ∩ supp_{R} D ∩ supp_{R} I .

The second equality holds by Corollary 5.3. Lemma 5.5 below then yields the first of the desired equalities; the second one holds by [7, Theorem 5.5(2)]. □

The following lemma provides an alternative description of the support of an object in T. Note that T need not be noetherian.

Lemma 5.5. Let X be an object in T and U a subset of supp_{R} T. If

supp_{R} Hom_{T}^{∗}(C, X) = U ∩ supp_{R} C

holds for each compact object C, then supp_{R} X = U.

Proof. It follows from [7, Theorem 5.2] that supp_{R} X ⊆ U.

Fix 𝔭 in U and choose a compact object D with 𝔭 in supp_{R} D. Then 𝔭 is in supp_{R}(D//𝔭), so the hypothesis yields that 𝔭 is in supp_{R} Hom_{T}^{∗}(D//𝔭, X). Hence 𝔭 belongs to supp_{R} X, by [7, Proposition 5.12]. □

For a compact object C, the functor Hom_{T}^{∗}(C, −) vanishes on Loc_{T}(Y) if and only if Hom^{∗}(C, Y) = 0. Using this observation, it is easy to verify that the theorem below is an extension of Corollary 5.3. Compare it also with [7, Corollary 5.8].

Theorem 5.6. Let T be an R-linear triangulated category that is noetherian and stratified by R. For any X and Y in T the conditions below are equivalent:

(1) Hom_{T}^{∗}(X, Y′) = 0 for any Y′ in Loc_{T}(Y);

(2) cl(supp_{R} X) ∩ supp_{R} Y = ∅.

Proof. (1) ⟹ (2): Let 𝔭 be a point in supp_{R} Y and C a compact object in T. Proposition 5.4 yields that T_{C}(E(R/𝔭)) is in Γ_{𝔭}T, and hence also in Loc_{T}(Y); the last assertion holds by Theorem 4.2. This explains the equality below:

Hom_{R}^{∗}(Hom_{T}^{∗}(C, X), E(R/𝔭)) ≅ Hom_{T}^{∗}(X, T_{C}(E(R/𝔭))) = 0 ,

while the isomorphism follows from the definition of T_{C}. Thus Hom_{T}^{∗}(C, X)_{𝔭} = 0. Since C was arbitrary, this means that 𝔭 is not in cl(supp_{R} X).

(2) ⟹ (1): One has supp_{R} Y′ ⊆ supp_{R} Y for Y′ in Loc_{T}(Y), since the functor Γ_{𝔭} is exact and preserves coproducts. The orthogonality property of supports, [7, Corollary 5.8] thus implies that if condition (2) holds, then Hom_{T}^{∗}(X, Y′) = 0. □

Recall that the left orthogonal subcategory of S, denoted ^{⊥}S, is the localizing subcategory {X ∈ T | Hom_{T}^{∗}(X, Y) = 0 for all Y ∈ S}. As a straightforward consequence of Theorem 5.6 one obtains a description of the support of the left orthogonal of a localizing category, answering a question raised by Rickard.^{1}

Corollary 5.7. For each localizing subcategory S of T the following equality holds: supp_{R}(^{⊥}S) = {𝔭 ∈ supp_{R} T | V(𝔭) ∩ supp_{R} S = ∅}. □

Remark 5.8. In the context of Theorem 5.6, for any compact object C one has

Hom_{T}^{∗}(C, Y) = 0 if and only if supp_{R} C ∩ supp_{R} Y = ∅ .

The next example shows that one cannot do away entirely with the hypothesis that C is compact; the point being that Hom_{T}^{∗}(X, Y) = 0 does not imply that Hom^{∗}(X, −) is zero on Loc_{T}(Y), unless X is compact.

Example 5.9. Let A be a complete local domain and Q its field of fractions. For example, take A to be the completion of ℤ at a prime p. It follows from a result of Jensen [22, Theorem 1] that Ext_{A}^{∗}(Q, A) = 0. Thus, with T the derived category of A, one gets supp_{A} Hom_{T}^{∗}(Q, A) = ∅ while supp_{A} Q ∩ supp_{A} A consists of the zero ideal. Note that Q is in Loc_{T}(A), so there is no contradiction with Theorem 5.6.

1After a talk by Iyengar at the workshop ‘Homological methods in group theory’, MSRI 2008.

6. Classifying thick subcategories

In this section we prove that when T is noetherian and stratified by R its thick subcategories of compact objects are parameterized by specialization closed subsets of supp_{R} T. As before, R is a graded-commutative noetherian ring and T is a compactly generated R-linear triangulated category.

Thick subcategories. One can deduce the next result from the classification of localizing subcategories, Theorem 4.2, as in [25, §3]. We give a different proof.

Theorem 6.1. Let T be a compactly generated R-linear triangulated category that is noetherian and stratified by R. The map

\{ \begin{matrix}Thick subcategories \\ of T^{c} \end{matrix} \} [figure: ^{supp^{R}}]\{ \begin{matrix}Specialization closed \\ subsets of supp_{R} T \end{matrix} \}

is bijective. The inverse map sends a specialization closed subset V of Spec R to the subcategory {C ∈ T^{c} | supp_{R} C ⊆ V}.

Observe that in the proof the injectivity of the map supp_{R} requires only that T satisfies the stratification condition (S2), while the surjectivity uses only the hypothesis that T is noetherian.

Proof. First we verify that supp_{R} C is specialization closed for any thick subcategory C of T^{c}. For any compact object C the R-module End_{T}^{∗}(C) is finitely generated, and this implies supp_{R} C = supp_{R} End_{T}^{∗}(C), by [7, Theorem 5.5]. Thus supp_{R} C is a closed subset of Spec R, and therefore supp_{R} C is specialization closed.

To verify that the map supp_{R} is surjective, let V be a specialization closed subset of supp_{R} T and set C = {C//𝔭 | C ∈ T^{c}, 𝔭 ∈ V}. One then has that Loc_{T}(C) = T_{V} by [7, Theorem 6.4], and therefore the following equalities hold

supp_{R} C = supp_{R} T_{V} = V ∩ supp_{R} T = V .

It remains to prove that supp_{R} is injective. Let C be a thick subcategory of T and set D = {D ∈ T^{c} | supp_{R} D ⊆ supp_{R} C}. We need to show that C = D. Evidently, an inclusion C ⊆ D holds. To establish the other inclusion, let L: T → T be the localization functor with Ker L = Loc_{T}(C); see Lemma 2.1 for its existence. Let D be an object in D. Each object C in C satisfies Hom_{T}^{∗}(C, LD) = 0, so Theorem 5.1 implies Supp_{R} C ∩ Supp_{R} LD = ∅. Hence LD = 0, that is to say, D belongs to Loc_{T}(C). It then follows from [25, Lemma 2.2] that D is in C. □

Smashing subcategories. Next we prove that when T is stratified and noetherian, the telescope conjecture [31] holds for T. In preparation for its proof, we record an elementary observation.

Lemma 6.2. Let 𝔭 ⊆ 𝔮 be prime ideals in Spec R. The injective hull E(R/𝔭) of R/𝔭 is a direct summand of a product of shifted copies of E(R/𝔮).

Proof. The shifted copies of E(R/𝔮) form a set of injective cogenerators for the category of 𝔮-local modules. This implies the desired result. □

A subset U of Spec R is said to be closed under generalization if Spec R \ U is specialization closed. More explicitly: 𝔮 ∈ U and 𝔭 ⊆ 𝔮 imply 𝔭 ∈ U.

Theorem 6.3. Let T be an R-linear triangulated category that is noetherian and stratified by R. There is then a bijection

\{ \begin{matrix}Localizing subcategories of T \\ closed under all products \end{matrix} \} [figure: ^{supp^{R}}]\{ \begin{matrix}Subsets of supp_{R} T \\ closed under generalization \end{matrix} \}

Moreover, if L: T → T is a localization functor that preserves arbitrary coproducts, then the localizing subcategory Ker L is generated by objects that are compact in T.

Remark 6.4. The inverse map of supp_{R} takes a generalization closed subset U of Spec R to the category of objects X of T with supp_{R} X ⊆ U; in other words, the category of L_{V}-local objects, where V = Spec R \ U.

Proof. Let S be a localizing subcategory of T that is closed under arbitrary products. We know from Theorem 4.2 that S is determined by its support supp_{R} S. Thus we need to show that it is closed under generalization.

Fix prime ideals 𝔭 ⊆ 𝔮 in supp_{R} T and suppose that 𝔮 is in supp_{R} S. It follows from Theorem 4.2 that Γ_{𝔮}T ⊆ S holds. Pick a compact object C such that supp_{R} C contains 𝔭; this is possible since supp_{R} T^{c} = supp_{R} T. Since T is noetherian, supp_{R} C is a closed subset of Spec R, by [7, Theorem 5.5], and hence contains also 𝔮. Let E(R/𝔮) be the injective hull of the R-module R/𝔮. Since T is noetherian, Proposition 5.4 yields that T_{C}(E(R/𝔮)) is in Γ_{𝔮}T and hence in S. The functor T_{C} preserves products, so Lemma 6.2 implies that T_{C}(E(R/𝔭)) is a direct summand of T_{C}(E(R/𝔮)) and hence it is also in S, because the latter is a localizing subcategory closed under products. Another application of Proposition 5.4 shows that supp_{R} T_{C}(E(R/𝔭)) = {𝔭}, so that 𝔭 ∈ supp_{R} S holds, as desired

Next let U be a generalization closed subset of Spec R and set V = Spec R \ U. Let S be the category of L_{V}-local objects, so that supp_{R} S = U holds, by [7, Corollary 5.7]. By construction, the category S is triangulated and closed under arbitrary products; it is localizing because the localization functor L_{V} preserves arbitrary coproducts, by [7, Corollary 6.5].

This completes the proof that supp_{R} induces the stated bijection.

Finally, let L: T → T be a localization functor that preserves arbitrary coproducts. The category of L-local objects, which always is closed under products, is then also a localizing subcategory of T. The first part of this proof shows that L ≅ L_{V} for some specialization closed subset V of Spec R, because the localization functor L is determined by the category of L-local objects. It remains to note that Ker L, which is the category T_{V}, is generated by compact objects, by [7, Theorem 6.4]. □

7. Tensor triangulated categories

In this section we discuss special properties of triangulated categories which hold when they have a tensor structure. The main result here is Theorem 7.2, which says that the local-global principle holds for such categories, when the action of the tensor product is also taken into account.

Let T = (T, ⊗, 𝟙) be a tensor triangulated category as defined in [7, §8]. In particular, T is a compactly generated triangulated category endowed with a symmetric monoidal structure; ⊗ is its tensor product and 𝟙 the unit of the tensor product. It is assumed that ⊗ is exact in each variable, preserves coproducts, and that 𝟙 is compact.

The symmetric monoidal structure ensures that the endomorphism ring End_{T}^{∗}(𝟙) is graded commutative. This ring acts on T via homomorphisms

End_{T}^{∗}(𝟙) ⟶^{X⊗−} End_{T}^{∗}(X) ,

In particular, any homomorphism R → End_{T}^{∗}(𝟙) of rings with R graded commutative induces an action of R on T. We say that an R action on T is canonical if it arises from such a homomorphism. In that case there are for each specialization closed subset V and point 𝔭 of Spec R natural isomorphisms

(7.1)  Γ_{V}X ≅ X ⊗ Γ_{V}𝟙 , L_{V}X ≅ X ⊗ L_{V}𝟙 , and Γ_{𝔭}X ≅ X ⊗ Γ_{𝔭}𝟙 .

These isomorphisms are from [7, Theorem 8.2, Corollary 8.3].^{2}

Tensor ideal localizing subcategories. A localizing subcategory S of T said to be tensor ideal if for each X ∈ T and Y ∈ S, the object X ⊗ Y, hence also Y ⊗ X, is in S. The smallest tensor ideal localizing subcategory containing a subcategory S is denoted Loc_{T}^{⊗}(S). Evidently there is always an inclusion Loc_{T}(S) ⊆ Loc_{T}^{⊗}(S); equality holds when the unit 𝟙 generates T.

The following result is proved in [8, Theorem 3.6] under the additional assumption that T has a single compact generator. The same argument carries over; except that, instead of [8, Proposition 3.5] use Proposition 2.7 above. We omit details.

Theorem 7.2. Let T be a tensor triangulated category with a canonical R-action. For each object X in T there is an equality

Loc_{T}^{⊗}(X) = Loc_{T}^{⊗} (Γ_{𝔭}X | 𝔭 ∈ Spec R) .

In particular, when 𝟙 generates T, the local global principle holds for T. □

Stratification. For each 𝔭 in Spec R, the localizing subcategory Γ_{𝔭}T, consisting of 𝔭-local and 𝔭-torsion objects, is tensor ideal; this is immediate from (7.1). We say that T is stratified by R when for each 𝔭, the category Γ_{𝔭}T is either zero or has no proper tensor ideal localizing subcategories. Note the analogy with condition (S2) in Section 4; the analogue of (S1) need not be imposed thanks to Theorem 7.2.

There are analogues, for tensor triangulated categories, of results in Sections 5 and 6; the proofs are similar, see also [7, §11]. One has in addition also the following ‘tensor product theorem’.

Theorem 7.3. Let T be a tensor triangulated category with a canonical R-action. If R stratifies T, then for any objects X, Y in T there is an equality

supp_{R}(X ⊗ Y) = supp_{R} X ∩ supp_{R} Y .

Proof. Fix a point 𝔭 in Spec R. From 7.1 it is easy to verify that there are isomorphisms Γ_{𝔭}(X ⊗ Y) ≅ Γ_{𝔭}X ⊗ Γ_{𝔭}Y ≅ Γ_{𝔭}X ⊗ Y. These will be used without further ado. They yield an inclusion:

supp_{R}(X ⊗ Y) ⊆ supp_{R} X ∩ supp_{R} Y .

When Γ_{𝔭}X ≠ 0 the stratification condition yields Γ_{𝔭}𝟙 ∈ Loc^{⊗}(Γ_{𝔭}X), and hence also Γ_{𝔭}Y ∈ Loc^{⊗}(Γ_{𝔭}X ⊗ Y). Thus when Γ_{𝔭}Y ≠ 0 also holds, Γ_{𝔭}(X ⊗ Y) ≠ 0 holds, which justifies the reverse inclusion. □

Example 7.4. Let G be a finite group, k a field of characteristic p, where p divides the order of G, and kG the group algebra. The homotopy category of complexes of injective kG-modules, K(Inj kG), is a compactly generated tensor triangulated category with a canonical action of the cohomology ring H^{∗}(G, k). One of the main results of [8], Theorem 9.7, is that K(Inj kG) is stratified by this action. The same is true also of the stable module category StMod kG; see [8, Theorem 10.3].

2For these results to hold, the R action should be canonical, for the R-linearity of the adjunction isomorphism Hom_{T}(X ⊗ Y, Z) ≅ Hom_{T}(X, Hom(Y, Z)) is used in the arguments.

8. Formal differential graded algebras

The goal of this section is to prove that the derived category of differential graded (henceforth abbreviated to ‘dg’) modules over a formal commutative dg algebra is stratified by its cohomology algebra, when that algebra is noetherian. This result specializes to one of Neeman’s [25] concerning rings, which may be viewed as dg algebras concentrated in degree 0.

For basic notions concerning dg algebras and dg modules over them we refer the reader to Mac Lane [24, §6.7]. A quasi-isomorphism between dg algebras A and B is a morphism φ: A → B of dg algebras such that H^{∗}(φ) is bijective; A and B are quasi-isomorphic if there is a chain of quasi-isomorphisms linking them. The multiplication on A induces one on its cohomology, H^{∗}(A). We say that A is formal if it is quasi-isomorphic to H^{∗}(A), viewed as a dg algebra with zero differential.

We write D(A) for the derived category of dg modules over a dg algebra A; it is a triangulated category, generated by the compact object A; see, for instance, [23].

A dg algebra A is said to be commutative if its underlying ring is graded commutative. In this case the derived tensor product of dg modules, denoted ⊗^{L}, endows D(A) with a structure of a tensor triangulated category, with unit A. One is thus in the framework of Section 7.

The next theorem generalizes [8, Theorem 5.2], which deals with the case of graded algebras of the form k[x_{1}, …, x_{n}], where k is a field and x_{1}, …, x_{n} are indeterminates, of even degree if the characteristic of k is not 2.

Theorem 8.1. Let A be a commutative dg algebra such that the ring H^{∗}(A) is noetherian. If A is formal, then D(A) is stratified by the canonical H^{∗}(A)-action.

In the proof we use a totalization functor from complexes over a graded ring to dg modules over the ring viewed as a dg algebra with differential zero; see [24, §10.9], where this functor is called condensation, and [23, §3.3].

Totalization. Let A be a graded algebra. For each graded A-module N and integer d we write N[d] for the graded A-module with N[d]^{i} = N^{d+i}, and multiplication the same as the one on N.

Let F be a complex of graded A-modules with differential δ; so each F^{i} is a graded A-module, δ^{i}: F^{i} → F^{i+1} are morphisms of graded A-modules, and δ^{i+1}δ^{i} = 0. We write F^{i,j} for the component of degree j in the graded module F^{i}. The totalization of F, denoted tot F, is the dg abelian group with

(tot F)^{n} = ⨁_{i+j=n} F^{i,j} for each n ∈ ℤ
∂(f) = δ^{i}(f) for each f ∈ F^{i,j}

We consider tot F as a graded A-module with multiplication defined by

a · f = (−1)^{di}af for each a ∈ A^{d} and f ∈ F^{i,j}.

A routine calculation shows that tot F is then a dg A-module, where A is viewed as dg algebra with zero differential, and that each morphism α: F → G of complexes of graded A-modules induces a morphism tot α: tot F → tot G of dg A-modules. Moreover, there are equalities of dg A-modules:

• tot A = A;

• tot N[d] = Σ^{d} tot N for each graded A-module N and integer d;

• tot Σ^{n}F = Σ^{n} tot F.

One thus gets an additive functor from the category of complexes of graded Amodules to the category of dg A-modules. It is easy to check that if the complex F is acyclic so is tot F.

Indeed, fix a cycle z in (tot F)^{n}, and write z = ∑_{i} z_{i} where z_{i} ∈ F^{i,n−i}. Since δ(z) = ∑_{i} δ^{i}(z_{i}) and δ^{i}(z_{i}) ∈ F^{i+1,n−i}, each z_{i} is a cycle in F^{i}. Since F is acyclic there exist elements w_{i} ∈ F^{i−1,j} with δ^{i−1}(w_{i}) = z_{i}; moreover, one may take w_{i} = 0 when z_{i} = 0. Note that the element w = ∑_{i} w_{i} is in (tot F)^{n−1} and δ(w) = z.

In conclusion, tot induces an exact functor

tot: D(GrMod A) ⟶ D(A) .

of triangulated categories; here D(GrMod A) is the derived category of graded Amodules, while D(A) is the derived category of dg A-modules.

Lemma 8.2. Let E be the Koszul complex on a sequence a = a_{1}, …, a_{c} of homogenous central elements in A. Then tot E ≅ Σ^{d}A//a in D(A), where d = ∑_{n} |a_{n}|.

Proof. Indeed, since tot preserves exact triangles, and both E and A//a can be obtained as iterated mapping cones, it suffices to verify the statement for the Koszul complex on a single element, say a. The desired result is then immediate from the properties of tot listed above. □

We require also some elementary results concerning transfer of stratification along exact functors; a detailed study is taken up in [9, Section 7].

Change of categories. As before R is a graded commutative noetherian ring and T is a compactly generated R-linear triangulated category. Let F: U → T be an equivalence of triangulated categories. Observe that U is then compactly generated; it is also R-linear with action given by the isomorphism of graded abelian groups

Hom_{U}^{∗}(X, Y) ≅ Hom_{T}^{∗}(F X, F Y)

induced by F, for all X, Y in U.

Proposition 8.3. The ring R stratifies U if and only if it stratifies T.

Proof. Using [7, Corollary 5.9], it is easy to verify that for each 𝔭 in Spec R and X in U, there is an isomorphism F (Γ_{𝔭}X) ≅ Γ_{𝔭}(F X), and that the induced functor Γ_{𝔭}U → Γ_{𝔭}T is an equivalence of triangulated categories. Given this, it is immediate from definitions that R stratifies U if and only if it stratifies T. □

When A → B is a quasi-isomorphism of dg algebras, B ⊗_{A}^{L} −: D(A) → D(B) is an equivalence of categories, with quasi-inverse the restriction of scalars; see, for example, [3, 3.6], or [23, 6.1]. The preceding result thus yields:

Corollary 8.4. Let A and B be quasi-isomorphic dg algebras. If D(A) is stratified by an action of R, then D(B) is stratified by the induced R-action. □

Proof of Theorem 8.1. Let R = H^{∗}(A). The category D(A) is tensor triangulated so it admits an R-action induced by the isomorphism R ≅ Hom_{D(A)}^{∗}(A, A). The dg algebras A and H^{∗}(A) are quasi-isomorphic, as A is formal, so it suffices to prove that D(H^{∗}(A)) is stratified by the induced R-action; see Corollary 8.4. It is easy to verify that the homomorphism R → Hom_{D(H^{∗}(A))}^{∗}(H^{∗}(A), H^{∗}(A)) = H^{∗}(A) induced by this R-action is bijective, and hence that D(H^{∗}(A)) is stratified by R if and only if it is stratified by the canonical H^{∗}(A)-action.

In summary, replacing A by H^{∗}(A) we may thus assume the differential of A is zero. Set D = D(A). Since A is a unit and a generator of this tensor triangulated category, its localizing subcategories are tensor closed. The local-global principle then holds for D, by Theorem 7.2. It remains to verify stratification condition (S2).

Fix a 𝔭 in Spec A. Since A is a compact generator for D, a dg A-module M is in Γ_{𝔭}D if and only if the A-module H^{∗}(M) = Hom_{D}^{∗}(A, M) is 𝔭-local and 𝔭-torsion. Hence for such an M the localization map M → M_{𝔭} is an isomorphism; here M_{𝔭} denotes the usual (homogenous) localization of M at 𝔭. Localizing A at 𝔭 we may thus assume that it is local with maximal ideal 𝔭; set k = A/𝔭, which is a graded field. Setting V = V(𝔭), one has an isomorphism of functors Γ_{𝔭} ≅ Γ_{V}.

Evidently, k is in Γ_{V}D, so to verify condition (S2) it suffices to verify that

(8.5)  Loc_{D}(M) = Loc_{D}(k)

holds for each M in Γ_{V}D with H^{∗}(M) ≠ 0.

It is enough to prove that (8.5) holds for M = Γ_{V}A. Indeed, applying the functor − ⊗_{A}^{L} M would then yield the second equality below:

Loc_{D}(M) = Loc_{D}(Γ_{V}A ⊗_{A}^{L} M) = Loc_{D}(k ⊗_{A}^{L} M) ,

while the first one holds, by (7.1), since M ≅ Γ_{V}M; in particular, H^{∗}(k ⊗_{A}^{L} M) ≠ 0. Since k is a graded field and the action of A on k ⊗_{A}^{L} M factors through k, this implies Loc_{D}(k ⊗_{A}^{L} M) = Loc_{D}(k). Combining with the equality above gives (8.5).

Now we verify (8.5) for M = Γ_{V}A. The dg module k is isomorphic to Γ_{V}A ⊗_{A}^{L} k and hence in Loc_{D}(Γ_{V}A). It remains to prove that Γ_{V}A is in Loc_{D}(k). Let a = a_{1}, …, a_{c} be a homogeneous set of generators for the ideal 𝔭, and let a^{2} denote the sequence a_{1}^{2}, …, a_{c}^{2}. It suffices to prove that

(8.6)  A//a^{2} ∈ Thick_{D}(k) ,

for then one has

Loc_{D}(A//𝔭) = Loc_{D}(Γ_{V}A) = Loc_{D}(Γ_{V(a^{2})}A) = Loc_{D}(A//a^{2}) ⊆ Loc_{D}(k)

where the first and third equalities are by Proposition 2.11, and the second holds because the radical of the ideal (a^{2}) equals 𝔭, so that V(a^{2}) = V.

Let tot: D(GrMod A) → D be the totalization functor described above and E in D(GrMod A) the Koszul complex on the sequence a^{2}; note that the elements a_{i} are central in A, since they are of even degree. The complex E is bounded, consists of finitely generated graded A-modules, and satisfies (a^{2}) · H^{∗}(E) = 0. Since k is a graded field, the subquotients of the filtration {0} ⊆ (a)H^{∗}(E) ⊆ H^{∗}(E) are thus finite direct sums of shifts of k. Hence there are inclusions

E ∈ Thick(H^{∗}(E)) ⊆ Thick(k)

in D(GrMod A); see, for example, [3, Theorem 6.2(3)]. Since tot is an exact functor, it follows that tot E is in Thick(tot k) in D. It remains to note that tot k = k and that tot E is isomorphic to a suspension of A//a^{2}, by Lemma 8.2.

This justifies (8.6) and hence completes the proof of the theorem. □

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Dave Benson, Institute of Mathematics, University of Aberdeen, King’s College, Aberdeen AB24 3UE, Scotland U.K.

Srikanth B. Iyengar, Department of Mathematics, University of Nebraska, Lincoln, NE 68588, U.S.A.

Henning Krause, Institut für Mathematik, Universität Paderborn, 33095 Paderborn, Germany.

Current address: Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany