A LOCAL-GLOBAL PRINCIPLE FOR SMALL TRIANGULATED CATEGORIES

DAVE BENSON, SRIKANTH B. IYENGAR, HENNING KRAUSE

Abstract. Local cohomology functors are constructed for the category of cohomological functors on an essentially small triangulated category T equipped with an action of a commutative noetherian ring. This is used to establish a local-global principle and to develop a notion of stratification, for T and the cohomological functors on it, analogous to such concepts for compactly generated triangulated categories.

Contents

1. Introduction  1
2. Cohomological functors  3
3. Cohomological localisation  7
4. Support  12
5. The local-global principle  14
6. Tensor triangulated categories  17
7. Stratification  18
8. Compactly generated triangulated categories  21
References  22

1. Introduction

In this paper we establish an analogue of the local-global principle from commutative algebra [7, Chap. II, §3] for an essentially small triangulated category, using the central action of a graded commutative ring. This has applications to the theory of support varieties in representation theory and in commutative algebra, that started with the work of Quillen [22] and Carlson [8].

Our paradigms for the local-global principle are the ones from [4] and from Stevenson’s work [23] for compactly generated (so “big”) triangulated categories. These play a crucial role in the classification of localising subcategories of the stable module category of the group algebra of a finite group [4] and of the singularity category of a locally complete intersection ring [24]. The local-global principle from [4] does yield an analogue (see Proposition 8.3) for the “small” category of compact objects, and that is useful in studying its thick subcategories, for example. The work presented here arose from a search for a more direct proof of this result.

2010 Mathematics Subject Classification. 18E30 (primary), 13D45, 16E35, 20J06.

Key words and phrases. cohomological functor, local cohomology, local-global principle, support.

Version from May 7, 2013.

The authors are grateful to MSRI at Berkeley for support while this work was in progress. The second author was partly supported by NSF grant DMS 1201889 and a Simons Fellowship.

Our reason for doing so, besides the obvious aesthetic one, is that there are small categories for which there is no canonical choice of a big category. And even if there were one, it is not clear that an action of a ring of operators on the small category extends to an action on the big one.

Following an idea of Grothendieck–Verdier [12], in this work we propose a different model for such constructions. Namely, given an essentially small triangulated category T, we consider the category Coh T of cohomological functors T^{op} → Ab into the category of abelian groups. Up to an equivalence, this is the category of ind-objects of T in the sense of [12]. The functor category contains a copy of T, because Yoneda’s lemma allows to identify an object X ∈ T with the representable functor Hom_{T}(−, X). While Coh T is no longer triangulated, it does carry an exact structure that is sufficient for our purposes; moreover, it admits filtered colimits. These are the basic ingredients we use for setting up our machinery.

Let us explain the main results in this work. Fix a noetherian graded commutative ring R acting centrally on T; the principal examples are listed in Example 3.1. This gives for each pair of objects X, Y in T an R-action on the graded abelian group

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

Let Spec R denote the set of homogeneous prime ideals. For each 𝔭 ∈ Spec R there is a localisation functor T → T_{𝔭} taking an object X to X_{𝔭}. The category T_{𝔭} has the same objects as T and there is a natural isomorphism

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

The formulation of the following local-global principle involves Koszul objects. Given an object X ∈ T and a homogeneous ideal 𝔞 of R, an iterated cone construction yields an object X//𝔞. While this object depends on a choice of a sequence of generators of 𝔞, the thick subcategory generated by it is does not; see Lemma 3.9. For 𝔭 ∈ Spec R set X(𝔭) = X_{𝔭}//𝔭.

Theorem 5.10 (Local-global principle). Let S be a thick subcategory of T. Then the following conditions are equivalent for an object X in T:

(1) X belongs to S.

(2) X_{𝔭} belongs to Thick(S_{𝔭}) for each 𝔭 ∈ Spec R.

(3) X(𝔭) belongs to Thick(S_{𝔭}) for each 𝔭 ∈ Spec R.

Motivated by this result, we define the support of an object X ∈ T to be the set

supp_{R} X = {𝔭 ∈ Spec R | X(𝔭) ≠ 0}.

When the R-module End_{T}^{∗}(X) is finitely generated, this coincides with the support, in the usual sense in commutative algebra, of the R-module End_{T}^{∗}(X); see Proposition 4.2. For each 𝔭 in Spec R, set

Γ_{𝔭}T = {X ∈ T_{𝔭} | End_{T_{𝔭}}^{∗}(X)_{𝔮} = 0 for all 𝔮 ⊉ 𝔭}.

This is a thick subcategory of T_{𝔭}. We say that T is stratified by the action of R if for each 𝔭 in Spec R, the category Γ_{𝔭}T admits no proper thick subcategory.

Theorem 7.4. Suppose that T is stratified by the action of R. For any pair of objects X, Y in T one has

X ∈ Thick(Y) ⟺ supp_{R} X ⊆ supp_{R} Y,
Hom_{T}^{∗}(X, Y) = 0 ⟺ (supp_{R} X) ∩ (supp_{R} Y) = ∅.

There is a partial converse when the endomorphism rings of objects in T are finitely generated R-modules: If T is not stratified, then there are objects in T having the same support but generating different thick subcategories; see Proposition 7.5.

The proofs of Theorems 5.10 and 7.4 involve the category of cohomological functors. Indeed, the R-action on T has an obvious extension to Coh T, and based on this we develop a theory of local cohomology and support for objects in Coh T, analogous to the one in [3] for compactly generated triangulated categories. This forms the foundation for much of this work.

In order to illustrate these results and techniques, it is shown that the category of perfect complexes over a commutative noetherian ring is stratified; this amounts to a classical theorem of Hopkins [13] and Neeman [19]. Applications of the local-global principle, Theorem 5.10, to the study of modules over locally complete intersections and over integral group rings, will appear elsewhere.

Most ideas in this paper are taken from our previous work [3, 4, 5]; see also the references given there for inspiration by other authors. However, the categorical setting in this work is fundamentally different and the systematic use of the category of cohomological functors in this context seems to be new. We do not work with a compactly generated triangulated category having arbitrary coproducts, but instead with an essentially small triangulated category. A brief comparison between these two approaches can be found in the final section. Otherwise, references to previous work are kept to a minimum.

2. Cohomological functors

In this section we introduce the category of cohomological functors on a triangulated category and study its basic properties. For instance, we discuss base change and a long exact sequence corresponding to a Verdier quotient.

Cohomological functors. Let T be an essentially small triangulated category with suspension Σ: T ⟶^{∼} T. Recall that a functor T^{op} → Ab into the category of abelian groups is cohomological if it takes exact triangles to exact sequences. We denote by Coh T the category of cohomological functors. Morphisms in Coh T are natural transformation and the Yoneda functor T → Coh T sending X ∈ T to

H_{X} = Hom_{T}(−, X)

is fully faithful. The suspension Σ extends to a functor Coh T ⟶^{∼} Coh T by taking F in Coh T to F ∘ Σ^{−1}; we denote this again by Σ.

It is convenient to view Coh T as a full subcategory of the category Mod T of all additive functors T^{op} → Ab. Note that (co)limits in Mod T are computed pointwise. For E and F in Mod T we write Hom(E, F) for the set of morphisms from E to F. Thus Hom(H_{X}, F) ≅ F (X) for X in T, by Yoneda’s lemma.

Any additive functor F: T^{op} → Ab can be written canonically as a colimit of representable functors

(2.1)  (colim_{H_{X}→F} H_{X}) ⟶^{∼} F

where the colimit is taken over the slice category T/F; see [12, Proposition 3.4]. Objects in T/F are morphisms H_{X} → F where X runs through the objects of T.

A morphism in T/F from H_{X} ⟶^{ϕ} F to H_{X′} ⟶^{ϕ′} F is a morphism α: X → X′ in T such that ϕ′H_{α} = ϕ.

A theorem of Lazard says that a module is flat if and only if it is a filtered colimit of finitely generated free modules; this has been generalised to functor categories by Oberst and Röhrl [21]. The following lemma shows that cohomological and flat functors agree; this is well-known, for instance from [18, Lemma 2.1].

Lemma 2.2. The cohomological functors T^{op} → Ab are precisely the filtered colimits of representable functors (in the category of additive functors T^{op} → Ab). In particular, the category Coh T has filtered colimits.

Sketch of proof. A filtered colimit of exact sequences is again exact. Thus a filtered colimit of representable functors is cohomological. Conversely, given a cohomological functor F, one easily checks that the slice category T/F is filtered. Thus (2.1) gives a presentation of F as a filtered colimit of representable functors. □

We say that a sequence of morphisms in Coh T is exact provided that evaluation at each object in T yields an exact sequence in Ab.

Lemma 2.3. The category Coh T is an exact category in the sense of Quillen; it admits enough projective and enough injective objects.

Proof. The cohomological functors form an extension closed subcategory of Mod T containing all projective objects and all injective objects. □

Exact functors. Let T and U be essentially small triangulated categories. A functor P: Coh T → Coh U is said to be exact if it takes exact sequences to exact sequences and if there is a natural isomorphism P ∘ Σ ⟶^{∼} Σ ∘ P.

An exact functor f: T → U induces a pair of functors

f^{∗}: Coh T ⟶ Coh U  and  f_{∗}: Coh U ⟶ Coh T

where f^{∗}(F) = colim_{H_{X}→F} H_{f(X)} and f_{∗}(G) = G ∘ f. The next lemma collects some of their basic properties. Recall that a triangulated subcategory is a full additive subcategory closed under forming cones and suspensions. The functor f is a quotient functor when it is equivalent to the canonical functor T → T/S given by a triangulated subcategory S ⊆ T.

Lemma 2.4. Let f: T → U be an exact functor between essentially small triangulated categories.

(1) The functor f^{∗} is a left adjoint of f_{∗}.

(2) The functors f^{∗} and f_{∗} are exact and preserve filtered colimits.

(3) If f is fully faithful, then f^{∗} is fully faithful and Id ⟶^{∼} f_{∗} ∘ f^{∗}.

(4) If f is a quotient functor, then f_{∗} is fully faithful and f^{∗} ∘ f_{∗} ⟶^{∼} Id.

Proof. (1) Given F ∈ Coh T and G ∈ Coh U, we claim that

Hom(f^{∗}F, G) ≅ Hom(F, f_{∗}G).

When F is representable this is immediate from Yoneda’s lemma. The general case then follows since F can be written as a colimit of representable functors.

(2) Clearly, f_{∗} is exact and preserves filtered colimits. A left adjoint, in particular, f^{∗}, automatically preserves colimits. The exactness of f^{∗} follows from the fact that f is exact; see [15, Lemma 2.2].

(3) We use the fact that for any pair (S, T) of adjoint functors, the left adjoint S is fully faithful iff the unit Id → T ∘ S is invertible; see [11, Proposition 1.3]. If f is fully faithful, then F ≅ (f_{∗} ∘ f^{∗})(F) for any representable functor F, and the general case follows by taking filtered colimits.

(4) If f is a quotient functor, then f_{∗} is fully faithful; see [11, Lemma 1.2]. Thus the counit f^{∗} ∘ f_{∗} → Id is invertible, by the argument dual to the one in (3). □

Remark 2.5. Observe that when f^{∗} is fully faithful so is f, since the Yoneda embedding of T into Coh T is fully faithful. However, there are examples where f_{∗} is fully faithful but f is not a quotient functor; see [16, Example 7.4].

Triangulated subcategories. Let S ⊆ T be a triangulated subcategory. We write i: S → T for the inclusion and q: T → T/S for the corresponding quotient functor. Henceforth we view Coh S and Coh T/S as full subcategories of Coh T, via i^{∗} and q_{∗} respectively. More specifically, there are identifications

Coh S = {F ∈ Coh T | F = colim_{α} H_{X_{α}} with all X_{α} ∈ S},
Coh T/S = {F ∈ Coh T | F |_{S} = 0}.

The second identification follows from the universal property of the quotient T/S [25, Chap. II, Cor. 2.2.11]. Here is a useful recognition criterion for objects in Coh S.

Lemma 2.6. Let S ⊆ T be a triangulated subcategory. Then F ∈ Coh T belongs to Coh S iff each morphism H_{X} → F with X ∈ T factors through H_{Y} for some Y ∈ S.

Proof. As in (2.1), we write F as a filtered colimit

F = colim_{H_{X}→F} H_{X}.

Then the assertion is an immediate consequence of the following lemma, applied with C′ and C the slice categories S/F and T/F, respectively. □

Lemma 2.7. Let i: C′ → C be a fully faithful functor with C a small filtered category. Suppose that for any X ∈ C there is an object Y ∈ C′ and a morphism X → iY. Then C′ is a small filtered category, and for any functor F: C → D into a category which admits filtered colimits, the natural morphism

colim_{Y ∈C′} F (iY) ⟶ colim_{X∈C} F (X)

is an isomorphism.

Proof. See [12, Proposition 8.1.3]. □

Localisation. Let S ⊆ T be a triangulated subcategory. With i: S → T and q: T → T/S the canonical functors, set

(2.8)  Γ = i^{∗} ∘ i_{∗}  and  L = q_{∗} ∘ q^{∗}.

These are exact functors on Coh T. By definition, for any F ∈ Coh T one has

(2.9)  Γ F = colim_{H_{S}→F} Hom_{T}(−, S)

where the colimit is taken over the slice category S/F, which is filtered because S is a triangulated subcategory.

Proposition 2.10. In Coh T each object F fits into a functorial exact sequence

(2.11) ⋯ ⟶ Σ^{−1}(LF) ⟶ Γ F ⟶ F ⟶ LF ⟶ Σ(Γ F) ⟶ ΣF ⟶ ⋯

Moreover, any exact sequence

(2.12)  ⋯ ⟶ Σ^{−1}F′′ ⟶ F′ ⟶ F ⟶ F′′ ⟶ ΣF′ ⟶ ΣF ⟶ ⋯

in Coh T with F′ ∈ Coh S and F′′ ∈ Coh T/S is isomorphic to (2.11).

For F = Hom_{T}(−, X), the sequence (2.11) specialises to

(2.13) ⋯ ⟶ Hom_{T/S}(−, Σ^{−1}X) ⟶ colim_{S→X} Hom_{T}(−, S) ⟶

⟶ Hom_{T}(−, X) ⟶ Hom_{T/S}(−, X) ⟶ ⋯ Proof. Fix an object X in T. Specialising (2.9) one gets that

Γ Hom_{T}(−, X) = colim_{S→X} Hom_{T}(−, S)

This functor fits into an exact sequence of cohomological functors of the form (2.13) since one has by definition

Hom_{T/S}(−, X) = colim_{X→Y} Hom_{T}(−, Y)

where X → Y runs through all morphisms with cone in S. The sequence is functorial and hence yields an exact sequence for each filtered colimit of representable functors. This justifies (2.11).

Given another sequence (2.12), we apply the exact functor L to see that the morphism F → F′′ is isomorphic to F → LF, since LF′ = 0 = Σ(LF′) and F′′ ⟶^{∼} LF′′. Analogously, an application of Γ shows that F′ → F is isomorphic to Γ F → F. This yields an isomorphism between (2.11) and (2.12). □

The functor L from (2.8) is a localisation functor^{1}, while Γ is a colocalisation functor^{2}, and the functorial exact sequence (2.11) is called the localisation sequence for S ⊆ T. In the following we consider the case that one of the natural morphisms Γ F → F and F → LF is an isomorphism.

Corollary 2.14. Let S ⊆ T be a triangulated subcategory and F ∈ Coh T. Then

F ∈ Coh S ⟺ Γ F ≅ F ⟺ LF = 0,
F ∈ Coh T/S ⟺ F ≅ LF ⟺ Γ F = 0.  □

A pair (U, V) of full subcategories of an additive category forms a torsion pair provided that the inclusion of U admits a right adjoint, the inclusion of V admits a left adjoint, U = {X | Hom(X, Y) = 0 for all Y ∈ V}, and V = {Y | Hom(X, Y) = 0 for all X ∈ U}.

Corollary 2.15. Let S ⊆ T be a triangulated subcategory. Then Coh S and Coh T/S form a torsion pair in Coh T. Thus

Coh T/S = {F ∈ Coh T | Hom(E, F) = 0 for all E ∈ Coh S},
Coh S = {F ∈ Coh T | Hom(F, G) = 0 for all G ∈ Coh T/S}.

Proof. We have already seen in Lemma 2.4 that the inclusion of Coh S admits a right adjoint while the inclusion of Coh T/S admits a left adjoint. For the first equality, observe that Hom(E, F) = 0 for all E ∈ Coh S means that F (X) = 0 for all X ∈ S. This is equivalent to Γ F = 0, and therefore to F ∈ Coh T/S. For the second equality, it remains to show that Hom(F, G) = 0 for all G ∈ Coh T/S implies F ∈ Coh S. The assumption on F implies LF = 0 since L = q_{∗} ∘ q^{∗} and

Hom(q^{∗}F, q^{∗}F) ≅ Hom(F, (q_{∗} ∘ q^{∗})F) = 0.

Thus F belongs to Coh S. □

In general, (co)localisation functors do note commute; see [3, Example 3.5]. The following lemma identifies some conditions under which they do.

Lemma 2.16. Let S_{1} ⊆ S_{2} ⊆ T be triangulated subcategories and let (Γ_{1}, L_{1}) and (Γ_{2}, L_{2}) be the corresponding pairs of (co)localisation functors on Coh T. The morphisms in (2.11) induce isomorphisms

Γ_{1}Γ_{2} ≅ Γ_{1} ≅ Γ_{2}Γ_{1}, L_{1}L_{2} ≅ L_{2} ≅ L_{2}L_{1}, Γ_{1}L_{2} = 0 = L_{2}Γ_{1}, Γ_{2}L_{1} ≅ L_{1}Γ_{2}.

1The natural morphism η: Id → L has the property that Lη is invertible and Lη = ηL.

2The natural morphism θ: Γ → Id has the property that Γ θ is invertible and Γ θ = θΓ.

Proof. Apply the localisation sequence (2.11). □

This has the following useful consequence.

Corollary 2.17. Given thick subcategories S_{1} and S_{2} of T, one has

S_{1} ⊆ S_{2}  ⟺  Coh S_{1} ⊆ Coh S_{2}.  □

3. Cohomological localisation

In this section we introduce cohomological localisation functors for categories of cohomological functors and explain how to compute these functors in terms of Koszul objects. These are analogues of results in [3, §§4–6].

Let T be an essentially small triangulated category. For objects X, Y in T set

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

More generally, each F in Mod T induces a functor F^{∗}: T^{op} → Ab^{∗} into the category of graded abelian groups, with

F^{n}(X) = F (Σ^{−n}X) for each n ∈ ℤ.

Central ring actions. Let R be a graded commutative ring; thus R is ℤ-graded and satisfies rs = (−1)^{|r||s|}sr for each pair of homogeneous elements r, s in R. We say that T is R-linear, or that R acts on T, if there is a homomorphism ϕ: R → Z^{∗}(T) of graded rings, where

Z^{∗}(T) = ⨁_{n∈ℤ}{η: Id_{T} → Σ^{n} | ηΣ = (−1)^{n}Ση}

is the graded centre of T. For each object X in T this yields a homomorphism ϕ_{X}: R → End_{T}^{∗}(X) of graded rings such that for all objects X, Y ∈ T the R-module structures on Hom_{T}^{∗}(X, Y) induced by ϕ_{X} and ϕ_{Y} agree up to a sign. Namely, for any homogeneous elements r ∈ R and α ∈ Hom_{T}^{∗}(X, Y), one has

ϕ_{Y} (r)α = (−1)^{|r||α|}αϕ_{X}(r).

Here are some examples.

Example 3.1. (1) Any triangulated category admits a canonical action of ℤ. (2) The derived category of a ring A has a canonical action of the centre of A.

(3) If A is an algebra over a field k, the derived category of A has a canonical action of the Hochschild cohomology of A over k.

(4) Given a finite dimensional Hopf algebra H over a field k (for example, the group algebra of a finite group), the derived category of H (and hence also the stable module category) has a canonical action of the k-algebra Ext_{H}^{∗}(k, k).

We fix an action of R on T. The following observations will be used repeatedly.

Remark 3.2. The R-action on T induces an action on any triangulated subcategory S ⊆ T and on the quotient T/S, compatible with the inclusion and quotient functors, respectively. It also extends to an action on Mod T.

Torsion objects. The set of homogeneous prime ideals of R is denoted Spec R. For a homogeneous ideal 𝔞 of R we set

V(𝔞) = {𝔭 ∈ Spec R | 𝔞 ⊆ 𝔭}.

Let V be a specialisation closed subset of Spec R; this condition means that if V contains a prime 𝔭, then it contains everything in V(𝔭). An R-module M is Vtorsion if M_{𝔭} = 0 for each 𝔭 ∈ Spec R \ V. Note that M is V(𝔞)-torsion if and only if each r ∈ 𝔞 and x ∈ M satisfy r^{n}x = 0 for n ≫ 0.

A functor F ∈ Coh T is V-torsion if F^{∗}(X) is V-torsion for all X ∈ T. The full subcategory of all V-torsion functors is denoted by (Coh T)_{V}. Analogously, an object X ∈ T is V-torsion if End_{T}^{∗}(X) is V-torsion, that is to say, if the functor Hom_{T}(−, X) is V-torsion. Set

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

Note that T_{V} is a thick subcategory of T. Recall that we view Coh(T_{V}) as a full subcategory of Coh T. It follows from the definitions that there is an inclusion

(3.3)  Coh(T_{V}) ⊆ (Coh T)_{V}.

Equality holds when R is noetherian; see Corollary 4.4, and also Propositions 3.6 and 3.10. Following the definition in (2.8), the inclusion T_{V} ⊆ T induces functors

Γ_{V}, L_{V}: Coh T ⟶ Coh T,

where Γ_{V} is a colocalisation functor and L_{V} is a localisation functor. Note that these functors are exact and preserve filtered colimits.

Inverting central elements. Given a homogeneous element r ∈ R of degree d and X ∈ T, we write X//r for the cone of the morphisms X ⟶^{r} Σ^{d}X. This definition yields the following exact sequence.

(3.4)  ⋯ ⟶ H_{X} ⟶^{±r} H_{Σ^{d}X} ⟶ H_{X//r} ⟶ H_{ΣX} ⟶^{±r} H_{Σ^{d+1}X} ⟶ ⋯

In particular, inverting r in T is equivalent to annihilating X//r for all X ∈ T.

Let Φ be a multiplicatively closed set of homogeneous elements in R. The following lemma describes the quotient functor for T that inverts the elements of Φ. We consider the specialisation closed set

Z(Φ) = {𝔭 ∈ Spec R | 𝔭 ∩ Φ ≠ ∅}.

Given an R-module M, we write M[Φ^{−1}] for the localisation of M with respect to Φ. Note that M is Z(Φ)-torsion iff M[Φ^{−1}] = 0.

Lemma 3.5. Let Φ be a multiplicatively closed set of homogeneous elements in R and T′ ⊆ T a subcategory satisfying Thick(T′) = T. Then there is an equality

T_{Z(Φ)} = Thick({X//r | X ∈ T′, r ∈ Φ}),

and the quotient functor T → T/T_{Z(Φ)} induces a natural isomorphism

Hom_{T}^{∗}(X, Y)[Φ^{−1}] ⟶^{∼} Hom_{T/T_{Z(Φ)}}^{∗}(X, Y)

for all objects X, Y in T.

Proof. Set S = Thick({X//r | X ∈ T′, r ∈ Φ}) and U = T/S. We claim: (1) If X or Y is in S, then Hom_{T}^{∗}(X, Y)[Φ^{−1}] = 0.

(2) For any X, Y in T, the natural morphism Hom_{T}^{∗}(X, Y) → Hom_{U}^{∗}(X, Y) induces an isomorphism

Hom_{T}^{∗}(X, Y)[Φ^{−1}] ⟶^{∼} Hom_{U}^{∗}(X, Y).

Indeed, (1) follows from (3.4). Given this, it follows from the exact sequence (2.13) that the morphism in (2) induces an isomorphism

Hom_{T}^{∗}(X, Y)[Φ^{−1}] ⟶^{∼} Hom_{U}^{∗}(X, Y)[Φ^{−1}].

On the other hand, Φ acts invertibly on Hom_{U}^{∗}(X, Y), that is to say,

Hom_{U}^{∗}(X, Y) ⟶^{∼} Hom_{U}^{∗}(X, Y)[Φ^{−1}].

It suffices to check this claim for all Y ∈ T′, and then it is clear from (3.4). Combining both isomorphisms yields (2), and completes the proof of the claims.

It remains to observe that X ∈ T is Z(Φ)-torsion iff End_{T}^{∗}(X)[Φ^{−1}] = 0; given (2) above, the latter condition translates to X = 0 in U. Thus T_{Z(Φ)} = S. □

Let Φ be a multiplicatively closed set. We define a functor

L_{Φ} : Coh T ⟶ Coh T

by taking F in Coh T to F [Φ^{−1}] given by F [Φ^{−1}]^{∗}(X) = F^{∗}(X)[Φ^{−1}] for X ∈ T. It is easy to verify that this is an exact localisation functor; the corresponding colocalisation functor is denoted Γ_{Φ}.

Proposition 3.6. There is a natural isomorphism L_{Φ} ⟶^{∼} L_{Z(Φ)} and hence

Coh(T_{Z(Φ)}) = (Coh T)_{Z(Φ)}.

Proof. Lemma 3.5 yields the isomorphism for representable functors, and the general case follows since L_{Φ} and L_{Z(Φ)} preserve filtered colimits. □

Localisation at a prime ideal. Let 𝔭 be a homogeneous prime ideal of R. Thus R \ 𝔭 is a multiplicatively closed subset and

Z(R \ 𝔭) = {𝔮 ∈ Spec R | 𝔮 ⊈ 𝔭}.

Set

T_{𝔭} = T/T_{Z(R\𝔭)}

and let X_{𝔭} denote the image of an object X in T under the natural functor T → T_{𝔭}. This quotient category is described in Lemma 3.5. Thus for all X, Y ∈ T there is a natural isomorphism

Hom_{T}^{∗}(X, Y)_{𝔭} ≅ Hom_{T_{𝔭}}^{∗}(X_{𝔭}, Y_{𝔭}).

It follows from Proposition 3.6 that the localisation functor

Coh T ⟶ Coh T, F ↦ F_{𝔭},

defined by (F_{𝔭})^{∗}(X) = F^{∗}(X)_{𝔭} is isomorphic to L_{Z(R\𝔭)}. A functor F ∈ Coh T is called 𝔭-local if F ≅ F_{𝔭}, and (Coh T)_{𝔭} denotes the full subcategory formed by all 𝔭-local functors. Proposition 3.6 yields an identification

(3.7)  Coh(T_{𝔭}) = (Coh T)_{𝔭}.

Koszul objects. 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.

Lemma 3.8. Let r ∈ R be a homogeneous element of degree d.

(1) For F ∈ Coh T, the colimit of the sequence

F ⟶^{r} Σ^{d}F ⟶^{r} Σ^{2d}F ⟶^{r} Σ^{3d}F ⟶^{r} ⋯

is naturally isomorphic to L_{V(r)}F.

(2) For X ∈ T, the colimit of the sequence

H_{Σ^{−1}X//r^{1}} ⟶ H_{Σ^{−1}X//r^{2}} ⟶ H_{Σ^{−1}X//r^{3}} ⟶ ⋯

is naturally isomorphic to Γ_{V(r)}H_{X}.

Proof. The colimit construction in (1) yields a functor Coh T → Coh T; we claim that it is isomorphic to L_{V(r)}. It suffices to prove this for representable functors as both functors preserve filtered colimits. When F = H_{X} one has an exact sequence

⋯ ⟶ colim H_{Σ^{−1}X//r^{n}} ⟶ H_{X} ⟶ colim H_{X_{n}} ⟶ colim H_{X//r^{n}} ⟶ ⋯

where r acts invertibly on colim H_{X_{n}} while colim H_{X//r^{n}} is V(r)-torsion. Thus the sequence is isomorphic to the localisation sequence for T_{V(r)} ⊆ T, by Propositions 2.10 and 3.6. □

Let 𝔞 be a finitely generated homogeneous ideal of R and X ∈ T. Pick a sequence of elements r_{1}, …, r_{n} in R that generate the ideal 𝔞 and define inductively

X_{0} = X  and  X_{i} = X_{i−1}//r_{i} for 1 ≤ i ≤ n.

We call X_{n} a Koszul object of X with respect to 𝔞, and denote it X//𝔞. This depends on a choice of a sequence of generators for 𝔞, so our notation is ambiguous. However, there is the following uniqueness result.

Lemma 3.9. There is an equality

Thick(X//𝔞) = {Y ∈ Thick(X) | End_{T}^{∗}(Y)_{𝔭} = 0 for all 𝔭 ⊉ 𝔞}.

Proof. When 𝔞 is generated by a single element the desired statement follows from Lemma 3.5, applied to T = Thick(X). An iteration settles the general case. □

Proposition 3.10. Let 𝔞 be a finitely generated homogeneous ideal of R. Then

T_{V(𝔞)} = Thick({X//𝔞 | X ∈ T})  and  Coh(T_{V(𝔞)}) = (Coh T)_{V(𝔞)}.

Moreover, the objects of T_{V(𝔞)} are precisely the direct summands of Koszul objects X//𝔟 with X ∈ T and 𝔟 an ideal of R satisfying \sqrt{𝔟} = \sqrt{𝔞}.

Proof. Set S = Thick({X//𝔞 | X ∈ T}). It suffices to show that

Coh(T_{V(𝔞)}) ⊆ (Coh T)_{V(𝔞)} ⊆ Coh S ⊆ Coh(T_{V(𝔞)}).

From this the first part of the assertion follows. The fact that all objects in T_{V(𝔞)} are direct summands of Koszul objects follows from the proof.

The first inclusion is by definition and the last follows from Lemma 3.9. To verify the inclusion in the middle, it suffices to show that for any F ∈ (Coh T)_{V(𝔞)} each morphism ϕ: H_{X} → F with X in T factors through a morphism H_{X} → H_{Y} with Y in S; see Lemma 2.6. To this end, let r_{1}, …, r_{n} be a sequence of elements that generate the ideal 𝔞. Starting with X_{0} = X and ϕ_{0} = ϕ, we construct factorisations

ϕ_{i−1} : H_{X_{i−1}} ⟶ H_{X_{i}} ⟶^{ϕ_{i}} F

for i = 1, 2, …, n. The assumption on F implies that each ϕ_{i−1} is annihilated by r_{i}^{α_{i}}, for some α_{i} ≥ 1. Thus we set

X_{i} = Σ^{−α_{i}|r_{i}|}X_{i−1}//r_{i}^{α_{i}}.

The object Y = X_{n} is the desired object; it belongs to S by Lemma 3.9. □

Composition laws. We show that cohomological localisation and colocalisation functors commute; see Lemma 2.16 for related commutation rules.

Lemma 3.11. Let Φ and Ψ be multiplicatively closed sets of homogeneous elements in R. Then

L_{Φ} ∘ L_{Ψ} ≅ L_{Ψ} ∘ L_{Φ},  L_{Φ} ∘ Γ_{Ψ} ≅ Γ_{Ψ} ∘ L_{Φ},  Γ_{Φ} ∘ Γ_{Ψ} ≅ Γ_{Ψ} ∘ Γ_{Φ}.

Proof. The first isomorphism is clear since localising R-modules with respect to Φ and Ψ commutes. We consider the exact localisation sequence

\xymatrix{𝔼_{Ψ} : & ⋯ \ar[r] & Γ_{Ψ} \ar[r] & Id \ar[r] & L_{Ψ} \ar[r] & ⋯}

and the pair of morphisms

(L_{Φ}𝔼_{Ψ}) ⟶ (L_{Φ}𝔼_{Ψ})L_{Φ} = L_{Φ}(𝔼_{Ψ}L_{Φ}) ⟵ (𝔼_{Ψ}L_{Φ}).

This yields the following commutative diagram with exact rows.

\xymatrix{⋯ \ar[r] & L_{Φ}Γ_{Ψ} \ar[r] \ar[d] & L_{Φ} \ar[r] \ar[d] & L_{Φ}L_{Ψ} \ar[r] \ar[d] & ⋯ \\ ⋯ \ar[r] & L_{Φ}Γ_{Ψ}L_{Φ} \ar[r] & L_{Φ}L_{Φ} \ar[r] & L_{Φ}L_{Ψ}L_{Φ} \ar[r] & ⋯ \\ ⋯ \ar[r] & Γ_{Ψ}L_{Φ} \ar[r] \ar[u] & L_{Φ} \ar[r] \ar[u] & L_{Ψ}L_{Φ} \ar[r] \ar[u] & ⋯}

In two of three columns the vertical morphisms are isomorphisms. Thus the five lemma implies that L_{Φ}Γ_{Ψ} ≅ Γ_{Ψ}L_{Φ}. A similar argument based on the pair of morphisms 𝔼_{Ψ}Γ_{Φ} ← Γ_{Φ}𝔼_{Ψ}Γ_{Φ} → Γ_{Φ}𝔼_{Ψ} is used to deduce the third isomorphism Γ_{Φ}Γ_{Ψ} ≅ Γ_{Ψ}Γ_{Φ} from the second. □

Lemma 3.12. Let 𝔞 and 𝔟 be finitely generated homogeneous ideals of R. Then

Γ_{V(𝔞)} ∘ Γ_{V(𝔟)} ≅ Γ_{V(𝔞)∩V(𝔟)}.

Proof. It suffices to show for 𝔞 generated by homogeneous elements r_{1}, …, r_{n} that

Γ_{V(𝔞)} = Γ_{V(r_{n})} ∘ … ∘ Γ_{V(r_{1})}.

Let 𝔞′ = (r_{1}, …, r_{n−1}). We know from Lemma 3.11 that the Γ_{V(r_{i})} commute. Thus Γ_{V(𝔞′)}Γ_{V(r_{n})} ≅ Γ_{V(r_{n})}Γ_{V(𝔞′)} by the induction hypothesis. Using Proposition 3.10, it follows that the image of Γ_{V(r_{n})}Γ_{V(𝔞′)} belongs to

Coh(T_{V(r_{n})}) ∩ Coh(T_{V(𝔞′)}) = (Coh T)_{V(r_{n})} ∩ (Coh T)_{V(𝔞′)}
= (Coh T)_{V(𝔞)}
= Coh(T_{V(𝔞)}).

Therefore L_{V(𝔞)}(Γ_{V(r_{n})}Γ_{V(𝔞′)}) = 0. On the other hand, Γ_{V(𝔞)}(Γ_{V(r_{n})}Γ_{V(𝔞′)}) = Γ_{V(𝔞)}. The exact localisation sequence (2.11) yields the following exact sequence

⋯ → Γ_{V(𝔞)}(Γ_{V(r_{n})}Γ_{V(𝔞′)}) → Γ_{V(r_{n})}Γ_{V(𝔞′)} → L_{V(𝔞)}(Γ_{V(r_{n})}Γ_{V(𝔞′)}) → ⋯

and therefore Γ_{V(𝔞)} ≅ Γ_{V(r_{n})}Γ_{V(𝔞′)}. □

Corollary 3.13. Let 𝔞 be a finitely generated homogeneous ideal, and 𝔭 a homogeneous prime ideal, of R. For each F ∈ Coh T, there is a natural isomorphism

(Γ_{V(𝔞)}F)_{𝔭} ≅ Γ_{V(𝔞)}(F_{𝔭}).

Proof. Apply Lemmas 3.11 and 3.12. □

4. Support

In this section, we define the support of a cohomological functor and establish some useful rules for computing it; the development parallels the one in [3, §5].

Let R be a graded commutative ring and T be an essentially small R-linear triangulated category. From now on we assume R to be noetherian.

Support. For each F in Coh T and 𝔭 in Spec R set

Γ_{𝔭}F = Γ_{V(𝔭)}(F_{𝔭}).

Then Γ_{𝔭} is an exact functor on Coh T that preserves filtered colimits. The subset

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

is called the support of F.

Proposition 4.1. Let F ∈ Coh T. Then supp_{R} F = ∅ if and only if F = 0.

Proof. Clearly, supp_{R} F = ∅ when F = 0. Suppose F is non-zero. Recall that if an R-module M is non-zero, then there exists a 𝔭 ∈ Spec R such that M_{𝔭} ≠ 0. Choose a prime 𝔭 that is minimal subject to the condition that F_{𝔭} ≠ 0. Then for all primes 𝔮 properly contained in 𝔭 and all X ∈ T, one has

F_{𝔭}(X)_{𝔮} ≅ F (X)_{𝔮} = 0.

Hence F_{𝔭}(X) is V(𝔭)-torsion, by [17, Theorem 6.5]; that is to say, F_{𝔭} is V(𝔭)-torsion. It then follows from Proposition 3.10 that F_{𝔭} is in Coh(T_{V(𝔭)}), so the natural map Γ_{𝔭}F → F_{𝔭} is an isomorphism. As F_{𝔭} ≠ 0, one gets that 𝔭 is in supp_{R} F. □

We can compute the support of the representable functors as follows.

Proposition 4.2. Let X be an object in T. Then

supp_{R} H_{X} ⊆ {𝔭 ∈ Spec R | End_{T}^{∗}(X)_{𝔭} ≠ 0},

and equality holds when End_{T}^{∗}(X) is finitely generated over R.

Proof. The inclusion holds because (H_{X})_{𝔭} = 0 iff End_{T}^{∗}(X)_{𝔭} = 0. Now suppose that End_{T}^{∗}(X)_{𝔭} ≠ 0 and that End_{T}^{∗}(X) is finitely generated. Then X_{𝔭} ≠ 0, and an application of Nakayama’s lemma gives X_{𝔭}//𝔭 ≠ 0; see [3, Lemma 5.11]. The functor H_{X_{𝔭}//𝔭} is V(𝔭)-torsion by Lemma 3.9. It is also 𝔭-local, so one gets

Γ_{𝔭}H_{X//𝔭} = Γ_{V(𝔭)}H_{X_{𝔭}//𝔭} = H_{X_{𝔭}//𝔭} ≠ 0.

Hence 𝔭 is in supp_{R} H_{X}. □

Composition laws. Computing support is compatible with cohomological localisation and colocalisation.

Proposition 4.3. Let V ⊆ Spec R be a specialisation closed subset. For each F ∈ Coh T the following equalities hold

supp_{R} Γ_{V}F = V ∩ supp_{R} F,
supp_{R} L_{V}F = (Spec R \ V) ∩ supp_{R} F.

Proof. If 𝔭 ∉ V then (Γ_{V}F)_{𝔭} = 0, since Γ_{V}F is V-torsion. Thus supp_{R} Γ_{V}F ⊆ V. If 𝔭 ∈ V then T_{V(𝔭)} ⊆ T_{V}, hence Γ_{V(𝔭)}Γ_{V} = Γ_{V(𝔭)}; see Lemma 2.16. This gives the second equality below:

Γ_{𝔭}(Γ_{V}F) = (Γ_{V(𝔭)}Γ_{V}F)_{𝔭} = (Γ_{V(𝔭)}F)_{𝔭} = Γ_{𝔭}F,

while the other two are by Corollary 3.13. Thus

supp_{R} Γ_{V}F = V ∩ supp_{R} Γ_{V}F = V ∩ supp_{R} F.

This proves the first equality; the proof of the second is similar. □

The following result says that a V-torsion functor is a colimit of representable functors defined by V-torsion objects.

Corollary 4.4. Let V ⊆ Spec R be specialisation closed. Then Coh(T_{V}) = Coh(T)_{V}.

Proof. It suffices to prove that Coh(T)_{V} ⊆ Coh(T_{V}); confer (3.3). Fix F ∈ (Coh T)_{V}. For any 𝔭 ∉ V, one has F_{𝔭} = 0, and hence Γ_{𝔭}F = 0, that is to say, supp_{R} F ⊆ V. Proposition 4.3 then implies that L_{V}F = 0 and it follows from (2.11) that the natural map Γ_{V}F → F is an isomorphism. This is the desired result. □

Corollary 4.5. Let V and W be specialisation closed subsets of Spec R. Then

Γ_{V}Γ_{W} ≅ Γ_{V∩W} ≅ Γ_{W}Γ_{V},  L_{V}L_{W} ≅ L_{V∪W} ≅ L_{W}L_{V},  Γ_{V}L_{W} ≅ L_{W}Γ_{V}.

Proof. Given Proposition 4.3, one can argue as for [3, Proposition 6.1]. □

Corollary 4.6. Let 𝔭 ∈ Spec R, and let V and W be specialisation closed subsets of Spec R such that V \ W = {𝔭}. Then

L_{W}Γ_{V} ≅ Γ_{𝔭} ≅ Γ_{V}L_{W}.

Proof. Given Proposition 4.3, one can argue as in the proof of [3, Theorem 6.2]. □

Colimits. The following result can be used to reduce computations involving specialisation closed subsets to those involving closed sets.

Lemma 4.7. Let V = ⋃_{α} V_{α} be a directed union of specialisation closed subsets of Spec R. Then

colim Γ_{V_{α}} ⟶^{∼} Γ_{V}  and  colim L_{V_{α}} ⟶^{∼} L_{V}.

Proof. We make repeated use of Proposition 4.1. Since Γ_{V_{α}}Γ_{V} = Γ_{V_{α}}, it follows from the localisation sequence (2.11) that the natural morphism colim Γ_{V_{α}} → Γ_{V} fits into an exact sequence

⋯ ⟶ colim Γ_{V_{α}} ⟶ Γ_{V} ⟶ colim L_{V_{α}}Γ_{V} ⟶ ⋯

We claim that colim L_{V_{α}}Γ_{V} = 0. Indeed, Γ_{𝔭} commutes with filtered colimits so the claim is that colim(Γ_{𝔭}L_{V_{α}}Γ_{V}) = 0 for each 𝔭 in Spec R. Proposition 4.3 yields

supp_{R}(Γ_{𝔭}L_{V_{α}}Γ_{V}F) ⊆ {𝔭} ∩ (Spec R \ V_{α}) ∩ V

for each F in Coh T. Thus, if 𝔭 ∉ V, then evidently Γ_{𝔭}L_{V_{α}}Γ_{V} = 0. Assume 𝔭 ∈ V. When 𝔭 ∈ V_{α} as well, it again follows from the equality above that Γ_{𝔭}L_{V_{α}}Γ_{V} = 0. Since V is directed union of the V_{α}, the desired vanishing follows.

This proves the claim. The exact sequence above then yields the isomorphism involving Γ_{V}. The assertion for L_{V} follows, using again (2.11). □

5. The local-global principle

Let R be a noetherian graded commutative ring and T be an essentially small R-linear triangulated category. In this section we establish a local-global principle for Coh T, analogous to the one in [4, §3]. A local-global principle for T then follows.

Localising subcategories. We call a full subcategory of Coh T localising if it is closed under forming coproducts, extensions, and suspensions. Here, F is an extension of F′ and F′′ if there is an exact sequence F′ → F → F′′ in Coh T. Any localising subcategory is closed under subobjects and quotient objects; this follows by specialising F′ = 0 or F′′ = 0. In particular, a localising subcategory is closed under filtered colimits. The smallest localising subcategory containing a subcategory C of T is denoted Loc(C).

The following lemma provides some basic properties of localising subcategories; they will be used without further mention. The argument is straightforward.

Lemma 5.1. Let P: Coh T → Coh U be an exact functor that preserves coproducts.

(1) If C ⊆ Coh U is localising, then {F ∈ Coh T | P (F) ∈ C} is a localising subcategory of Coh T. In particular, the kernel

Ker(P) = {F ∈ Coh T | P (F) = 0}

is a localising subcategory of Coh T.

(2) Any subcategory C ⊆ Coh T satisfies P (Loc(C)) ⊆ Loc(P (C)). □

We provide important examples of localising subcategories.

Lemma 5.2. Let S ⊆ T be a triangulated subcategory. Then Coh S and Coh T/S are localising when viewed as subcategories of Coh T.

Proof. From Corollary 2.14 it follows that Coh S equals the kernel of the functor L, while Coh T/S equals the kernel of the functor Γ. It remains to observe that both functors are exact and preserve coproducts, by Lemma 2.4. □

Proposition 5.3. Let V be a specialisation closed subset of Spec R and F ∈ Coh T. Then Γ_{V}F and L_{V}F belong to Loc(F).

Proof. From the localisation sequence (2.11) it follows that it suffices to prove this for Γ_{V}. The assertion follows from Lemmas 3.8 and 3.12 when V = V(𝔞) for some ideal 𝔞. The general case then follows from Lemma 4.7. □

Lemma 5.4. Let X ∈ T and 𝔞 be a homogeneous ideal of R. Then

Loc(H_{X}) = Coh Thick(X)  and  Loc(Γ_{V(𝔞)}H_{X}) = Loc(H_{X//𝔞}).

Proof. Both assertions use the following observation:

Y ∈ Thick(X) ⟹ H_{Y} ∈ Loc(H_{X}).

From this it follows that Coh Thick(X) ⊆ Loc(H_{X}), while the reverse inclusion is by Lemma 5.2. This settles the first of the desired equalities.

For the second one, note that Γ_{V(r)}H_{X} ∈ Loc(H_{X//r}) for any homogeneous element r ∈ R, by Lemma 3.8. Thus an induction on the number of generators of 𝔞 shows that Γ_{V(𝔞)}H_{X} ∈ Loc(H_{X//𝔞}). Conversely, H_{X//𝔞} belongs to Loc(H_{X}), by the observation above, so Γ_{V(𝔞)}H_{X//𝔞} ≅ H_{X//𝔞} belongs to Loc(Γ_{V(𝔞)}H_{X}). □

The local-global principle. The following result is the analogue of a local-global principle for compactly generated triangulated categories [4, Theorem 3.1].

Theorem 5.5 (Local-global principle). Let F ∈ Coh T. Then

Loc(F) = Loc({Γ_{𝔭}F | 𝔭 ∈ Spec R}) = Loc({F_{𝔭} | 𝔭 ∈ Spec R}).

Proof. It follows from Proposition 5.3 that Γ_{𝔭}F and F_{𝔭} belong to Loc(F).

Now we set C = Loc({Γ_{𝔭}F | 𝔭 ∈ Spec R}) and prove that F ∈ C. Consider the set of specialisation closed subsets W of Spec R such that Γ_{W}F ∈ C. This set is closed under directed unions by Lemma 4.7. Thus it has a maximal element, say V, by Zorn’s lemma. We claim that V = Spec R. To this end assume V ≠ Spec R and choose a prime ideal 𝔭 maximal in the subset Spec R \ V. The subset V ∪ {𝔭} is then specialisation closed. Consider the localisation sequence with respect to V:

⋯ ⟶ Γ_{V}Γ_{V∪{𝔭}}F ⟶ Γ_{V∪{𝔭}}F ⟶ L_{V}Γ_{V∪{𝔭}}F ⟶ ⋯

Corollaries 4.5 and 4.6 yield that Γ_{V} ≅ Γ_{V}Γ_{V∪{𝔭}} and Γ_{𝔭} ≅ L_{V}Γ_{V∪{𝔭}}. Hence the terms on the left and on the right of the localisation sequence are in C, and hence so is the one in the middle, Γ_{V∪{𝔭}}F. This contradicts the maximality of V and yields the first equality.

The second equality follows from the first since Γ_{𝔭}F = Γ_{V(𝔭)}(F_{𝔭}) ∈ Loc(F_{𝔭}) for each prime 𝔭, by Proposition 5.3. □

Recall that the R-action on T induces an action on any triangulated subcategory S ⊆ T and on T/S. It is not hard to see that the induced functor S_{𝔭} → T_{𝔭} is exact, full, and faithful. For this reason, we view S_{𝔭} as a triangulated subcategory of T_{𝔭}.

Lemma 5.6. Let S ⊆ T be a triangulated subcategory and 𝔭 ∈ Spec R. Then the canonical functor T_{𝔭} → (T/S)_{𝔭} induces an equivalence of triangulated categories

T_{𝔭}/S_{𝔭} ⟶^{∼} (T/S)_{𝔭}.

Proof. Consider the commutative diagrams below. The one on the left is clear from the constructions and induces the one on the right.

[figure: T; f; p; T/S; q; T_{𝔭}; f_{𝔭}; (T/S)_{𝔭}; Coh T; f_{∗}; p_{∗}; Coh T/S; q_{∗}; Coh(T_{𝔭}); (f_{𝔭})_{∗}; Coh(T/S)_{𝔭}]

Since f and q are quotient functors, so is their composition qf. Then f_{𝔭}p and p are quotient functors, and from this it is not hard to verify that so is f_{𝔭}. We claim that its kernel consists precisely of direct summands of objects of S_{𝔭}, and the statement would then follow.

As to the claim: In the display above, the functors in the square on the right are all fully faithful, since they are induced by quotient functors. Hence we view all the categories in the diagram as subcategories of Coh T. Recall from Corollary 2.14 that Ker(f^{∗}) = Coh S. Using (3.7) it follows that Ker(f_{𝔭}^{∗}) = Coh(S_{𝔭}). In particular, for an object X ∈ T_{𝔭}, one has f_{𝔭}(X) = 0 iff f_{𝔭}^{∗}(H_{X}) = 0 iff X is a direct summand of an object in S_{𝔭}; the second equivalence is by Lemma 2.6. This justifies the claim. □

For 𝔭 ∈ Spec R we set Γ_{𝔭}T = (T_{𝔭})_{V(𝔭)}. This yields the following diagram

[figure: T; T_{𝔭}; Γ_{𝔭}T]

and henceforth we make the identification

Coh Γ_{𝔭}T = (Coh T)_{𝔭} ∩ (Coh T)_{V(𝔭)}.

Corollary 5.7. Taking a localising subcategory C ⊆ Coh T to the family

(C ∩ Coh Γ_{𝔭}T)_{𝔭∈Spec R}

induces a bijection between – the localising subcategories of Coh T, and

– the families (C(𝔭))_{𝔭∈Spec R} with C(𝔭) a localising subcategory of Coh Γ_{𝔭}T.

Proof. The inverse map takes a family (C(𝔭))_{𝔭∈Spec R} to the smallest localising subcategory of Coh T containing all C(𝔭). □

Remark 5.8. There is an analogue of Corollary 5.7 for thick subcategories of T since each thick subcategory S ⊆ T is determined by the localising subcategory Coh S ⊆ Coh T; see Lemma 5.2.

Consequences of the local-global principle. For X ∈ T and 𝔭 ∈ Spec R, we set X(𝔭) = (X//𝔭)_{𝔭} and identify this with (X_{𝔭})//𝔭. Note that Lemma 5.4 implies

(5.9)  Loc(Γ_{𝔭}H_{X}) = Loc(H_{X(𝔭)}).

The following is the local-global principle for T announced in the introduction.

Theorem 5.10 (Local-global principle). Let S be a thick subcategory of T. Then the following conditions are equivalent for an object X in T:

(1) X belongs to S.

(2) X_{𝔭} belongs to Thick(S_{𝔭}) for each 𝔭 ∈ Spec R.

(3) X(𝔭) belongs to Thick(S_{𝔭}) for each 𝔭 ∈ Spec R.

Proof. Evidently (1) ⇒ (2) and (2) ⇒ (3).

Assume (3) holds. We work in Coh T. For each 𝔭 ∈ Spec R, the hypothesis implies the first inclusion below:

H_{X(𝔭)} ∈ Coh(S_{𝔭}) = (Coh S)_{𝔭} ⊆ Coh S.

The equality is by Proposition 3.6. Thus Γ_{𝔭}H_{X} ∈ Coh S for all 𝔭, by (5.9). It follows from Theorem 5.5 that H_{X} belongs to Coh S, and hence that X ∈ S. □

Theorem 5.11. For any pair of objects X, Y in T the following holds:

Hom_{T}^{∗}(X, Y) = 0 ⟺ Hom_{T_{𝔭}}^{∗}(X_{𝔭}, Y_{𝔭}) = 0 for all 𝔭 ∈ Spec R
⟺ Hom_{T_{𝔭}}^{∗}(X(𝔭), Y (𝔭)) = 0 for all 𝔭 ∈ Spec R.

Proof. Let S = Thick(X). Then Theorem 5.5 yields the following equivalences:

Hom_{T}^{∗}(X, Y) = 0 ⟺ H_{Y} ∈ Coh T/S
⟺ H_{Y_{𝔭}} ∈ (Coh T/S)_{𝔭} for all 𝔭 ∈ Spec R
⟺ Γ_{𝔭}H_{Y} ∈ (Coh T/S)_{𝔭} for all 𝔭 ∈ Spec R.

Using the identification Coh(T_{𝔭}/S_{𝔭}) = Coh(T/S)_{𝔭} = (Coh T/S)_{𝔭} from Lemma 5.6 and the identity (5.9), we obtain

Hom_{T}^{∗}(X, Y) = 0 ⟺ Hom_{T_{𝔭}}^{∗}(X_{𝔭}, Y_{𝔭}) = 0 for all 𝔭 ∈ Spec R
⟺ Hom_{T_{𝔭}}^{∗}(X_{𝔭}, Y (𝔭)) = 0 for all 𝔭 ∈ Spec R.

In the last condition, X_{𝔭} can be replaced by X(𝔭). This follows from the general fact that for any homogenous ideal 𝔞 of R and any pair of objects U, V in T

Hom_{T}^{∗}(U, V) = 0 ⟺ Hom_{T}^{∗}(U//𝔞, V) = 0

when Hom_{T}^{∗}(U, V) is V(𝔞)-torsion; for a proof use (3.4) or see [3, Lemma 5.11]. □

6. Tensor triangulated categories

Let (T, ⊗, 1) be a tensor triangulated category that is essentially small. The tensor product ⊗: T × T → T is then symmetric monoidal, exact in each variable, and admits a unit 1. The tensor product on T extends to a tensor product Coh T×Coh T → Coh T that we denote again by ⊗. We list the basic (and defining) properties. For objects X, Y ∈ T and F, G ∈ Coh T we have:

(1) H_{X} ⊗ H_{Y} ≅ H_{X⊗Y}.

(2) F ⊗ − and − ⊗ G are exact and preserve filtered colimits.

(3) F ⊗ G ≅ G ⊗ F.

Lemma 6.1. (Coh T, ⊗, H_{1}) is a symmetric monoidal category. □

Strongly monoidal functors. A functor f between symmetric monoidal categories is called strongly monoidal if there are isomorphisms

1 ⟶^{∼} f(1)  and  f(X) ⊗ f(Y) ⟶^{∼} f(X ⊗ Y)

that are natural and compatible with the monoidal structures. We have the following projection formula.

Lemma 6.2. Let f: T → U be a strongly monoidal exact functor between tensor triangulated categories. For F ∈ Coh T and G ∈ Coh U, there is a natural morphism

α_{F,G}: F ⊗ f_{∗}(G) ⟶ f_{∗}(f^{∗}(F) ⊗ G).

This is an isomorphism when T is generated as a triangulated category by 1.

Proof. Observe that f^{∗}: Coh T → Coh U is strongly monoidal. This is clear for representable functors; the general case follows by taking filtered colimits in one argument and then in the other. The morphism α_{F,G} is the adjoint of the composition

f^{∗}(F ⊗ f_{∗}(G)) ⟶^{∼} f^{∗}(F) ⊗ f^{∗}f_{∗}(G) → f^{∗}(F) ⊗ G.

Observe that the objects F such that α_{F,G} is an isomorphism for all G form a localising subcategory of Coh T containing H_{1}. If 1 generates T, then Loc(H_{1}) = Coh T, by Lemma 5.4. Thus α_{F,G} is an isomorphism for all F and G. □

Cohomological localisation. Let R be a noetherian graded commutative ring acting on T. The cohomological (co)localisation functors arising from this action can be expressed as tensor functors.

Proposition 6.3. Let V be a specialisation closed subset of Spec R. Then

Γ_{V} ≅ Γ_{V}H_{1} ⊗ −  and  L_{V} ≅ L_{V}H_{1} ⊗ −.

Proof. A simple calculation shows that one isomorphism implies the other. For instance, when L_{V} ≅ L_{V}H_{1} ⊗ − then L_{V}(Γ_{V}H_{1} ⊗ −) = 0. This yields a morphism Γ_{V}H_{1} ⊗ − → Γ_{V} making the following diagram commutative.

[figure: ⋯; Γ_{V}H_{1} ⊗ −; Id; L_{V}H_{1} ⊗ −; ⋯; ⋯; Γ_{V}; Id; L_{V}; ⋯]

The five lemma then shows that this is an isomorphism.

It follows from the description of L_{V(r)} in Lemma 3.8 that the assertion holds for a closed set V(r) given by some r ∈ R. Lemma 3.12 then implies the assertion for a closed set V(𝔞) given by a finitely generated ideal 𝔞 of R, and Lemma 4.7 implies the assertion for an arbitrary specialisation closed subset. □

7. Stratification

Let R be a noetherian graded commutative ring and T be an essentially small R-linear triangulated category. In this section we study the stratification of T and Coh T; this is the analogue of stratification for compactly generated triangulated categories introduced in [4, §4] and inspired by [14, §6].

Stratification. The triangulated category T is called minimal if T admits no proper thick subcategory. This means if S ⊆ T is a thick subcategory then S = 0 or S = T. Analogously, Coh T is said to be minimal if Coh T admits no proper localising subcategory. Clearly, T is minimal when Coh T is minimal.

Definition 7.1. We say that T is stratified by the action of R if Γ_{𝔭}T is minimal for each 𝔭 ∈ Spec R. In the same vein, Coh T is stratified by the action of R if Coh Γ_{𝔭}T is minimal for each 𝔭 ∈ Spec R.

In each case stratification yields a classification of thick or localising subcategories in terms of subsets of Spec R; see Corollary 5.7.

Remark 7.2. Suppose that T is minimal. Then T is stratified by any R-action, and in particular, by the canonical action of ℤ. Moreover, there is a unique prime 𝔭 ∈ Spec R such that Γ_{𝔭}T ≠ 0. Clearly, this implies Γ_{𝔮}T = 0 for all 𝔮 ≠ 𝔭 in Spec R.

Consequences of stratification. It is convenient to set supp_{R} X = supp_{R} H_{X} for each object X ∈ T. Observe that (5.9) implies

supp_{R} X = {𝔭 ∈ Spec R | X(𝔭) ≠ 0},

and this can be reformulated in terms of the following identity which is an immediate consequence of Lemma 3.9

(7.3)  Thick(X(𝔭)) = Thick(X_{𝔭}) ∩ Γ_{𝔭}T = Γ_{𝔭} Thick(X).

Theorem 7.4. Suppose that T is stratified by the action of R. Given objects X, Y in T, we have

X ∈ Thick(Y) ⟺ supp_{R} X ⊆ supp_{R} Y,
Hom_{T}^{∗}(X, Y) = 0 ⟺ (supp_{R} X) ∩ (supp_{R} Y) = ∅.

Proof. For the first assertion set S = Thick(Y). The local-global principle from Theorem 5.10 gives the first equivalence:

X ∈ Thick(Y) ⟺ X(𝔭) ∈ Γ_{𝔭}S for all 𝔭 ∈ Spec R
⟺ supp_{R} X ⊆ supp_{R} Y.

The second equivalence uses the minimality of Γ_{𝔭}T and the identity (7.3). For the second assertion recall from Theorem 5.11 that

Hom_{T}^{∗}(X, Y) = 0 ⟺ Hom_{T_{𝔭}}^{∗}(X(𝔭), Y (𝔭)) = 0 for all 𝔭 ∈ Spec R.

The minimality of Γ_{𝔭}T implies for objects U, V in Γ_{𝔭}T that Hom_{T_{𝔭}}^{∗}(U, V) ≠ 0 iff U ≠ 0 ≠ V. Thus Hom_{T_{𝔭}}^{∗}(X(𝔭), Y (𝔭)) = 0 iff 𝔭 ∉ (supp_{R} X) ∩ (supp_{R} Y). □

Theorem 7.4 has a converse when endomorphism rings are finitely generated.

Proposition 7.5. Suppose that for each object X in T the endomorphism ring End_{T}^{∗}(X) is finitely generated over R. If T is not stratified by R, then there are objects X, Y ∈ T such that supp_{R} X = supp_{R} Y but Thick(X) ≠ Thick(Y).

Proof. Assume Γ_{𝔭}T is not minimal. Thus there are non-zero objects X_{𝔭}, Y_{𝔭} in Γ_{𝔭}T such that Thick(X_{𝔭}) ≠ Thick(Y_{𝔭}). In T consider the objects X′ = X//𝔭 and Y′ = Y//𝔭. Then supp_{R} X′ = V(𝔭) = supp_{R} Y′ by Proposition 4.2. On the other hand, Lemma 3.9 gives the equalities below:

Thick(X′_{𝔭}) = Thick(X_{𝔭}) ≠ Thick(Y_{𝔭}) = Thick(Y′_{𝔭}),

so that Thick(X′) ≠ Thick(Y′). □

The Hom vanishing statement in Theorem 7.4 can be strengthened when morphism spaces are finitely generated over R. For an R-module M, we write

Supp_{R} M = {𝔭 ∈ Spec R | M_{𝔭} ≠ 0}.

The following theorem can be used to explain results on the symmetry of Hom vanishing, as studied in [1, 6].

Theorem 7.6. Let X and Y be objects in T.

(1) If Hom_{T}^{∗}(X, Y) is finitely generated over R, then

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

(2) If T is stratified by the action of R, then

Supp_{R} Hom_{T}^{∗}(X, Y) ⊇ (supp_{R} X) ∩ (supp_{R} Y).

Proof. (1) Let 𝔭 ∈ Supp_{R} Hom_{T}^{∗}(X, Y). Thus Hom_{T_{𝔭}}^{∗}(X_{𝔭}, Y_{𝔭}) ≠ 0. The assumption implies that this is finitely generated, and an application of Nakayama’s lemma gives

Hom_{T_{𝔭}}^{∗}(X(𝔭), Y (𝔭)) ≠ 0;

use (3.4) or see [3, Lemma 5.11] for a proof. Thus 𝔭 ∈ (supp_{R} X) ∩ (supp_{R} Y). (2) Let 𝔭 ∈ (supp_{R} X) ∩ (supp_{R} Y). Then stratification implies

Hom_{T_{𝔭}}^{∗}(X(𝔭), Y (𝔭)) ≠ 0,

and therefore

Hom_{T}^{∗}(X, Y)_{𝔭} ≅ Hom_{T_{𝔭}}^{∗}(X_{𝔭}, Y_{𝔭}) ≠ 0.

Thus 𝔭 ∈ Supp_{R} Hom_{T}^{∗}(X, Y). □

Perfect complexes. Let A be a noetherian commutative ring. We denote by D(A) the derived category of the category of A-modules. An object in D(A) is called perfect if it is isomorphic to a bounded complex of finitely generated projective A-modules; these form a thick subcategory denoted by D^{per}(A). For X ∈ D(A) set

H_{X} = Hom_{A}(−, X)|_{D^{per}(A)}.

The ring R = A acts canonically on T = D^{per}(A) and we show that Coh T is stratified by this action. The residue fields play a special role. For 𝔭 ∈ Spec R let k(𝔭) = A_{𝔭}/𝔭_{𝔭}, viewed as a complex concentrated in degree zero.

Lemma 7.7. Let 𝔭 ∈ Spec R. Then Coh Γ_{𝔭}T = Loc(H_{k(𝔭)}).

Proof. Since H_{k(𝔭)} is 𝔭-local and V(𝔭)-torsion when evaluated at any object from T, it belongs to Coh Γ_{𝔭}T; this justfies one inclusion. For the other one, it is convenient to identify T_{𝔭} and D^{per}(A_{𝔭}). Thus an object X in Γ_{𝔭}T is a perfect complex over A_{𝔭} such that its cohomology is of finite length over A_{𝔭}. It follows that

X ∈ Thick(k(𝔭)) ⊆ D(A_{𝔭}).

This is easily shown by an induction on the number of integers n such H^{n}(X) ≠ 0; see for example [10, Example 3.5]. Thus H_{X} ∈ Loc(H_{k(𝔭)}). □

Theorem 7.8. Let A be a commutative noetherian ring. Then Coh D^{per}(A) is stratified by the canonical action of A.

Proof. Fix 𝔭 ∈ Spec A. We need to show that Coh Γ_{𝔭}T is minimal, which is equivalent to Loc(F) = Loc(H_{k(𝔭)}) for each non-zero F ∈ Coh Γ_{𝔭}T, by Lemma 7.7. This is clear for F = Γ_{𝔭}H_{A}. This gives the second of the following equalities:

Loc(F) = Loc(F ⊗ Γ_{𝔭}H_{A}) = Loc(F ⊗ H_{k(𝔭)}).

The first one is by Proposition 6.3. Let f denote the functor

− ⊗_{A} k(𝔭): D^{per}(A) ⟶ D^{per}(k(𝔭)).

Then F ⊗ H_{k(𝔭)} ≅ f_{∗}f^{∗}(F) by Lemma 6.2. Thus F ⊗ H_{k(𝔭)} is a direct sum of suspensions of H_{k(𝔭)}, since every object in D^{per}(k(𝔭)) is a direct sum of suspensions of k(𝔭). It follows that Loc(F) = Loc(H_{k(𝔭)}). □

The following result is due to Hopkins [13] and Neeman [19].

Corollary 7.9. Let S ⊆ D^{per}(A) be a thick subcategory. Then

S = {X ∈ D^{per}(A) | supp_{A} X ⊆ V}

for some specialisation closed subset V ⊆ Spec A.

Proof. The assertion follows from Theorems 7.4 and 7.8, using the fact that supp_{A} X is specialisation closed for each X ∈ D^{per}(A) by Proposition 4.2. □

Remark 7.10. Theorem 7.8 generalises with same proof in two directions as follows.

(1) Let A be a commutative differential graded algebra such that the ring H^{∗}(A) is noetherian. If A is formal, then Coh D^{per}(A) is stratified by the canonical H^{∗}(A)- action. This is an analogue of Theorem 8.1 in [4] that asserts that D(A) is stratified by H^{∗}(A).

(2) Let A be a graded commutative noetherian ring. More precisely, we fix an abelian grading group G endowed with a symmetric bilinear form

(−, −): G × G ⟶ ℤ/2,

and A admits a decomposition

A = ⨁_{g∈G} A_{g}

such that the multiplication satisfies A_{g}A_{h} ⊆ A_{g+h} for all g, h ∈ G and xy = (−1)^{(g,h)}yx for all homogeneous elements x ∈ A_{g}, y ∈ A_{h}. We consider localising subcategories of Coh D^{per}(A) and D(A) that are closed under twists. Then Coh D^{per}(A) is stratified by the canonical A-action, and this is an analogue of Corollary 5.7 in [9] that asserts that D(A) is stratified by A.

8. Compactly generated triangulated categories

Let R be a noetherian graded commutative ring and T be a compactly generated R-linear triangulated category. The subcategory of compacts, T^{c}, is an essentially small triangulated category and has an induced R-action. In [3] we developed a theory of local cohomology and support for T. In this section, we use the restricted Yoneda functor

T ⟶ Coh T^{c}, X ↦ H_{X} = Hom_{T}(−, X)|_{T^{c}},

to compare it with the one for Coh T^{c} introduced in this article.

Remark 8.1. The morphisms annihilated by the functor T → Coh T^{c} are called phantom maps. In the context of the stable module category T = StMod kG of a finite group G, these were studied by Benson and Gnacadja. In particular, in [2, §4] it is shown that there are filtered systems in T^{c} = stmod kG that do not lift to mod kG. As a consequence, there are objects in Coh T^{c} that are not in the image of T → Coh T^{c}, namely the filtered colimit of the corresponding representable functors. In the context of the derived category of a commutative noetherian ring, examples of filtered systems that do not lift can be found in Neeman [20].

Cohomological localisation. Given a specialisation closed subset V of Spec R, there is an exact localisation functor L̄_{V}: T → T such that L̄_{V}X = 0 iff H_{X} is V-torsion; see [3, §4]. The corresponding colocalisation functor is denoted by Γ̄_{V}. Thus each X ∈ T fits into an exact localisation triangle

Γ̄_{V}X ⟶ X ⟶ L̄_{V}X ⟶ .

The following proposition says that notions developed in [3] for compactly generated triangulated categories are determined by analogous concepts for the category of cohomological functors which are discussed in this work. This applies, for instance, to the notion of support.

Proposition 8.2. Let V ⊆ Spec R be specialisation closed and X ∈ T. Then

H_{Γ̄_{V} X} ≅ Γ_{V}H_{X}  and  H_{L̄_{V} X} ≅ L_{V}H_{X}.

Proof. It follows from Proposition 2.10 that the long exact sequence

⋯ ⟶ H_{Σ^{−1}(L̄_{V} X)} ⟶ H_{Γ̄_{V} X} ⟶ H_{X} ⟶ H_{L̄_{V} X} ⟶ H_{Σ(Γ̄_{V} X)} ⟶ ⋯

is isomorphic to the localisation sequence (2.11) for T_{V}^{c} ⊆ T^{c}, applied to H_{X}. □

Localising subcategories. A full triangulated subcategory of T is localising if it is closed under forming coproducts. Following [4, §3], we say that the local-global principle holds for T, if for each object X ∈ T we have

Loc(X) = Loc({Γ̄_{𝔭}X | 𝔭 ∈ Spec R}).

This local-global principle has been established in a number of relevant cases. For instance, it holds when R has finite Krull dimension [4, Corollary 3.5], or when T admits a model [23, Theorem 6.9].

We obtain an alternative proof of Theorem 5.10, provided the local-global principle holds for T.

Proposition 8.3. The local-global principle for T implies the principle for T^{c}.

Proof. We verify that conditions (1)–(3) of Theorem 5.10 are equivalent. Evidently (1) ⇒ (2) and (2) ⇒ (3).

Assume (3) holds. Given 𝔭 ∈ Spec R, it follows from [4, Theorem 3.1] that S_{𝔭} is contained in Loc(S). Thus

Loc_{T}(Γ̄_{𝔭}X) = Loc(X(𝔭)) ⊆ Loc(S_{𝔭}) ⊆ Loc(S),

where the equality holds by [4, Lemma 3.8]. Now the local-global principle for T yields that X belongs to Loc(S). It remains to observe that this implies X ∈ S, because X is compact and S is a subcategory of compact objects. □

Cohomological localising subcategories. Any localising subcategory of Coh T^{c} induces one of T via the restricted Yoneda functor. However, we do not know if this is a bijection between the corresponding localising subcategories. This changes when one restricts to localising subcategories that are defined cohomologically.

We call a localising subcategory S ⊆ T cohomological if there is a cohomological functor F: T → A such that

(1) A is an abelian category with exact filtered colimits,

(2) F preserves coproducts, and

(3) S equals the full subcategory of objects in T annihilated by F. Analogously, a localising subcategory C ⊆ Coh T^{c} is called cohomological if there is an exact functor F: Coh T^{c} → A such that

(1) A is an abelian category with exact filtered colimits,

(2) F preserves coproducts, and

(3) C equals the full subcategory of objects in Coh T^{c} annihilated by F.

Example 8.4. An intersection of cohomological localising subcategories is cohomological. Given a subset U ⊆ Spec R, the localising subcategories

{X ∈ T | supp_{R} X ⊆ U}  and  {X ∈ Coh T^{c} | supp_{R} X ⊆ U}

are cohomological.

Proposition 8.5. Taking a localising subcategory C ⊆ Coh T^{c} to {X ∈ T | H_{X} ∈ C} induces a bijection between

– the cohomological localising subcategories of Coh T^{c}, and

– the cohomological localising subcategories of T.

Proof. To describe the inverse map, let F: T → A be a cohomological functor which preserves coproducts. This extends essentially uniquely to an exact and coproduct preserving functor F̄: Coh T^{c} → A by sending a filtered colimit of representable functors colim_{α} H_{X_{α}} to colim_{α} F (X_{α}). It remains to observe that for each X ∈ T we have F (X) = 0 iff F̄(H_{X}) = 0. Thus the localising subcategories determined by F and F̄ correspond to each other under the above assignment. □

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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, Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany.