THE STABLE DERIVED CATEGORY OF A NOETHERIAN SCHEME

HENNING KRAUSE

Abstract. For a noetherian scheme, we introduce its unbounded stable derived category. This leads to a recollement which reflects the passage from the bounded derived category of coherent sheaves to the quotient modulo the subcategory of perfect complexes. Some applications are included, for instance an analogue of maximal Cohen-Macaulay approximations, a construction of Tate cohomology, and an extension of the classical Grothendieck duality. In addition, the relevance of the stable derived category in modular representation theory is indicated.

1. Introduction

Let 𝕏 be a separated noetherian scheme and denote by Qcoh 𝕏 the category of quasicoherent sheaves on 𝕏. We consider the derived category D(Qcoh 𝕏) and two full subcategories

D^{perf}(coh 𝕏) ⊆ D^{b}(coh 𝕏) ⊆ D(Qcoh 𝕏)

which are of particular interest. Here, D^{b}(coh 𝕏) denotes the bounded derived category of coherent sheaves, and D^{perf}(coh 𝕏) denotes the subcategory of perfect complexes.

Now let Inj 𝕏 be the full subcategory of injective objects in Qcoh 𝕏, and denote by K(Inj 𝕏) its homotopy category. The composite

Q: \xymatrix{K(Inj 𝕏) \ar[r]^{inc} & K(Qcoh 𝕏) \ar[r]^{can} & D(Qcoh 𝕏)}

gives rise to a localization sequence

\xymatrix{S(Qcoh 𝕏) \ar[r]^{I} & K(Inj 𝕏) \ar[r]^{Q} & D(Qcoh 𝕏)}

where S(Qcoh 𝕏) denotes the full subcategory of all acyclic complexes in K(Inj 𝕏). Thus Q induces an equivalence

K(Inj 𝕏)/S(Qcoh 𝕏) ⟶^{∼} D(Qcoh 𝕏).

Next we recall that an object X in some category with coproducts is compact if every map X → ∐_{i} Y_{i} into an arbitrary coproduct factors through a finite coproduct. For instance, an object in D(Qcoh 𝕏) is compact if and only if it is isomorphic to a perfect complex. It is well known that the derived category D(Qcoh 𝕏) is compactly generated, that is, there is a set of compact objects which generate D(Qcoh 𝕏) [27]. To formulate our main result, let us denote by K^{c}(Inj 𝕏) and S^{c}(Qcoh 𝕏) the full subcategories of compact objects in K(Inj 𝕏) and S(Qcoh 𝕏) respectively.

Theorem 1.1. Let 𝕏 be a separated noetherian scheme.

2000 Mathematics Subject Classification. Primary: 14F05, 18E30; Secondary: 16E45, 16G50, 55U35.

Version from April 2004.

(1) The functors I, Q have left adjoints I_{λ}, Q_{λ} and right adjoints I_{ρ}, Q_{ρ} respectively. We have therefore a recollement

\xymatrix{S(Qcoh 𝕏) \ar[r] & K(Inj 𝕏) \ar[r] \ar[l] \ar[l] & D(Qcoh 𝕏). \ar[l] \ar[l]}

(2) The triangulated category K(Inj 𝕏) is compactly generated, and Q induces an equivalence K^{c}(Inj 𝕏) → D^{b}(coh 𝕏).

(3) The sequence

\xymatrix{D(Qcoh 𝕏) \ar[r]^{Q_{λ}} & K(Inj 𝕏) \ar[r]^{I_{λ}} & S(Qcoh 𝕏)}

is a localization sequence. Therefore S(Qcoh 𝕏) is compactly generated, and I_{λ} ^{∘} Q_{ρ} induces (up to direct factors) an equivalence

D^{b}(coh 𝕏)/D^{perf}(coh 𝕏) ⟶^{∼} S^{c}(Qcoh 𝕏).

Note that this theorem is a special case of a general result about Grothendieck categories. All we need is a locally noetherian Grothendieck category A, for instance A = Qcoh 𝕏, such that D(A) is compactly generated. There is a surprising consequence which seems worth mentioning.

Corollary 1.2. Let 𝕏 be a separated noetherian scheme. Then a product of acyclic complexes of injective objects in Qcoh 𝕏 is acyclic.

We call the category S(Qcoh 𝕏) the stable derived category of Qcoh 𝕏. A first systematic study of the bounded stable derived category

D^{b}(coh 𝕏)/D^{perf}(coh 𝕏)

can be found in work of Buchweitz [12]. Unfortunately this beautiful paper has never been published; see however [11]. He identifies for a Gorenstein ring Λ the bounded derived category of finitely generated Λ-modules modulo perfect complexes

D^{b}(mod Λ)/D^{perf}(mod Λ)

with the stable category of maximal Cohen-Macaulay Λ-modules and with the category of acyclic complexes of finitely generated projective Λ-modules. The same identification appears in [31] for selfinjective algebras and plays an important role in modular representation theory of finite groups; see also [23]. The approach in the present paper differs from that of Buchweitz substantially because we work in the unbounded setting and we use injective objects instead of projectives. This has some advantages. For instance, we have in any Grothendieck category enough injectives but often not enough projectives. On the other hand, we obtain a recollement in the unbounded setting which does not exists in the bounded setting. In fact, the celebrated theory of maximal Cohen-Macaulay approximations [2] is described as ‘decomposition’ [2] or ‘glueing’ [12], but finds a natural interpretation as ‘recollement’ in the sense of [6] if one passes to the unbounded setting. To be precise, the recollement

\xymatrix{S(Mod Λ) \ar[r] & K(Inj Λ) \ar[r] \ar[l]^{I_{λ}} \ar[l]_{I_{ρ}} & D(Mod Λ) \ar[l]^{Q_{λ}} \ar[l]_{Q_{ρ}}}

induces for any Gorenstein ring Λ the Gorenstein injective approximation functor

T: Mod Λ ⟶^{can} D(Mod Λ) ⟶^{I_{λ} ∘ Q_{ρ}} S(Mod Λ) ⟶^{Z^{0}} \underline{Mod} Λ

where \underline{Mod} Λ denotes the stable category modulo injective objects. For any Λ-module A, the Gorenstein injective approximation A → T A is the ‘dual’ of the maximal Cohen-Macaulay approximation which is based on projective resolutions. Let us stress again that this approach generalizes to any locally noetherian Grothendieck category A provided that D(A) is compactly generated.

Next we explain the connection between Gorenstein injective approximations and Tate cohomology. We fix a locally noetherian Grothendieck category A and pass from the stable derived category S(A) to the full subcategory T(A) of totally acyclic complexes. An object in A is by definition Gorenstein injective if it is of the form Ker(X^{0} → X^{1}) for some X in T(A). The inclusion J: T(A) → K(Inj A) has a left adjoint J_{λ}. Given an object A in A with injective resolution iA, we may think of J_{λ}iA as a complete injective resolution of A. This leads to the following definition of Tate cohomology groups

Êxt_{A}^{n}(A, B) = H^{n} Hom_{K(A)}(A, Σ^{n}(J_{λ}iB))

for any A, B in A and n ∈ ℤ. This cohomology theory is symmetric in the sense that for any A in A, we have Êxt_{A}^{∗}(A, −) = 0 iff Êxt_{A}^{∗}(−, A) = 0 iff Êxt_{A}^{0}(A, A) = 0. Let X denote the class of all objects A such that Êxt_{A}^{∗}(A, −) vanishes, and let Y be the class of Gorenstein injective objects in A.

Theorem 1.3. Let A be a locally noetherian Grothendieck category and suppose that D(A) is compactly generated.

(1) X = {A ∈ A | Ext_{A}^{1}(A, B) = 0 for all B ∈ Y}.

(2) Y = {B ∈ A | Ext_{A}^{1}(A, B) = 0 for all A ∈ X }.

(3) Every object A in A fits into exact sequences

0 → Y_{A} → X_{A} → A → 0 and 0 → A → Y^{A} → X^{A} → 0

in A with X_{A}, X^{A} in X and Y_{A}, Y^{A} in Y.

(4) X ∩ Y = Inj A.

After explaining some historical backround, let us mention more recent work on stable derived categories. For instance, Beligiannis develops a general theory of ‘stabilization’ in the framework of relative homological algebra [4], and Jørgensen studies the category of ‘spectra’ for a module category [19]. Also, Orlov discusses the category

D^{b}(coh 𝕏)/D^{perf}(coh 𝕏)

under the name ‘triangulated category of singularities’ and points out some connection with the Homological Mirror Symmetry Conjecture [29]. In any case, our notation S(Qcoh 𝕏) reflects this terminology.

Our main results suggests that the homotopy category K(Inj 𝕏) deserves some more attention. We may think of this category as the ‘compactly generated completion’ of the category D^{b}(coh 𝕏). In fact, the category coh 𝕏 of coherent sheaves carries a natural DG structure and its derived category D_{dg}(coh 𝕏) is equivalent to K(Inj 𝕏). This follows from Keller’s work [22] and complements a recent result of Bondal and van den Bergh [9] which says that D(Qcoh 𝕏) is equivalent to D_{dg}(A) for some DG algebra A.

As another application of our main result, let us mention that the adjoint pair of functors Rf_{∗} and f^{!} which establish the Grothendieck duality for a morphism f: 𝕏 → 𝕐 between schemes [16, 27], can be extended to a pair of adjoint functors between K(Inj 𝕏) and K(Inj 𝕐).

Theorem 1.4. Let f: 𝕏 → 𝕐 be a morphism between separated noetherian schemes. Denote by Rf_{∗}: D(Qcoh 𝕏) → D(Qcoh 𝕐) the right derived direct image functor and by f^{!} its right adjoint. Then there is an adjoint pair of functors R̂f_{∗} and f̂^{!} making the following diagram commutative.

\xymatrix{D(Qcoh 𝕏) \ar[d]^{Q_{λ}} \ar[r]^{Rf_{∗}} & D(Qcoh 𝕐) & D(Qcoh 𝕐) \ar[d]^{Q_{ρ}} \ar[r]^{f^{!}} & D(Qcoh 𝕏) \\ K(Inj 𝕏) \ar[r]^{R̂f_{∗}} & K(Inj 𝕐) \ar[u]_{Q} & K(Inj 𝕐) \ar[r]^{f̂^{!}} & K(Inj 𝕏) \ar[u]_{Q}}

Again, this theorem is really a lot more general. It is irrelevant that the functor f_{∗} comes from a morphism f: 𝕏 → 𝕐. All we need is that f_{∗} and its right derived functor Rf_{∗} preserve coproducts. On the other hand, there is a strengthened version of Theorem 1.4 which uses the special properties of f_{∗}. I am grateful to Amnon Neeman for pointing out the following.

Theorem 1.5 (Neeman). Let f: 𝕏 → 𝕐 be a morphism between separated noetherian schemes. Then R̂f_{∗} sends acyclic complexes to acyclic complexes. Thus we have an adjoint pair of functors between S(Qcoh 𝕏) and S(Qcoh 𝕐), making the following diagrams commutative.

\xymatrix{S(Qcoh 𝕏) \ar[d]^{Sf_{∗}} \ar[r]^{I} \ar[d] & K(Inj 𝕏) \ar[r]^{Q} \ar[d]^{R̂f_{∗}} & D(Qcoh 𝕏) \ar[d]^{Rf_{∗}} \\ S(Qcoh 𝕐) \ar[r]^{I} & K(Inj 𝕐) \ar[r]^{Q} & D(Qcoh 𝕐) \\ S(Qcoh 𝕐) \ar[d] \ar[d]^{Sf^{!}} & K(Inj 𝕐) \ar[l]_{I_{ρ}} \ar[d]^{f̂^{!}} & D(Qcoh 𝕐) \ar[l]_{Q_{ρ}} \ar[d]^{f^{!}} \\ S(Qcoh 𝕏) & K(Inj 𝕏) \ar[l]_{I_{ρ}} & D(Qcoh 𝕏) \ar[l]_{Q_{ρ}}}

It seems an interesting project to study the functor Sf_{∗}, for instance to find out when it is an equivalence. The following result demonstrates the geometric content of this question; it generalizes a result of Orlov for the bounded stable derived category [29].

Theorem 1.6. Let 𝕐 be a seperated noetherian scheme of finite Krull dimension. If f: 𝕏 → 𝕐 denotes the inclusion of an open subscheme which contains all singular points of 𝕐, then Sf_{∗}: S(Qcoh 𝕏) → S(Qcoh 𝕐) is an equivalence.

Despite the title of this paper and the algebraic geometric formulation of the main results, there is another source of serious interest in stable categories. Take a finite group G and a field k. A classical object in modular representation theory is the stable module category \underline{Mod} kG of the group algebra kG. We shall see that this stable category is equivalent to the stable derived category of the full module category Mod kG. Using a slightly different setting, Hovey, Palmieri, and Strickland studied the functor

I_{λ}: K(Inj kG) ⟶ S(Mod kG) ≅ \underline{Mod} kG

in their work on axiomatic stable homotopy theory [18]. Note that K(Inj kG) carries a commutative tensor product and the (graded) endomorphism ring of its unit is nothing but the group cohomology ring H^{∗}(G, k). Therefore, K(Inj kG) seems to be the right object for studying representations of G via methods from commutative algebra.

Having stated some of the main results, let us sketch the outline of this paper. The paper deals with locally noetherian Grothendieck categories and covers therefore various applications, for instance in algebraic geometry or representation theory. Thus we fix a locally noetherian Grothendieck category A and study the recollement

(1.1)  \xymatrix{S(A) \ar[r] & K(Inj A) \ar[r] \ar[l] \ar[l] & D(A). \ar[l] \ar[l]}

More specifically, we begin in Section 2 with the basic properties of the homotopy category K(Inj A). The recollement (1.1) is established in Sections 3 and 4. In Section 5, we discuss the essential properties of the stable derived category S(A). Then we extend derived functors in Section 6, and Section 7 is devoted to studying Gorenstein injective approximations and Tate cohomology. In the final Section 8, we indicate the relevance of the stable derived category in modular representation theory. An appendix provides additional material about DG categories.

2. The homotopy category of injectives

We fix a locally noetherian Grothendieck category A. Thus A is an abelian Grothendieck category and has a set A_{0} of noetherian objects which generate A, that is, every object in A is a quotient of a coproduct of objects in A_{0}. We denote by noeth A the full subcategory formed by the noetherian objects in A, and Inj A denotes the full subcategory of injective objects. Note that Inj A is closed under taking coproducts.

We write K(A) for the homotopy category and D(A) for the derived category of unbounded complexes in A; for their definitions and basic properties, we refer to [33]. We do not distinguish between an object in A and the corresponding complex concentrated in degree zero in the homotopy category K(A). The inclusion noeth A → A induces a fully faithful functor

D^{b}(noeth A) ⟶ D(A)

which identifies D^{b}(noeth A) with the full subcategory of objects X in D(A) such that H^{n}X is noetherian for all n and H^{n}X = 0 for almost all n ∈ ℤ.

In this section, we study the basic properties of the homotopy category K(Inj A). We shall see that this category solves a completion problem for the triangulated category D^{b}(noeth A). Let us begin with some elementary observations.

Lemma 2.1. Let A be an object in A and denote by iA an injective resolution. Then the natural map

(2.1)  Hom_{K(A)}(iA, X) ⟶ Hom_{K(A)}(A, X)

is an isomorphism for all X in K(Inj A). Therefore iA is a compact object in K(Inj A) if A is noetherian.

Proof. Denote for any n ∈ ℤ by σ^{⩾n}X the truncation satisfying

(σ^{⩾n}X)^{p} = \begin{cases}X^{p} & if p ⩾ n, \\ 0 & if p < n. \end{cases}

We complete the map A → iA to an exact triangle

aA ⟶ A ⟶ iA ⟶ Σ(aA)

and obtain

0 = Hom_{K(A)}(aA, σ^{⩾−1}X) ≅ Hom_{K(A)}(aA, X)

since aA is acyclic and concentrated in non-negative degrees. Thus Hom_{K(A)}(iA, X) ≅ Hom_{K(A)}(A, X).

Now assume that A is noetherian. Clearly, A is a compact object in A and therefore a compact object in K(A). The isomorphism (2.1) shows that iA is a compact object in K(Inj A). □

Lemma 2.2. Let X be a non-zero object in K(Inj A). Then there exists a noetherian object A in A such that Hom_{K(A)}(A, Σ^{n}X) ≠ 0 for some n ∈ ℤ.

Proof. Suppose first H^{n}X ≠ 0 for some n. Choose a noetherian object A and a map A → Z^{n}X inducing a non-zero map A → H^{n}X. We obtain a chain map A → Σ^{n}X which induces a non-zero element in Hom_{K(A)}(A, Σ^{n}X).

Now suppose H^{n}X = 0 for all n. We can choose n such that Z^{n}X is non-injective. Using Baer’s criterion, there exists a noetherian object A in A such that Ext_{A}^{1}(A, Z^{n}X) is non-zero. Now observe that

Hom_{K(A)}(A, Σ^{n+p}X) ≅ Ext_{A}^{p}(A, Z^{n}X)

for all p ⩾ 1. Thus Hom_{K(A)}(A, Σ^{n+1}X) ≠ 0. This completes the proof. □

Let T be a triangulated category with arbitrary coproducts. Recall that an object X in T is compact if Hom_{T} (X, −) preserves all coproducts. The triangulated category is compactly generated if there is a set T_{0} of compact objects such that Hom_{T} (X, Σ^{n}Y) = 0 for all X ∈ T_{0} and n ∈ ℤ implies Y = 0 for every object Y in T .

Proposition 2.3. Let A be a locally noetherian Grothendieck category, and let K^{c}(Inj A) denote the full subcategory of compact objects in K(Inj A).

(1) The triangulated category K(Inj A) is compactly generated.

(2) The canonical functor K(A) → D(A) induces an equivalence

K^{c}(Inj A) ⟶^{∼} D^{b}(noeth A).

Proof. It follows from Lemma 2.1 and Lemma 2.2 that K(Inj A) is compactly generated. A standard argument shows that K^{c}(Inj A) equals the thick subcategory of K(Inj A) which is generated by the injective resolutions of the noetherian objects in A; see [26, 2.2]. The equivalence K^{+}(Inj A) → D^{+}(A) restricts to an equivalence K^{+,b}(Inj A) → D^{b}(A) and identifies K^{c}(Inj A) with D^{b}(noeth A). □

Note that we obtain a functor D^{b}(noeth A) → K(Inj A) which identifies D^{b}(noeth A) with the full subcategory of compact objects. Therefore the formation of the category K(Inj A) solves a completion problem which we explain by an analogy. The category A is a completion of noeth A in the following sense.

• A is an additive category with filtered colimts.

• The inclusion noeth A → A identifies noeth A with the full subcategory of finitely presented objects.

• A coincides with the smallest subcategory which contains all finitely presented

objects and is closed under forming filtered colimits. Recall that an object X in A is finitely presented if the functor Hom_{A}(X, −) preserves filtered colimits. Similarly, we have the following for T = K(Inj A).

• T is a triangulated category with coproducts.

• The functor D^{b}(noeth A) → T identifies D^{b}(noeth A) with the full subcategory of compact objects.

• T coincides with the smallest subcategory which contains all compact objects

and is closed under forming triangles and coproducts. The category A is, up to an equivalence, uniquely determined by noeth A. It would be interesting to know to what extent K(Inj A) is uniquely determined by D^{b}(noeth A).

Example 2.4. Suppose there is a noetherian object A in A such that D^{b}(noeth A) is generated by A, that is, there is no proper thick subcategory containing A. Take an injective resolution iA and denote by End_{A}(A) the endomorphism DG algebra of iA. Then Hom_{A}(iA, −) inducs an equivalence between K(Inj A) and the derived category D_{dg}(End_{A}(A)) of DG End_{A}(A)-modules; see [22]. If one replaces a single generator by a set of generating objects, then one obtains an analogue which involves a DG category instead of a DG algebra. In particular, noeth A carries the structure of a DG category such that K(Inj A) and D_{dg}(noeth A) are equivalent. We refer to Appendix A for details.

Example 2.5. Let G be a finite p-group and k be a field of characteristc p > 0. We consider the category A = Mod kG of modules over the group algebra kG. Take an injective resolution ik of the trivial representation k, and denote by End_{kG}(k) the endomorphism DG algebra of ik. Then its derived category D_{dg}(End_{kG}(k)) is equivalent to K(Inj A). The tensor product ⊗_{k} on A restricts to a product on Inj A and induces therefore a (total) tensor product on K(Inj A). On the other hand, the E_{∞}-structure of End_{kG}(k) induces a product on D_{dg}(End_{kG}(k)). We conjecture that these products are naturally isomorphic.

Example 2.6. Let Λ be a finite dimensional algebra over a field k. Then E = Hom_{k}(Λ^{op}, k) is an injective cogenerator for A = Mod Λ, and Hom_{Λ}(E, −) induces an equivalence Inj A → Proj A since Hom_{Λ}(E, E) ≅ Λ. Thus the homotopy category K(Proj A) is compactly generated. For more on K(Proj A), see [19, 20].

3. A localization sequence

Let A be a locally noetherian Grothendieck category and let

K_{ac}(Inj A) = K(Inj A) ∩ K_{ac}(A).

In this section, we prove that the canonical functors

I: K_{ac}(Inj A) ⟶^{inc} K(Inj A) and Q: K(Inj A) ⟶^{inc} K(A) ⟶^{can} D(A)

form a localization sequence

(3.1)  \xymatrix{K_{ac}(Inj A) \ar[r]^{I} & K(Inj A) \ar[r]^{Q} & D(A).}

Let us start with some preparations. In particular, we need to give the definition of a localization sequence.

Definition 3.1. We say that a sequence

\xymatrix{T′ \ar[r]^{F} & T \ar[r]^{G} & T′′}

of exact functors between triangulated categories is a localization sequence if the following holds.

(L1) The functor F has a right adjoint F_{ρ}: T → T′ satisfying F_{ρ} ^{∘} F ≅ Id_{T′}.

(L2) The functor G has a right adjoint G_{ρ}: T′′ → T satisfying G^{∘} G_{ρ} ≅ Id_{T′′}.

(L3) Let X be an object in T . Then GX = 0 if and only if X ≅ F X′ for some

X′ ∈ T′. The sequence (F, G) of functors is called colocalization sequence if the sequence (F^{op}, G^{op}) of opposite functors is a localization sequence.

The basic properties of a localization sequence are the following [33, II.2].

(1) The functors F and G_{ρ} are fully faithful.

(2) Identify T′ = Im F and T′′ = Im G_{ρ}. Given objects X, Y ∈ T , then

X ∈ T′ ⟺ Hom_{T} (X, T′′) = 0,
Y ∈ T′′ ⟺ Hom_{T} (T′, Y) = 0.

(3) Identify T′ = Im F. Then the functor G induces an equivalence T /T′ → T′′.

(4) Let X be an object in T . Then there is an exact triangle

(F ^{∘} F_{ρ})X ⟶ X ⟶ (G_{ρ} ^{∘} G)X ⟶ Σ((F ^{∘} F_{ρ})X)

which is functorial in X.

(5) The sequence

\xymatrix{T′′ \ar[r]^{G_{ρ}} & T \ar[r]^{F_{ρ}} & T′}

is a colocalization sequence. The next lemma is well known; it provides useful criteria for a sequence to be a

localization sequence. Recall that a full subcategory of a triangulated category is thick if it is a triangulated subcategory which is closed under taking direct factors.

Lemma 3.2. Let T be a triangulated category and S be a thick subcategory. Then the following are equivalent.

(1) The sequence S ⟶^{inc} T ⟶^{can} T /S is a localization sequence.

(2) The inclusion functor S → T has a right adjoint.

(3) The quotient functor T → T /S has a right adjoint.

Proof. Condition (1) implies (2) and (3). Also, (2) and (3) together imply (1). Thus we need to show that (2) and (3) are equivalent. Let us write F: S → T and G: T → T /S for the functors which are involved.

(2) ⇒ (3): Let T′′ denote the full subcategory of objects Y in T such that Hom_{T} (S, Y) = 0. We obtain a left adjoint L: T → T′′ for the inclusion by completing for each X in T the natural map F_{ρ}X → X to an exact triangle

F_{ρ}X ⟶ X ⟶ LX ⟶ Σ(F_{ρ}X).

The functor L annihilates S and factors therefore through G via an exact functor L̄: T /S → T′′. The composite of L̄ with the inclusion T′′ → T is a right adjoint for G.

(3) ⇒ (2): We obtain a right adjoint F_{ρ}: T → S for the inclusion F by completing for each X in T the natural map X → (G_{ρ} ^{∘} G)X to an exact triangle

F_{ρ}X ⟶ X ⟶ (G_{ρ} ^{∘} G)X ⟶ Σ(F_{ρ}X).

Note that F_{ρ}X belongs to S since G(F_{ρ}X) = 0. □

We need to construct left and right adjoints for functors starting in a compactly generated triangulated category. Our basic tool for this is the following result which is due to Neeman.

Proposition 3.3. Let F: S → T be an exact functor between triangulated categories, and suppose S is compactly generated.

(1) There is a right adjoint T → S if and only if F preserves all coproducts.

(2) There is a left adjoint T → S if and only if F preserves all products.

Proof. For (1), see [27, Theorem 4.1]. The proof of (2) is analogous and uses covariant Brown representability [28, Theorem 8.6.1]; see also [24]. □

We record a similar result for later use.

Proposition 3.4. Let T be a compactly generated triangulated category and S_{0} be a set of objects in T . Denote by U the full subcategory of objects Y in T such that Hom_{T} (Σ^{n}X, Y) = 0 for all X ∈ S_{0} and n ∈ ℤ. Then the inclusion U → T has a left adjoint.

Proof. The localizing subcategory S generated by S_{0} is well generated and the inclusion S → T has therefore a right adjoint; see [28]. We obtain a localization sequence S ⟶^{inc} T ⟶^{can} T /S by Lemma 3.2, and the right adjoint of the canonical functor T → T /S identifies T /S with U. □

There is a useful criterion when a left adjoint preserves compactness.

Lemma 3.5. Let F: S → T be an exact functor between compactly generated triangulated categories which has a right adjoint G. Then F preserves compactness if and only if G preserves coproducts.

Proof. See [27, Theorem 5.1]. □

The following result establishes the localization sequence for the homotopy category of injective objects.

Proposition 3.6. Let A be a locally noetherian Grothendieck category. Then the canonical functors K_{ac}(Inj A) → K(Inj A) and K(Inj A) → D(A) form a localization sequence

\xymatrix{K_{ac}(Inj A) \ar[r]^{I} & K(Inj A) \ar[r]^{Q} & D(A).}

Proof. We know from Proposition 2.3 that K(Inj A) is compactly generated. In addition, we use Lemma 3.2 and Proposition 3.3. The inclusion J: K(Inj A) → K(A) preserves products and has therefore a left adjoint J_{λ} satisfying J_{λ} ^{∘} J ≅ Id_{K(Inj A)}. We obtain a localization sequence

\xymatrix{K \ar[r]^{inc} & K(A) \ar[r]^{J_{λ}} & K(Inj A)}

where K denotes the kernel of J_{λ}. Thus

Hom_{K(A)}(X, Y) = 0 for all X ∈ K and Y ∈ K(Inj A).

This implies K ⊆ K_{ac}(A) and gives the following commutative diagram of exact functors.

[figure: K; inc; inc; K(A); J_{λ}; K(Inj A); F; K_{ac}(A); inc; K(A); can; D(A)]

The functor F is induced by the canonical functor K(A) → D(A), and we have F ≅ Q since J_{λ} ^{∘} J ≅ Id_{K(Inj A)}. Moreover, F preserves coproducts and has therefore a right adjoint F_{ρ}. The composite J ^{∘} F_{ρ} is a right adjoint for the canonical functor K(A) → D(A). This implies F ^{∘} F_{ρ} ≅ Id_{D(A)}. On the other hand, K_{ac}(Inj A) is the kernel of F. Thus we conclude that the sequence (3.1) is a localization sequence. □

We add some useful remarks which are immediate consequences.

Remark 3.7. Let J_{λ}: K(A) → K(Inj A) be the left adjoint of the inclusion K(Inj A) → K(A). Then the composite Q^{∘} J_{λ} is naturally isomorphic to the canonical functor K(A) → D(A).

Remark 3.8. The right adjoint Q_{ρ} of Q induces an equivalence

D^{b}(noeth A) ⟶^{∼} K^{c}(Inj A)

which is a quasi-inverse for the equivalence K^{c}(Inj A) → D^{b}(noeth A) induced by Q.

Let us denote by K_{inj}(A) the full subcategory of complexes Y in K(Inj A) such that Hom_{K(A)}(X, Y) = 0 for all acyclic complexes X in K(A). Following Spaltenstein’s terminology [32], the objects in K_{inj}(A) are precisely the K-injective complexes having injective components. There are various results about K-injective resolutions in the literature; see for instance [32, 10]. The following is certainly not the most general one; however it is sufficient in our context.

Corollary 3.9. The inclusion K_{inj}(A) → K(A) has a left adjoint i: K(A) → K_{inj}(A) which has the following properties.

(1) Every object X in K(A) fits into an exact triangle

aX ⟶ X ⟶ iX ⟶ Σ(aX)

such that aX is an acyclic complex.

(2) The functor i: K(A) → K_{inj}(A) induces an equivalence

D(A) = K(A)/K_{ac}(A) ⟶^{∼} K_{inj}(A).

(3) We have for all X, Y in K(A)

Hom_{D(A)}(X, Y) ≅ Hom_{K(A)}(X, iY).

Proof. Put iX = Q_{ρ}X for each X in K(A), where Q_{ρ} denotes the right adjoint of Q: K(Inj A) → D(A). □

The functor

R: D(A) = K(A)/K_{ac}(A) ⟶^{∼} K_{inj}(A) ⟶^{inc} K(A)

provides a right adjoint for the canonical functor K(A) → D(A). Let us mention as an application that the right derived functor of any additive functor F: A → B is obtained as composite

RF: D(A) ⟶^{R} K(A) ⟶^{K(F)} K(B) ⟶^{can} D(B).

Example 3.10. Suppose every object in A has finite injective dimension. Then the functor K(Inj A) → D(A) is an equivalence since K_{ac}(Inj A) = 0. In particular, the compact objects in D(A) are precisely those from D^{b}(noeth A).

Example 3.11. Suppose products in A are exact. For instance, let A be a module category. Then one can show that K_{inj}(A) is the smallest triangulated subcategory of K(A) which is closed under taking products and contains the injective objects of A (viewed as complexes concentrated in degree zero).

4. A recollement

In this section, we provide a criterion for A such that the sequence

\xymatrix{K_{ac}(Inj A) \ar[r]^{I} & K(Inj A) \ar[r]^{Q} & D(A)}

induces a recollement

\xymatrix{K_{ac}(Inj A) \ar[r] & K(Inj A) \ar[r] \ar[l] \ar[l] & D(A) \ar[l] \ar[l]}

in the sense of [6]. It is important to note that one cannot expect a recollement

(4.1)  \xymatrix{K_{ac}(A) \ar[r] & K(A) \ar[r] \ar[l] \ar[l] & D(A) \ar[l] \ar[l]}

without severe restrictions on A; see Example 4.9. In fact, a recollement (4.1) implies that a product of exact sequences in A remains exact.

We begin with a lemma.

Lemma 4.1. Let A be a locally noetherian Grothendieck category. Then a compact object in D(A) belongs to D^{b}(noeth A).

Proof. Suppose X is compact in D(A). We need to show that H^{n}X is noetherian for all n, and that H^{n}X vanishes for almost all n in ℤ. We have for any injective object E in A an isomorphism

Hom_{D(A)}(X, E) ≅ Hom_{A}(H^{0}X, E).

Therefore Hom_{A}(H^{0}X, −) preserves coproducts in Inj A. This implies that each H^{n}X is noetherian; see [30]. Now fix for each n an injective envelope H^{n}X → E(H^{n}X) and consider the induced map

α: X ⟶ ∏_{n∈ℤ}Σ^{−n}E(H^{n}X)

in D(A). The canonical map

∐_{n∈ℤ}Σ^{−n}E(H^{n}X) ⟶ ∏_{n∈ℤ}Σ^{−n}E(H^{n}X)

is an isomorphism in D(A), and therefore α factors though a finite number of factors in

∏_{n∈ℤ}Σ^{−n}E(H^{n}X).

Thus H^{n}X vanishes for almost all n in ℤ, and the proof is complete. □

We denote by D^{c}(A) the full subcategory of D(A) which is formed by all compact objects.

Theorem 4.2. Let A be a locally noetherian Grothendieck category and suppose D(A) is compactly generated. Then the canonical functor Q: K(Inj A) → D(A) has a left adjoint and therefore the sequence

\xymatrix{K_{ac}(Inj A) \ar[r]^{I} & K(Inj A) \ar[r]^{Q} & D(A)}

is a colocalization sequence.

Proof. Let K be the localizing subcategory of K(Inj A) which is generated by all compact objects X in K(Inj A) such that QX is compact in D(A). We claim that Q|_{K}: K → D(A) is an equivalence. First note that K and D(A) are both compactly generated. We have seen in Lemma 4.1 that

D^{c}(A) ⊆ D^{b}(noeth A),

and Q induces an equivalence

K^{c}(Inj A) ⟶^{∼} D^{b}(noeth A),

by Proposition 2.3. Thus Q induces an equivalence between the subcategories of compact objects in K and D(A). Then a standard argument shows that Q|_{K} is an equivalence since Q preserves all coproducts. Now fix a left adjoint L: D(A) → K. We claim that the composite

D(A) ⟶^{L} K ⟶^{inc} K(Inj A)

is a left adjoint for Q. To see this, consider for objects X in D(A) and Y in K(Inj A) the natural map

α_{X,Y} : Hom_{K(Inj A)}(LX, Y) ⟶ Hom_{D(A)}(QLX, QY) ⟶^{∼} Hom_{D(A)}(X, QY)

which is induced by Q. If X and Y are compact, then α_{X,Y} is bijective by Proposition 2.3. We use a standard argument to show that α_{X,Y} is bijective for arbitrary X and Y. Fix a compact object X. Then the objects Y such that α_{X,Y} is bijective form a triangulated subcategory which is closed under taking coproducts and contains all compact objects. Thus α_{X,Y} is bijective for all Y because K(Inj A) is compactly generated. Now fix any object Y. The same argument shows that α_{X,Y} is bijective for all X because D(A) is compactly generated. We conclude that Q has a left adjoint. Moreover, Lemma 3.2 implies that I and Q form a colocalization sequence. □

Following Beilinson, Bernstein, and Deligne [6], we say that a sequence

(4.2)  \xymatrix{T′ \ar[r] & T \ar[r] & T′′}

of exact functors between triangulated categories induces a recollement

\xymatrix{T′ \ar[r] & T \ar[r] \ar[l] \ar[l] & T′′ \ar[l] \ar[l]}

if the sequence (4.2) is a localization sequence and a colocalization sequence in the sense of Definition 3.1.

Corollary 4.3. Let A be a locally noetherian Grothendieck category and suppose D(A) is compactly generated. Then the sequence

\xymatrix{K_{ac}(Inj A) \ar[r]^{I} & K(Inj A) \ar[r]^{Q} & D(A)}

induces a recollement

\xymatrix{K_{ac}(Inj A) \ar[r] & K(Inj A) \ar[r] \ar[l] \ar[l] & D(A). \ar[l] \ar[l]}

Corollary 4.4. Let A be a locally noetherian Grothendieck category and suppose D(A) is compactly generated. Then a product of acyclic complexes of injective objects in A is acyclic.

Let us give a criterion for A such that the derived category D(A) is compactly generated.

Lemma 4.5. Let A be a locally noetherian Grothendieck category. Suppose there is a set A_{0} of objects in A which are compact when viewed as objects in D(A). If A_{0} generates A, then D(A) is compactly generated by A_{0}.

The lemma is an immediate consequence of the following statement.

Lemma 4.6. Let A be a locally noetherian Grothendieck category and fix a set A_{0} of generating objects. Let X be a complex in A such that H^{0}X ≠ 0. Then there exists some object A in A_{0} such that

Hom_{K(A)}(A, X) ≠ 0 and Hom_{D(A)}(A, X) ≠ 0.

Proof. Choose A in A_{0} and a map A → Z^{0}X such that the composite with Z^{0}X → H^{0}X is non-zero. This induces a non-zero element in

H^{0}(Hom_{A}(A, X)) ≅ Hom_{K(A)}(A, X).

The second assertion follows from the first since for any object A in A we have

Hom_{D(A)}(A, X) ≅ Hom_{K(A)}(A, iX)

and H^{0}(iX) ≅ H^{0}X. □

We give examples of Grothendieck categories such that objects in A become compact objects in D(A).

Example 4.7. Let Λ be an associative ring. Denote by A = Mod Λ the category of (right) Λ-modules and by proj Λ the full subcategory of finitely generated projective Λ-modules. Then every object in proj Λ is compact when viewed as object in D(A). Thus the inclusion D^{b}(proj Λ) → D(A) identifies D^{b}(proj Λ) with the full subcategory of compact objects in D(A). Suppose now that Λ is right noetherian. Then the fully faithful functor Q_{λ}: D(A) → K(Inj A) identifies D(A) with the localizing subcategory of K(Inj A) which is generated by the injective resolution iΛ of Λ.

Let us return to the completion problem for triangulated categories which has been adressed in Section 2. Keeping the analogy between the completion with respect to filtered colimits and the completion with respect to triangles und coproducts, we obtain the following diagram for a right noetherian ring Λ. The vertical arrows denote completions and the horizontal ones the appropriate inclusions.

\xymatrix{proj Λ \ar[d] \ar[r] & mod Λ \ar[d] & D^{b}(proj Λ) \ar[d] \ar[r] & D^{b}(mod Λ) \ar[d] \\ Flat Λ \ar[r] & Mod Λ & D(Mod Λ) \ar[r]^{Q_{λ}} & K(Inj Λ)}

Here, Flat Λ denotes the full subcategory of flat Λ-modules, which is the closure of proj Λ under forming filtered colimits.

Example 4.8. Let 𝕏 be a quasi-compact and separated scheme, and let L be a locally free sheaf of finite rank. Then

Hom_{D(Qcoh 𝕏)}(L, −) ≅ Hom_{D(Qcoh 𝕏)}(O_{𝕏}, L^{∨} ⊗_{O_{𝕏}} −) ≅ H^{0}(L^{∨} ⊗_{O_{𝕏}} −),

where L^{∨} = Hom_{O_{𝕏}}(L, O_{𝕏}). Thus L is a compact object in D(Qcoh 𝕏); see [27]. If 𝕏 has an ample family of line bundles, then the locally free sheaves of finite rank generate Qcoh 𝕏.

Our final example shows that products in Qcoh 𝕏 need not to be exact. I learned this example from Bernhard Keller; it is included as an illustration for Corollary 4.4

Example 4.9. Let k be a field and 𝕏 = ℙ_{k}^{1} the projective line with homogeneous coordiante ring S = k[x_{0}, x_{1}]. For each n ∈ ℤ, we have an exact sequence

S(n − 1) ⨿ S(n − 1) ⟶ S(n) ⟶ k(n) ⟶ 0

of S-modules which gives rise to an epimorphism O(n−1)⨿O(n−1) → O(n) in Qcoh 𝕏. We claim that the product

π: ∏_{n∈ℤ}(O(n − 1) ⨿ O(n − 1)) ⟶ ∏_{n∈ℤ}O(n)

is not an epimorphism. In fact, the cokernel of π is isomorphic to the sheaf corresponding to the module ∏_{n∈ℤ} k(n). But this module is not torsion. For instance, it contains ∐_{n⩾0} k(n) as a finitely generated submodule which is not of finite length. We conclude that π has a non-trivial cokernel.

5. The stable derived category

Let A be a locally noetherian Grothendieck category. We suppose that D(A) is compactly generated.

Definition 5.1. The stable derived category S(A) of A is by definition the full subcategory of K(A) which is formed by all acyclic complexes of injective objects in A. The full subcategory of compact objects is denoted by S^{c}(A).

In this section, we show that the stable derived category is compactly generated, and the description of the category of compact objects justifies our terminology. Our basic tool is the (co)localization sequence

\xymatrix{S(A) \ar[r]^{I} & K(Inj A) \ar[r]^{Q} & D(A)} .

Thus we use the fact that I and Q have left adjoints I_{λ}, Q_{λ} and right adjoints I_{ρ}, Q_{ρ}. The stabilization functor is by definition the composite

S: D(A) ⟶^{I_{λ} ∘ Q_{ρ}} S(A).

We begin with the following lemma.

Lemma 5.2. Let A be a locally noetherian Grothendieck category and suppose D(A) is compactly generated. The functors Q_{λ}, Q_{ρ}: D(A) → K(Inj A) admit a natural transformation η: Q_{λ} → Q_{ρ}, and η is an isomorphism when restricted to the subcategory of compact objects in D(A).

Proof. Each object in D(A) is isomorphic to one in the image of Q, and each map QX → QY is a fraction of maps in the image of Q. Thus the natural transformation

Q_{λ} ^{∘} Q ⟶ Id_{K(Inj A)} ⟶ Q_{ρ} ^{∘} Q

induces a natural transformation η: Q_{λ} → Q_{ρ}. Now observe that Q(η) induces an isomorphism

Q^{∘} Q_{λ} ⟶^{∼} Q^{∘} Q_{ρ}.

We know from Proposition 2.3 that Q induces an equivalence

K^{c}(Inj A) ⟶^{∼} D^{b}(noeth A).

On the other hand,

Q_{λ}(D^{c}(A)) ⊆ K^{c}(Inj A)

since a left adjoint preserves compactness if the right adjoint preserves coproducts; see Lemma 3.5. Also,

Q_{ρ}(D^{c}(A)) ⊆ K^{c}(Inj A),

since D^{c}(A) ⊆ D^{b}(noeth A) by Lemma 4.1, and

Q_{ρ}(D^{b}(noeth A)) = K^{c}(Inj A)

by Remark 3.8. We conclude that η|_{D^{c}(A)} is an isomorphism. □

Proposition 5.3. Let A be a locally noetherian Grothendieck category, and suppose D(A) is compactly generated. Then we have a localization sequence

(5.1)  \xymatrix{D(A) \ar[r]^{Q_{λ}} & K(Inj A) \ar[r]^{I_{λ}} & S(A)}

which induces the following commutative diagram.

[figure: D^{c}(A); inc; D^{b}(noeth A); ≀; Q_{ρ}|_{D^{b}(noeth A)}; can; D^{b}(noeth A)/D^{c}(A); F; D^{c}(A); inc; K^{c}(Inj A); inc; S^{c}(A); inc; D(A); Q_{λ}; K(Inj A); I_{λ}; S(A)]

Proof. It follows from Theorem 4.2 that the sequence (5.1) is a localization sequence. Let us explain the commutativity of the diagram. First observe that a left adjoint preserves compactness if the right adjoint preserves coproducts; see Lemma 3.5. Therefore I_{λ} and Q_{λ} preserve compactness, and this explains the commutativity of the lower squares. Now observe that

D^{c}(A) ⊆ D^{b}(noeth A),

by Lemma 4.1, and that Q_{ρ}|_{D^{b}(noeth A)} is a quasi-inverse for Q|_{K^{c}(Inj A)}. It follows from Lemma 5.2 that the upper left hand square commutes. The functor F is by definition the unique functor making the upper right hand square commutative. It exists because I_{λ} ^{∘} Q_{λ} = 0. □

We have seen that the stable derived category S(A) is a localization of the homotopy category K(Inj A). This has some interesting consequences.

Corollary 5.4. The stable derived category S(A) is compactly generated, and the functor I_{λ} ^{∘} Q_{ρ}: D(A) → S(A) induces (up to direct factors) an equivalence

F: D^{b}(noeth A)/D^{c}(A) ⟶^{∼} S^{c}(A).

Proof. We know from Proposition 2.3 that K(Inj A) is compactly generated. This property carries over to S(A) since I_{λ} sends a set of compact generators of K(Inj A) to a set of compact generators of S(A). The functor Q_{λ} identifies D(A) with the localizing subcategory of K(Inj A) which is generated by all compact objects in the image of Q_{λ}. Now apply the localization theorem of Neeman-Ravenel-Thomason-Trobaugh-Yao [26]. This result describes the category of compact objects of the quotient S(A), up to direct factors, as the quotient of the compact objects in K(Inj A) modulo those from the localizing subcategory. To be precise, F is fully faithful and every object in S^{c}(A) is a direct factor of some object in the image of F. □

Corollary 5.5. The composite

A ⟶^{can} D(A) ⟶^{I_{λ} ∘ Q_{ρ}} S(A)

preserves all coproducts and annihilates the objects in A ∩ D^{c}(A).

Proof. The diagram in Proposition 5.3 shows that I_{λ} ^{∘} Q_{ρ} annihilates A ∩ D^{c}(A). To show that I_{λ} ^{∘} Q_{ρ} preserves all coproducts, observe that Q_{ρ} sends an object in A to an injective resolution. A coproduct of injective resolutions is again an injective resolution, and the left adjoint I_{λ} preserves all coproducts. This finishes the proof. □

Using the stabilization functor S: D(A) → S(A), we define for objects X, Y in D(A) and n ∈ ℤ the stable cohomology group

\underline{Ext}_{A}^{n}(X, Y) = Hom_{K(A)}(SX, Σ^{n}(SY)).

Note that in both arguments each exact sequence in A induces a long exact sequence in stable cohomology. We do not go into details but refer to our discussion of Tate cohomology in Section 7. In fact, both cohomology theories coincide in case A satisfies some appropriate Gorenstein property, and we shall see explicit formulae for the Tate cohomology groups.

Example 5.6. Suppose A is a module category. Then the stabilization functor annihilates all modules of finite projective dimension. Similarly, if A is a category of quasi-coherent sheaves, then the stabilization functor annihilates all sheaves having a finite resolution with locally free sheaves.

Given a noetherian scheme 𝕏, the stable derived category S(Qcoh 𝕏) vanishes if 𝕏 is regular. Nonetheless, a classical result of Bernstein-Gelfand-Gelfand [8] shows that stable derived categories are relevant when one studies regular schemes. This is sketched in the following example.

Example 5.7. Let Λ be a Koszul algebra and Λ^{!} its Koszul dual. Then we have under appropriate assumptions an equivalence K(Inj Λ) →^{∼} K(Inj Λ^{!}) which induces an equivalence D^{b}(mod Λ) →^{∼} D^{b}(mod Λ^{!}) when restricted to the full subcategories of compact objects [7, 22]. Note that we consider the categories of graded modules over Λ and Λ^{!} respectively. The classical example is the symmetric algebra Λ = SV of a d-dimensional space V over a field k, where Λ^{!} is the exterior algebra ⋀ V^{∗} of the dual space V^{∗}. The equivalence K(Inj Λ) →^{∼} K(Inj Λ^{!}) takes an injective resolution ik of Λ_{0} = k to Λ^{!} and identifies the localizing subcategory K generated by ik with the localizing subcategory generated by Λ^{!}, which is D(Mod Λ^{!}). Note that the quotient K(Inj Λ)/K identifies with the derived category of the quotient Mod Λ/(Mod Λ)_{0}, where (Mod Λ)_{0} denotes the subcategory of torsion modules. This quotient is equivalent to Qcoh ℙ_{k}^{d−1} by Serre’s Theorem. Thus we obtain an equivalence

D(Qcoh ℙ_{k}^{d−1}) ⟶^{∼} S(Mod ⋀ k^{d}).

Note that S(Mod ⋀ k^{d}) is equivalent to the stable module category \underline{Mod} ⋀ k^{d} because the exterior algebra is self-injective; see Example 7.16. Passing to the subcategory of compact objects, one obtains the equivalence

D^{b}(coh ℙ_{k}^{d−1}) ⟶^{∼} \underline{mod} ⋀ k^{d}

of Bernstein-Gelfand-Gelfand [8], where \underline{mod} ⋀ k^{d} denotes the stable category of all finite dimensional ⋀ k^{d}-modules. This example generalizes to non-commutative algebras, for instance, to Artin-Schelter regular algebras [21].

6. Extending derived functors

An additive functor F: A → B between locally noetherian Grothendieck categories admits a right derived functor RF: D(A) → D(B). In this section, we extend this to a functor R̂F: K(Inj A) → K(Inj B) and investigate its right and left adjoints. As an application, we consider for F the direct image functor f_{∗}: Qcoh 𝕏 → Qcoh 𝕐 corresponding to a morphism f: 𝕏 → 𝕐 between noetherian schemes. We use the following functors

I: K(Inj A) ⟶^{inc} K(A) and Q: K(Inj A) ⟶^{inc} K(A) ⟶^{can} D(A)

simultanously for A and B. Moreover, we use the fact that both functors have left and right adjoints.

Theorem 6.1. Let F: A → B be an additive functor between locally noetherian Grothendieck categories. Suppose D(A) and D(B) are compactly generated. Then the composite

R̂F: K(Inj A) ⟶^{I} K(A) ⟶^{K(F)} K(B) ⟶^{I_{λ}} K(Inj B)

makes the following diagram commutative.

\xymatrix{D(A) \ar[d]^{Q_{ρ}} \ar[r]^{RF} & D(B) \\ K(Inj A) \ar[r]^{R̂F} & K(Inj B) \ar[u]_{Q}}

(1) Suppose F preserves coproducts. Then R̂F preserves coproducts and has therefore a right adjoint (R̂F)_{ρ}.

(2) Suppose F and RF preserve coproducts. Then RF has a right adjoint (RF)_{ρ} making the following diagram commutative.

\xymatrix{D(A) \ar[d]^{Q_{λ}} \ar[r]^{RF} & D(B) & D(B) \ar[d]^{Q_{ρ}} \ar[r]^{(RF)_{ρ}} & D(A) \\ K(Inj A) \ar[r]^{R̂F} & K(Inj B) \ar[u]_{Q} & K(Inj B) \ar[r]^{(R̂F)_{ρ}} & K(Inj A) \ar[u]_{Q}}

Proof. The composite

K(A) ⟶^{I_{λ}} K(Inj A) ⟶^{Q} D(A)

is naturally isomorphic to the canonical functor K(A) → D(A); see Remark 3.7. Clearly, I^{∘} Q_{ρ} is its right adjoint. We denote by RF the right derived functor of F and have

RF = Q^{∘} I_{λ} ^{∘} K(F)^{∘} I^{∘} Q_{ρ}.

Using the definition R̂F = I_{λ} ^{∘} K(F)^{∘} I, we obtain RF = Q^{∘} R̂F ^{∘} Q_{ρ}.

(1) Suppose F preserves coproducts. Then K(F) preserves coproducts. It follows that R̂F preserves coproducts since I and I_{λ} preserve coproducts. Now apply Proposition 3.3 to obtain a right adjoint for R̂F.

(2) Suppose F and RF preserve coproducts. Then RF has a right adjoint by Proposition 3.3. Next we show that

Q^{∘} R̂F ^{∘} Q_{λ} ≅ Q^{∘} R̂F ^{∘} Q_{ρ}.

We have a natural transformation Q_{λ} → Q_{ρ} which is induced from the natural transformation Q_{λ} ^{∘} Q → Q_{ρ} ^{∘} Q. Now apply Q^{∘} R̂F to get a natural transformation

μ: Q^{∘} R̂F ^{∘} Q_{λ} ⟶ Q^{∘} R̂F ^{∘} Q_{ρ}.

It is shown in Lemma 5.2 that Q_{λ} → Q_{ρ} is an isomorphism when restricted to compact objects in D(A). On the other hand, Q^{∘} R̂F ^{∘} Q_{λ} and Q^{∘} R̂F ^{∘} Q_{ρ} both preserve coproducts by our assumption on RF. It follows that μ is an isomorphism since D(A) is compactly generated. Clearly, Q^{∘}(R̂F)_{ρ} ^{∘} Q_{ρ} is a right adjoint for Q^{∘} R̂F ^{∘} Q_{λ}. This completes the proof. □

The extended derived functor and its right adjoint admit some alternative description. I am indebted to Bernhard Keller for providing this remark.

Remark 6.2. It is possible to express R̂F as the tensor functor and its right adjoint (R̂F)_{ρ} as the Hom functor with respect to a bimodule of DG categories; see [22, 6.4]. This depends on the appropriate choice of DG categories A_{0} and B_{0} such that K(Inj A) ≅ D_{dg}(A_{0}) and K(Inj B) ≅ D_{dg}(B_{0}) respectively.

Next we consider as an example a morphism f: 𝕏 → 𝕐 between separated noetherian schemes. Let f_{∗}: Qcoh 𝕏 → Qcoh 𝕐 denote the direct image functor. Note that the right derived functor Rf_{∗}: D(Qcoh 𝕏) → D(Qcoh 𝕐) preserves coproducts [27, Lemma 1.4]. Thus Rf_{∗} and its right adjoint Grothendieck duality functor f^{!} extend to functors between K(Inj 𝕏) and K(Inj 𝕐), by Theorem 6.1. This is the statement of Theorem 1.4 from the introduction. In fact, the situation is in this case much nicer. I am grateful to Amnon Neeman for pointing out that the functor R̂f_{∗} and its right adjoint (R̂f_{∗})_{ρ} make the following diagram commutative.

(6.1) \xymatrix{K(Inj 𝕏) \ar[d]^{Q} \ar[r]^{R̂f_{∗}} & K(Inj 𝕐) \ar[d]^{Q} & K(Inj 𝕐) \ar[r]^{(R̂f_{∗})_{ρ}} & K(Inj 𝕏) \\ D(Qcoh 𝕏) \ar[r]^{Rf_{∗}} & D(Qcoh 𝕐) & D(Qcoh 𝕐) \ar[u]_{Q_{ρ}} \ar[r]^{f^{!}} & D(Qcoh 𝕏) \ar[u]_{Q_{ρ}}}

This is essentially the statement of Theorem 1.5 from the introduction. The proof which is due to Neeman uses the following lemma.

Lemma 6.3. Keep the assumptions from Theorem 6.1 and consider the following diagram.

\xymatrix{K(Inj A) \ar[d]^{Q} \ar[r]^{R̂F} & K(Inj B) \ar[d]^{Q} \\ D(A) \ar[r]^{RF} & D(B)}

There is a natural transformation Q^{∘} R̂F → RF ^{∘} Q which is an isomorphism if and only if F sends every acyclic complex of injective objects to an acyclic complex.

Proof. We apply the the localization sequence

\xymatrix{K_{ac}(Inj A) \ar[r]^{I} & K(Inj A) \ar[r]^{Q} & D(A)}

from Proposition 3.6. Let X be an object in K(Inj A) and consider the triangle

(I^{∘} I_{ρ})X ⟶ X ⟶ (Q_{ρ} ^{∘} Q)X ⟶ Σ(I^{∘} I_{ρ})X

in K(Inj A). Now apply Q^{∘} R̂F which gives a map

(Q^{∘} R̂F)X ⟶ (RF ^{∘} Q)X

since Q^{∘} R̂F ^{∘} Q_{ρ} ≅ RF, by Theorem 6.1. Clearly, this map is an isomorphism if and only if Q^{∘} R̂F annihilates (I^{∘} I_{ρ})X. □

We include a simple example which illustrates the preceding lemma.

Example 6.4. Let k be a field and Λ = k[t]/(t^{2}). We take the functor

F: Mod Λ ⟶ Mod k, X ↦ Hom_{Λ}(k, X),

and observe that the following diagram does not commute.

\xymatrix{K(Inj Λ) \ar[d]^{Q} \ar[r]^{R̂F} & K(Inj k) \ar[d]_{≀}^{Q} \\ D(Mod Λ) \ar[r]^{RF} & D(Mod k)}

For instance, we have QX = 0 and (R̂F)X ≠ 0 if we take for X the acyclic complex

⋯ ⟶^{t} Λ ⟶^{t} Λ ⟶^{t} Λ ⟶^{t} ⋯

in K(Inj Λ).

Proof of Theorem 1.5. We need to show that both squares in (6.1) commute. Then we use the localization sequence

\xymatrix{S(Qcoh 𝕏) \ar[r]^{I} & K(Inj 𝕏) \ar[r]^{Q} & D(Qcoh 𝕏)}

from Proposition 3.6 and obtain from R̂f_{∗} and (R̂f_{∗})_{ρ} an adjoint pair of functors between S(Qcoh 𝕏) and S(Qcoh 𝕐).

In order to show the commutativity of (6.1), we apply Lemma 6.3 and need to show that f_{∗} sends an acyclic complex X of injective objects to an acyclic complex. The question is local in 𝕐 and we may assume 𝕐 affine. Cover 𝕏 by a finite number of affines. Then f_{∗} can be computed using the Čech cohomology of the cover. If there are n open sets in the cover, then for any quasi-coherent sheaf A we have

R^{n+1}f_{∗}A = 0.

Now take our acyclic complex X of injective sheaves on 𝕏. Then the sequence

0 ⟶ X^{0} ⟶ X^{1} ⟶ X^{2} ⟶ ⋯

is an injective resolution of the kernel A of the map X^{0} → X^{1}. Applying f_{∗}, the sequence computes for us R^{i}f_{∗}A, which vanishes if i ⩾ n + 1. Thus f_{∗}X is acyclic above degree n, but by shifting we conclude that it is acyclic everywhere.

Having shown the commutativity of the left hand square, the commutativity of the right hand square follows, because it is obtained by taking right adjoints. Thus the proof is complete. □

Next we investigate for an exact functor F: A → B an extension L̂F of the derived functor LF: D(A) → D(B). For this we need some assumptions, and it is convenient to introduce the following notation. As before, A and B denote locally noetherian Grothendieck categories. Let f: noeth A → noeth B be an additive functor. Then there is, up to isomorphism, a unique functor f^{∗}: A → B which extends f and preserves filtered colimits. This has a right adjoint f_{∗}: B → A if and only if f is right exact. Note that f is exact iff f^{∗} is exact iff f_{∗} sends injective objects to injective objects. Here is an example.

Example 6.5. Let f: 𝕏 → 𝕐 be a morphism between noetherian schemes. Then the inverse image functor f^{∗}: Qcoh 𝕐 → Qcoh 𝕏 sends coherent sheaves to coherent sheaves and preserves filtered colimits. Moreover, the direct image functor f_{∗} is a right adjoint of f^{∗}. Our notation is therefore consistent if we identify the morphism f: 𝕏 → 𝕐 with the functor coh 𝕐 → coh 𝕏.

Theorem 6.6. Let A and B locally noetherian Grothendieck categories such that D(A) and D(B) are compactly generated. Let f: noeth A → noeth B be an exact functor. Then R̂f_{∗} has a left adjoint L̂f^{∗} which induces a functor Sf^{∗} making the following diagram commutative.

(6.2) [figure: K^{c}(Inj A); ∼; D^{b}(noeth A); D^{b}(f); S(A); I; Sf^{∗}; K(Inj A); Q; L̂f^{∗}; D(A); Lf^{∗}; K^{c}(Inj B); ∼; D^{b}(noeth B); S(B); I; K(Inj B); Q; D(B)]

If Rf_{∗} preserves coproducts, then in addition the following diagram commutes.

[figure: D^{b}(noeth A)/D^{c}(A); F_{A}; \overline{D^{b}(f)}; K^{c}(Inj A); D^{c}(A); S(A); Sf^{∗}; K(Inj A); I_{λ}; L̂f^{∗}; D(A); Q_{λ}; Lf^{∗}; D^{b}(noeth B)/D^{c}(B); F_{B}; K^{c}(Inj B); D^{c}(B); S(B); K(Inj B); I_{λ}; D(B); Q_{λ}]

Note that F_{A} and F_{B} induce, up to direct factors, equivalences onto the full subcategories of compact objects in S(A) and S(B) respectively. Thus \overline{D^{b}(f)} determines the functor Sf^{∗}.

Proof. The exactness of f implies the exactness of f^{∗}. Thus f_{∗} sends injective objects to injective objects and we have the following commutative diagram.

(6.3)  [figure: K(Inj B); J; R̂f_{∗}; K(B); K(f_{∗}); K(Inj B); J; K(B)]

The right adjoint f_{∗} preserves products and we have therefore a left adjoint for R̂f_{∗}, by Proposition 3.3, which we denote by L̂f^{∗}. We obtain the following diagram

\xymatrix{K(A) \ar[r]^{J_{λ}} \ar[d]^{K(f^{∗})} & K(Inj A) \ar[r]^{Q} \ar[d]^{L̂f^{∗}} & D(A) \ar[d]^{Lf^{∗}} \\ K(B) \ar[r]^{J_{λ}} & K(Inj B) \ar[r]^{Q} & D(B)}

and claim it is commutative. The left hand square commutes because it is obtained from (6.3) by taking left adjoints. The outer square commutes because the composite Q^{∘} J_{λ} is naturally isomorphic to the canonical functor K(A) → D(A); see Remark 3.7. We conclude the commutativity of the right hand square, using that J_{λ} ^{∘} J ≅ Id_{K(Inj A)}. Clearly, L̂f^{∗} sends acyclic complexes to acyclic complexes and we obtain the functor Sf^{∗} making the diagram (6.2) commutative.

Now assume in addition that Rf_{∗} preserves coproducts. We use the fact that a left adjoint preserves compactness if the right adjoint preserves coproducts; see Lemma 3.5. The functor R̂f_{∗} preserves coproducts since f_{∗} preserves coproducts. Thus Lf^{∗} and L̂f^{∗} preserve compactness. Note that Q_{λ} identifies D(A) with the localizing subcategory of K(Inj A) which is generated by

D^{c}(A) ⊆ D^{b}(noeth A) ≅ K^{c}(Inj A) ⊆ K(Inj A).

Of course, the same applies for B. We obtain the following diagram

\xymatrix{S(A) \ar[d]^{S} & K(Inj A) \ar[l]_{I_{λ}} \ar[d]^{L̂f^{∗}} & D(A) \ar[l]_{Q_{λ}} \ar[d]^{Lf^{∗}} \\ S(B) & K(Inj B) \ar[l]_{I_{λ}} & D(B) \ar[l]_{Q_{λ}}}

where the right hand square commutes. The horizontal sequences are localization sequences by Theorem 4.2, and L̂f^{∗} induces a functor S: S(A) → S(B) making the left hand square commutative. Moreover, we have

S = S^{∘} I_{λ} ^{∘} I = I_{λ} ^{∘} L̂f^{∗} ^{∘} I = I_{λ} ^{∘} I^{∘} Sf^{∗} = Sf^{∗}.

The functors F_{A} and F_{B} are both induced by I_{λ}, and the commutativity

Sf^{∗} ^{∘} F_{A} = F_{B} ^{∘} \overline{D^{b}(f)}

is easily checked; see Corollary 5.4. This completes the proof. □

Remark 6.7. Let f: 𝕏 → 𝕐 be a morphism between noetherian schemes and suppose f^{∗} is exact. Then Sf^{∗} is the left adjoint of Sf_{∗} which appears in Theorem 1.5.

Next we investigate the inclusion f: 𝕏 → 𝕐 of an open subscheme. In this case, the adjoint pair of functors f_{∗} and f^{∗} between Qcoh 𝕏 and Qcoh 𝕐 restricts to an adjoint pair of functors between Inj 𝕏 and Inj 𝕐; see [15, VI]. Moreover, f^{∗} ^{∘} f_{∗} ≅ Id_{Qcoh 𝕏}. Thus we can identify R̂f_{∗} = f_{∗} and L̂f^{∗} = f^{∗}. Note that both functors send acyclic complexes with injective components to acyclic complexes. This is clear for f^{∗} because it is exact, and follows for f_{∗} from Theorem 1.5. We denote for each sheaf A in Qcoh 𝕐 by Supp A the support of A and observe that f^{∗} annihilates A if and only Supp A is contained in 𝕐 \ 𝕏. In fact, the natural map A → (f_{∗} ^{∘} f^{∗})A induces a split exact sequence

0 ⟶ A′ ⟶ A ⟶ (f_{∗} ^{∘} f^{∗})A ⟶ 0

if A is injective. In particular, the support of A′ is contained in 𝕐 \ 𝕏.

Now fix a complex X in K(Inj 𝕐). The support of X is by definition

Supp X = ⋃_{n∈ℤ}Supp X^{n}.

We write X_{𝕏} = (f_{∗} ^{∘} f^{∗})X, and the natural map X → X_{𝕏} induces an exact triangle

(6.4)  X_{𝕐\𝕏} ⟶ X ⟶ X_{𝕏} ⟶ Σ(X_{𝕐\𝕏})

in K(Inj 𝕐) where the support of X_{𝕐\𝕏} is contained in 𝕐 \ 𝕏.

Lemma 6.8. Let 𝕐 be a seperated noetherian scheme and f: 𝕏 → 𝕐 be the inclusion of an open subscheme. If X is a complex in K(Inj 𝕐), then f^{∗}X = 0 if and only if X is isomorphic to a complex with support contained in 𝕐 \ 𝕏.

Proof. We have f^{∗}X = 0 if and only if the first map in the triangle (6.4) is an isomorphism. □

It is well known that f^{∗} induces an equivalence

D(Qcoh 𝕐)/D_{𝕐\𝕏}(Qcoh 𝕐) ⟶^{∼} D(Qcoh 𝕏),

where D_{𝕐\𝕏}(Qcoh 𝕐) denotes the full subcategory of all complexes in D(Qcoh 𝕐) such that the support of the cohomology is contained in 𝕐 \ 𝕏. We obtain an analogue for K(Inj 𝕐) and S(Qcoh 𝕐) if we define

K_{𝕐\𝕏}(Inj 𝕐) = {X ∈ K(Inj 𝕐) | Supp X ⊆ 𝕐 \ 𝕏},
S_{𝕐\𝕏}(Qcoh 𝕐) = {X ∈ S(Qcoh 𝕐) | Supp X ⊆ 𝕐 \ 𝕏}.

Proposition 6.9. Let 𝕐 be a seperated noetherian scheme and f: 𝕏 → 𝕐 be the inclusion of an open subscheme. Then f^{∗} induces equivalences

K(Inj 𝕐)/K_{𝕐\𝕏}(Inj 𝕐) ⟶^{∼} K(Inj 𝕏),
S(Qcoh 𝕐)/S_{𝕐\𝕏}(Qcoh 𝕐) ⟶^{∼} S(Qcoh 𝕏).

Proof. We have f^{∗} ^{∘} f_{∗} ≅ Id_{Qcoh 𝕏}, and this carries over to complexes of injectives. On the other hand, we have for X in K(Inj 𝕐) a natural map X → (f_{∗} ^{∘} f^{∗})X which induces an isomorphism in K(Inj 𝕐)/K_{𝕐\𝕏}(Inj 𝕐), by Lemma 6.8. □

Let us give a more elaborate formulation of Proposition 6.9. The functor R̂f_{∗} = f_{∗}: K(Inj 𝕏) → K(Inj 𝕐) admits a left and a right adjoint. Therefore R̂f_{∗} induces a recollement

\xymatrix{K(Inj 𝕏) \ar[r] & K(Inj 𝕐) \ar[r] \ar[l] \ar[l] & K_{𝕐\𝕏}(Inj 𝕐). \ar[l] \ar[l]}

This recollement is compatible with the recollement

\xymatrix{S(Qcoh 𝕐) \ar[r] & K(Inj 𝕐) \ar[r] \ar[l] \ar[l] & D(Qcoh 𝕐), \ar[l] \ar[l]}

and we obtain the following diagram.

\xymatrix{S(Qcoh 𝕏) \ar[d] \ar[r] & K(Inj 𝕏) \ar[d] \ar[r] \ar[l] \ar[l] & D(Qcoh 𝕏) \ar[d] \ar[l] \ar[l] \\ S(Qcoh 𝕐) \ar[d] \ar[u] \ar[u] \ar[r] & K(Inj 𝕐) \ar[d] \ar[r] \ar[u] \ar[u] \ar[l] \ar[l] & D(Qcoh 𝕐) \ar[d] \ar[u] \ar[u] \ar[l] \ar[l] \\ S_{𝕐\𝕏}(Qcoh 𝕐) \ar[u] \ar[u] \ar[r] & K_{𝕐\𝕏}(Inj 𝕐) \ar[r] \ar[u] \ar[u] \ar[l] \ar[l] & D_{𝕐\𝕏}(Qcoh 𝕐) \ar[u] \ar[u] \ar[l] \ar[l]}

In this diagram, each row and each column is a recollement. Moreover, the diagram is commutative if one restricts to arrows in south and east direction. All other commutativity relations follow by taking left adjoints or right adjoints.

Proposition 6.9 tells us precisely when the inclusion of a subscheme induces an equivalence for the stable derived category. In [29], Orlov observed that the bounded stable derived category of a noetherian scheme depends only on the singular points. We extend this result to the unbounded stable derived category, using a completely different proof.

Corollary 6.10. Let 𝕐 be a seperated noetherian scheme of finite Krull dimension. If f: 𝕏 → 𝕐 denotes the inclusion of an open subscheme which contains all singular points of 𝕐, then Sf^{∗}: S(Qcoh 𝕐) → S(Qcoh 𝕏) is an equivalence.

Proof. We apply Proposition 6.9 and need to show that S_{𝕐\𝕏}(Qcoh 𝕐) = 0. But this is clear from our assumptions on 𝕏 and 𝕐. □

Orlov’s result [29, Proposition 1.14] is an immediate consequence if one restricts the equivalence Sf^{∗} to compact objects; see Theorem 6.6.

Corollary 6.11. Let 𝕐 be a seperated noetherian scheme of finite Krull dimension. If f: 𝕏 → 𝕐 denotes the inclusion of an open subscheme which contains all singular points of 𝕐, then f^{∗} induces (up to direct factors) an equivalence

D^{b}(coh 𝕐)/D^{perf}(coh 𝕐) ⟶ D^{b}(coh 𝕏)/D^{perf}(coh 𝕏).

7. Gorenstein injective approximations and Tate cohomology

Let A be a locally noetherian Grothendieck category and suppose that the derived category D(A) is compactly generated. In this section, we study the category of complete injective resolutions. We assign functorially to each complex of injectives a complete resolution. This yields Gorenstein injective approximations and Tate cohomology groups for objects in A. The classical definition of Tate cohomology is based on complete projective resolutions. Our approach is essentially the same, using however resolutions with injective instead of projective components. Another aspect in this section is the interplay between the stable derived category S(A) and the stable category \underline{A} modulo injective objects, which is obtained from A by identifying two maps if their difference factors through some injective object. Given objects A, B in A, we write

\underline{Hom}_{A}(A, B) = Hom\underline{A}(A, B).

The functor

K(Inj A) ⟶ \underline{A}, X ↦ Z^{0}X = Ker(X^{0} → X^{1})

provides a link between the stable categories S(A) and \underline{A}. In particular, we obtain an explicit description of the stabilization functor

S: A ⟶^{can} D(A) ⟶^{I_{λ} ∘ Q_{ρ}} S(A)

provided that A has some appropriate Gorenstein property.

Most of the concepts in this section are classical, but seem to be new in this setting and this generality. We refer to the end of this section for historical remarks and references to the literature.

Let us start with the relevant definitions. A complex X in Inj A is called totally acyclic if Hom_{A}(A, X) and Hom_{A}(X, A) are acyclic complexes of abelian groups for all A in Inj A. We denote by K_{tac}(Inj A) the full subcategory of all totally acyclic complexes in K(Inj A). Following [13], we call an object A in A Gorenstein injective if it is of the form Z^{0}X for some X in K_{tac}(Inj A). We write GInj A for the full subcategory formed by all Gorenstein injective objects.

Lemma 7.1. Let A be an abelian category and let X, Y be objects in K(Inj A). Suppose H^{n}X = 0 for all n > 0 and Y is totally acyclic. Then the canonical map

σ_{X,Y} : Hom_{K(Inj A)}(X, Y) ⟶ \underline{Hom}_{A}(Z^{0}X, Z^{0}Y)

is bijective.

Proof. Fix a map α: Z^{0}X → Z^{0}Y in A. We need to extend α to a chain map ᾱ: X → Y such that Z^{0}ᾱ = α. We use the assumption on X to extend α in non-negative degrees, and the assumption on Y allows to extend α in negative degrees. Thus σ_{X,Y} is surjective. To show that σ_{X,Y} is injective, let ϕ: X → Y be a chain map such that Z^{0}ϕ factors through some injective object. A similar argument as before yields a chain homotopy X → Y which shows that ϕ is null-homotopic. Thus the proof is complete. □

Let us denote by GInj \underline{A} the full subcategory of \underline{A} formed by the objects in GInj A. Observe that GInj A is a Frobenius category with respect to the class of exact sequences from A. With respect to this exact structure, an object A in GInj A is projective iff A is injective iff A belongs to Inj A. Thus the category GInj \underline{A} carries a triangulated structure. The shift takes an object A to the cokernel ΣA of a monomorphism A → E into an injective object E. The exact triangles are induced from short exact sequences in A.

Proposition 7.2. Let A be an abelian category. Then the functor

K_{tac}(Inj A) ⟶ GInj \underline{A}, X ↦ Z^{0}X,

is an equivalence of triangulated categories.

Proof. We need to show that the functor is fully faithful and surjective on isomorphism classes of objects. The last property is clear from the definition of GInj A. The functor is fully faithful by Lemma 7.1. Finally, observe that an exact triangle of complexes comes, up to isomorphism, from a sequence of complexes which is split exact in each degree. Thus we obtain an exact sequence in A and an exact triangle in \underline{A} if we apply Z^{0}. □

The following lemma is crucial because it provides the existence of complete injective resolutions.

Lemma 7.3. Let A be a locally noetherian Grothendieck category and suppose that D(A) is compactly generated. Then the inclusion J: K_{tac}(Inj A) → K(Inj A) has a left adjoint

J_{λ}: K(Inj A) ⟶ K_{tac}(Inj A).

Proof. The inclusion I: K_{ac}(Inj A) → K(Inj A) has a left adjoint I_{λ} by Theorem 4.2. Thus it is sufficient to show that the inclusion K_{tac}(Inj A) → K_{ac}(Inj A) has a left adjoint. Let us fix an injective cogenerator E in A. By definition, K_{tac}(Inj A) consists of all objects X in K_{ac}(Inj A) such that

Hom_{K(Inj A)}(Σ^{n}E, X) ≅ Hom_{K_{ac}(Inj A)}(I_{λ}(Σ^{n}E), X)

vanishes for all n ∈ ℤ. The category K_{ac}(Inj A) is compactly generated by Corollary 5.4, and we can apply Proposition 3.4 to obtain a left adjoint for the inclusion K_{tac}(Inj A) → K_{ac}(Inj A). □

Given an object A in A with injective resolution iA, we call the natural map

iA ⟶ J_{λ}iA

a complete injective resolution of A. If we apply the functor Z^{0} to this map, we obtain a Gorenstein injective approximation of A.

Theorem 7.4. Let A be a locally noetherian Grothendieck category and suppose that D(A) is compactly generated. Then the inclusion GInj \underline{A} → \underline{A} has a left adjoint

T: \underline{A} ⟶ GInj \underline{A}.

Thus we have for each object A in A a natural map A → T A which induces a bijection

\underline{Hom}_{A}(T A, B) ⟶^{∼} \underline{Hom}_{A}(A, B) for all B ∈ GInj A.

Proof. Fix an object A in A and choose an injective resolution iA. We put

T A = Z^{0}(J_{λ}iA),

and this induces a functor T: \underline{A} → GInj \underline{A}. Let B in GInj A and fix a totally acyclic complex tB such that Z^{0}tB = B. The natural map iA → J_{λ}iA induces a map A → T A in \underline{A} which makes the following square commutative.

\xymatrix{Hom_{K(Inj A)}(J_{λ}iA, tB) \ar[d]^{Z^{0}} \ar[r]^{∼} & Hom_{K(Inj A)}(iA, tB) \ar[d]^{Z^{0}} \\ \underline{Hom}_{A}(T A, B) \ar[r] & \underline{Hom}_{A}(A, B)}

The vertical maps are bijective by Lemma 7.1, and we conclude that T is a left adjoint for the inclusion GInj \underline{A} → \underline{A}. □

Next we use complete injective resolutions to define Tate cohomology groups for objects in A.

Definition 7.5. Given objects A, B in A and n ∈ ℤ, the Tate cohomology group is

Êxt_{A}^{n}(A, B) = H^{n} Hom_{A}(A, J_{λ}iB)

Remark 7.6. The correct term for this cohomology theory would be ‘injective Tate cohomology’ in order to distinguish it from the usual ‘projective Tate cohomology’ which is defined via complete projective resolutions. For simplicity, we drop the extra adjective ‘injective’. Note that confusion is not possible because we do not consider projective Tate cohomology in this paper.

Tate cohomology is natural in both arguments because the formation of complete injective resolutions is functorial. In addition, we have a comparison map

Ext_{A}^{n}(A, B) ⟶ Êxt_{A}^{n}(A, B),

which is induced by the map iB → J_{λ}iB. There is an alternative description of Tate cohomology which is based on the left adjoint T: \underline{A} → GInj \underline{A}.

Proposition 7.7. Given objects A, B in A and n ∈ ℤ, there is a natural isomorphism

Êxt_{A}^{n}(A, B) ≅ \underline{Hom}_{A}(A, Σ^{n}(T B)).

Proof. Using Lemmas 2.1 and 7.1, we have the following sequence of isomorphisms

H^{n} Hom_{A}(A, J_{λ}iB) ≅ Hom_{K(A)}(A, Σ^{n}(J_{λ}iB))
≅ Hom_{K(A)}(iA, Σ^{n}(J_{λ}iB))
≅ \underline{Hom}_{A}(A, Σ^{n}(T B)).

□

We have a more conceptual definition of Tate cohomology for objects in D(A) which uses the composite

U: D(A) ⟶^{J_{λ} ∘ Q_{ρ}} K_{tac}(Inj A).

Thus we define for objects X, Y in D(A) and n ∈ ℤ

Êxt_{A}^{n}(X, Y) = Hom_{K(A)}(UX, Σ^{n}(UY)).

Note that this definition is consistent with the original definition of Tate cohomology if we take objects in A and view them as complexes concentrated in degree 0. This follows from the fact that Q_{ρ}X is nothing but an injective resolution of X^{0} when X is concentrated in degree 0. From now on, we will use one of the alternative descriptions of Tate cohomology whenever this is convenient.

Next we show that each exact sequence in A induces a long exact sequence in Tate cohomology. This is based on the following simple lemma.

Lemma 7.8. The left adjoint T: \underline{A} → GInj \underline{A} has the following properties. (1) An exact sequence 0 → A′ → A → A′′ → 0 in A induces an exact triangle

T A′ ⟶ T A ⟶ T A′′ ⟶ Σ(T A′) in GInj \underline{A}.

(2) Let A, B be in A and n ∈ ℤ. The natural map A → T A induces an isomorphism

Êxt_{A}^{n}(T A, B) ≅ Êxt_{A}^{n}(A, B).

Proof. (1) We have an exact triangle A′ → A → A′′ → Σ(A′) in D(A). Now use that the exact functor Z^{0} ^{∘} J_{λ} ^{∘} Q_{ρ} computes T.

(2) The adjointness property of T implies \underline{Hom}_{A}(T A, T B) ≅ \underline{Hom}_{A}(A, T B). □

Proposition 7.9. Let 0 → B′ → B → B′′ → 0 be an exact sequence in A. Then we have for A and C in A the following long exact sequences.

⋯ ⟶ Êxt_{A}^{n}(A, B′) ⟶ Êxt_{A}^{n}(A, B) ⟶ Êxt_{A}^{n}(A, B′′) ⟶

⟶ Êxt_{A}^{n+1}(A, B′) ⟶ Êxt_{A}^{n+1}(A, B) ⟶ Êxt_{A}^{n+1}(A, B′′) ⟶ ⋯

⋯ ⟶ Êxt_{A}^{n}(B′′, C) ⟶ Êxt_{A}^{n}(B, C) ⟶ Êxt_{A}^{n}(B′, C) ⟶

⟶ Êxt_{A}^{n+1}(B′′, C) ⟶ Êxt_{A}^{n+1}(B, C) ⟶ Êxt_{A}^{n+1}(B′, C) ⟶ ⋯

Proof. We apply Lemma 7.8 and use the fact that \underline{Hom}_{A}(T A, −) and \underline{Hom}_{A}(−, T C) are cohomological functors. □

We compute Tate cohomology for Gorenstein injective objects.

Proposition 7.10. Let A, B be objects in A and suppose B is Gorenstein injective. Then the comparison map

Ext_{A}^{n}(A, B) ⟶ Êxt_{A}^{n}(A, B)

is an isomorphism for n > 0 and induces an isomorphism

\underline{Hom}_{A}(A, B) ⟶^{∼} Êxt_{A}^{0}(A, B) for n = 0.

Proof. Our assumption implies T B = B. The case n = 0 is clear. For n = 1, choose an exact sequence 0 → B → E → ΣB → 0 with E injective and apply Hom_{A}(A, −). The cokernel of

Hom_{A}(A, E) ⟶ Hom_{A}(A, ΣB)

is isomorphic Ext_{A}^{1}(A, B); it is isomorphic to \underline{Hom}_{A}(A, ΣB) since B is Gorenstein injective. For n > 1, use dimension shift. □

Next we describe those objects A in A such that Êxt_{A}^{∗}(A, −) vanishes. For instance, Tate cohomology vanishes for all objects having finite projective or finite injective dimension.

Proposition 7.11. For an object A in A, the following are equivalent.

(1) Êxt_{A}^{∗}(A, −) = 0.

(2) Êxt_{A}^{0}(A, A) = 0.

(3) Êxt_{A}^{∗}(−, A) = 0.

(4) Ext_{A}^{1}(A, B) = 0 for all B ∈ GInj A.

(5) \underline{Hom}_{A}(A, B) = 0 for all B ∈ GInj A.

Proof. Use the isomorphism Êxt_{A}^{∗}(−, −) ≅ \underline{Hom}_{A}^{∗}(T −, T −) and the fact that Êxt_{A}^{1}(−, B) can be computed via Ext_{A}^{1}(−, T B). □

The following result formulates our analogue of the maximal Cohen-Macaulay approximation in the sense of Auslander and Buchweitz [2]. Note that a Gorenstein injective object is the ‘dual’ of a maximal Cohen-Macaulay object which one defines in a category with enough projectives. Let

X = {A ∈ A | Ext_{A}^{1}(A, B) = 0 for all B ∈ GInj A} and Y = GInj A.

Theorem 7.12. Let A be a locally noetherian Grothendieck category and suppose that D(A) is compactly generated.

(1) Every object A in A fits into exact sequences

0 → Y_{A} → X_{A} → A → 0 and 0 → A → Y^{A} → X^{A} → 0

in A with X_{A}, X^{A} in X and Y_{A}, Y^{A} in Y.

(2) The map A ↦ X_{A} induces a right adjoint for the inclusion \underline{X} → \underline{A}.

(3) The map A ↦ Y^{A} induces a left adjoint for the inclusion \underline{Y} → \underline{A}.

(4) X ∩ Y = Inj A.

Note that this is essentially the statement of Theorem 1.3 from the introduction, since X is precisely the subcategory of objects A in A such that the Tate cohomology functor Êxt_{A}^{∗}(A, −) vanishes.

Proof. We use the basic properties of Tate cohomology.

(1) Fix an object A in A and a complete injective resolution

iA ⟶ J_{λ}iA = yA.

We complete this map to an exact triangle

(7.1)  iA ⟶ yA ⟶ xA ⟶ Σ(iA).

in K(Inj A) and have therefore a sequence 0 → iA → yA → xA → 0 of complexes which is split exact in each degree. Applying Z^{0} : K(Inj A) → \underline{A} produces an exact sequence

0 ⟶ A ⟶^{α} Y^{A} ⟶^{β} X^{A} ⟶ 0

in A. Clearly, Y^{A} belongs to Y. On the other hand, Êxt_{A}^{∗}(α, −) is an isomorphism. Thus Êxt_{A}^{∗}(X^{A}, −) vanishes and X^{A} belongs to X. The second sequence ending in A is obtained by rotating the triangle (7.1).

(2) We consider the exact sequence

0 ⟶ Y_{A} ⟶^{μ} X_{A} ⟶^{ν} A ⟶ 0

and need to show that \underline{Hom}_{A}(X, ν) is bijective for all X in X. To see this, let ϕ: X → A be a map with X in X. The map ϕ factors through ν since Ext_{A}^{1}(X, Y_{A}) = 0. Therefore \underline{Hom}_{A}(X, ν) is surjective. To show that \underline{Hom}_{A}(X, ν) is injective, let ψ: X → X_{A} be

a map such that ν^{∘} ψ has a factorization X →^{ϕ′} E →^{ϕ′′} A with injective E. We obtain a factorization ϕ′′ = ν^{∘} χ, since E belongs to X. We have ν^{∘}(ψ −χ^{∘} ϕ′) = 0, and ψ −χ^{∘} ϕ′ needs to factor through μ. Therefore ψ − χ^{∘} ϕ′ factors through some injective object, since \underline{Hom}_{A}(X, Y_{A}) = 0. We conclude that ψ factors through an injective object. Thus the map \underline{Hom}_{A}(X, ν) is bijective.

(3) We consider the exact sequence

0 ⟶ A ⟶^{α} Y^{A} ⟶^{β} X^{A} ⟶ 0

and need to show that \underline{Hom}_{A}(α, Y) is bijective for all Y in Y. But this is clear from the long exact sequence for Tate cohomology, since \underline{Hom}_{A}(−, Y) ≅ Êxt_{A}^{0}(−, Y) and Êxt_{A}^{∗}(X^{A}, −) vanishes.

(4) Clearly, Inj A is contained in X ∩ Y. Now let A be in X ∩ Y. Thus

\underline{Hom}_{A}(A, A) ≅ Êxt_{A}^{0}(A, A) = 0.

Therefore the identity map A → A factors through an injective object. We conclude that A is injective. □

Let us comment on the interplay between the stable category \underline{A} and the stable derived category S(A). We have already seen that the definition of Tate cohomology is possible in both settings. It is more elementary in \underline{A}, but more conceptual using the category of complete injective resolutions K_{tac}(Inj A) which is a subcategory of S(A). The same phenomenon appears when one studies Gorenstein injective approximations. The proof of Theorem 7.12 we have given uses the category of complexes K(Inj A). There is an alternative proof which avoids complexes and uses instead the left adjoint T: \underline{A} → GInj \underline{A}.

Gorenstein rings and schemes play an important role in applications and have a number of interesting homological properties. It is therefore important to formulate a Gorenstein property for a locally noetherian Grothendieck category A. Let us denote by Σ^{∞}A the full subcategory of objects A in A which fit into an exact sequence

⋯ ⟶ E_{2} ⟶ E_{1} ⟶ E_{0} ⟶ A ⟶ 0

with E_{n} injective for all n. We say that A has the injective Gorenstein property if the equivalent conditions in the following proposition are satisfied. This property has been studied by Beligiannis in [4].

Proposition 7.13. Let A be a locally noetherian Grothendieck category and suppose that D(A) is compactly generated. Then the following are equivalent.

(1) Ext_{A}^{1}(A, B) = 0 for all A ∈ Inj A and B ∈ Σ^{∞}A.

(2) Every acyclic complex in Inj A is totally acyclic.

(3) GInj A = Σ^{∞}A.

(4) S: A ⟶^{can} D(A) ⟶^{I_{λ} ∘ Q_{ρ}} S(A) annihilates all injective objects.

(5) S induces an equivalence GInj \underline{A} → S(A).

Proof. The conditions (1) - (3) are pairwise equivalent. This follows from the formula

Ext_{A}^{n}(A, Z^{0}Y) ≅ Hom_{K(A)}(A, Σ^{n}Y) ≅ H^{n} Hom_{A}(A, Y)

where A is any object in A and Y is an acyclic complex in Inj A. The first isomorphism is valid for all n ⩾ 1, and the second for all n ∈ ℤ.

Now observe that I_{λ} annihilates precisely those objects X in K(Inj A) such that Hom_{K(Inj A)}(X, Y) = 0 for every acyclic complex Y in Inj A. On the other hand, A → D(A) and Q_{ρ} are faithful. Thus (1) - (3) are equivalent to (4). Also, (5) implies (4). So, it remains to show that (1) - (4) imply (5).

Suppose (2) and (4) hold. We have already seen in Proposition 7.2 that

S(A) ⟶ GInj \underline{A}, X ↦ Z^{0}X,

is an equivalence, since every acyclic complex is totally acyclic. On the other hand, S annihilates all injective objects and induces therefore a functor \underline{A} → S(A). The composite with Z^{0} : S(A) → GInj \underline{A} is precisely the right adjoint of the inclusion GInj \underline{A} → \underline{A} constructed in Theorem 7.4. Thus (Z^{0} ^{∘} S)A ≅ A for all A in GInj A. □

Corollary 7.14. Let A be a locally noetherian Grothendieck category and suppose that D(A) is compactly generated. If A has the injective Gorenstein property, then the composite

GInj A ⟶^{inc} A ⟶^{can} D(A) ⟶^{I_{λ} ∘ Q_{ρ}} S(A) ⟶^{Z^{0}} GInj \underline{A}

is naturally isomorphic to the canonical projection GInj A → GInj \underline{A}.

We are now in the position that we can describe the stabilization functor S: A → S(A), provided that A has the injective Gorenstein property. We use the left adjoint T: \underline{A} → GInj \underline{A} of the inclusion GInj \underline{A} → \underline{A}. For A in A, choose any acyclic complex X of injective objects such that Z^{0}X ≅ T A. Then SA ≅ X.

Example 7.15. Let Λ be a ring and suppose Λ is Gorenstein, that is, Λ is two-sided noetherian and Λ has finite injective dimension as left and right Λ-module. In this case, the category Mod Λ has the injective Gorenstein property. This follows from the fact that every injective Λ-module has finite projective dimension if Λ is Gorenstein; see Example 5.6. Given a Λ-module A, Tate cohomology Êxt_{Λ}^{∗}(A, −) vanishes iff A has finite injective dimension iff A has finite projective dimension. Note that for Gorenstein rings, the classical Tate cohomology defined via complete projective resolutions coincides with our Tate cohomology, which is defined via complete injective resolutions; see [5].

Example 7.16. Let Λ be a ring and suppose that projective and injective Λ-modules coincide. Then every Λ-module is Gorenstein injective. In particular, S(Mod Λ) is equivalent to the stable category \underline{A} of A = Mod Λ. Given a Λ-module A, there is an exact triangle

pA ⟶ iA ⟶ tA ⟶ Σ(pA)

in K(Inj A) where pA denotes a projective, iA an injective, and tA a Tate resolution of A. This triangle is isomorphic to the canonical triangle

(Q_{λ} ^{∘} Q)Ā ⟶ Ā ⟶ (I^{∘} I_{λ})Ā ⟶ Σ(Q_{λ} ^{∘} Q)Ā

where Ā = Q_{ρ}A.

Example 7.17. Let 𝕏 be a noetherian scheme and suppose that every injective object E in Qcoh 𝕏 admits a finite resolution

0 ⟶ L_{r} ⟶ ⋯ ⟶ L_{2} ⟶ L_{1} ⟶ L_{0} ⟶ E ⟶ 0

with L_{n} locally free for each n. Then the category Qcoh 𝕏 has the injective Gorenstein property.

Historical remarks. Gorenstein injective approximations and Tate cohomology have a long history. Auslander and Bridger [1] introduce the stable module category and assign to each module a G-dimension. Over Gorenstein rings, the modules of G-dimension 0 are precisely the maximal Cohen-Macaulay or Gorenstein projective modules. Auslander and Buchweitz establish maximal Cohen-Macaulay approximations in [2], and there is an alternative unpublished approach by Buchweitz [12] which involves the derived category. Enochs and his collaborators drop finiteness conditions on modules and prove the existence of Gorenstein projective and Gorenstein injective approximations for arbitrary modules, for instance over Gorenstein rings [14]. Further generalizations can be found in work of Beligiannis [4]. Jørgensen [19] constructs Gorenstein projective approximations for artin algebras via Bousfield localization, using the category of complete projective resolutions. Papers of Hovey [17] and Beligiannis and Reiten [5] employ the formalism of model category structures and cotorsion pairs.

The exposition of Buchweitz [12] discusses the close connection between maximal Cohen-Macaulay approximations and Tate cohomology over Gorenstein rings. For more general settings, we refer to the work of Beligiannis and Reiten [5]. A paper of Mislin explains Tate cohomology via satellites [25]. Another exposition of Tate cohomology over noetherian rings can be found in a paper of Avramov and Martsinkovsky [3].

8. Tensor products in modular representation theory

Let G be a finite group and k be a field. The stable module category \underline{Mod} kG of the group algebra kG plays an important role in modular representation theory. In this section, we show that this category is equivalent to the stable derived category S(Mod kG) and study its tensor product.

It is convenient to work in a slightly more general setting. Thus we fix a finite dimensional cocommutative Hopf algebra Λ over a field k and consider the module category A = Mod Λ. Note that projective and injective modules over Λ coincide. The tensor product ⊗_{k} over k induces a tensor product on A which extends to a tensor product on K(A). Similarly, Hom_{k}(−, −) induces products on A and K(A). Note that we have a natural isomorphism

(8.1)  Hom_{K(A)}(X ⊗_{k} Y, Z) ≅ Hom_{K(A)}(X, Hom_{k}(Y, Z))

for all X, Y, Z in K(A). The subcategories K(Inj A) and S(A) inherit tensor products from K(A) because of the following elementary fact.

Lemma 8.1. The subcategories K(Inj A) and S(A) are tensor ideals in K(A). More preciseley,

(1) X ∈ K(Inj A) and Y ∈ K(A) imply X ⊗_{k} Y ∈ K(Inj A);

(2) X ∈ S(A) and Y ∈ K(A) imply X ⊗_{k} Y ∈ S(A).

Now consider k as a Λ-module and view it as a complex concentrated in degree zero; it is the unit of the tensor product in K(A). This complex fits into exact triangles

ak ⟶ k ⟶ ik ⟶ Σ(ak) and pk ⟶ ik ⟶ tk ⟶ Σ(pk)

in K(A), where ik denotes an injective resolution and pk a projective resolution of k in A. We consider the canonical maps k → ik and pk → ik. Thus ak and tk are acyclic complexes. In fact, tk is a Tate resolution of k which is obtained by splicing together pk and ik.

We have seen in previous sections that the inclusions

S(A) ⟶ K(Inj A) and K(Inj A) ⟶ K(A)

have left adjoints. Next we provide explicit descriptions of these adjoints. In particular, we see that they preserve the tensor product ⊗_{k}. Note that Hovey, Palmieri, and Strickland pointed out the relevance of these categories in their work on axiomatic stable homotopy theory [18]; we refer to this work for further details and applications.

Theorem 8.2. Let Λ be a finite dimensional cocommutative Hopf algebra over a field k, and let A = Mod Λ.

(1) The functor

− ⊗_{k} ik: K(A) ⟶ K(Inj A)

is a left adjoint for the inclusion K(Inj A) → K(A).

(2) The functor

− ⊗_{k} tk: K(Inj A) ⟶ S(A)

is a left adjoint for the inclusion S(A) → K(Inj A).

(3) The functor

A ⟶^{can} K(A) ⟶^{−⊗tk} S(A)

induces an equivalence \underline{A} → S(A) with quasi-inverse Z^{0} : S(A) → \underline{A}.

Proof. (1) Fix an object X in K(A). The map X ≅ X ⊗_{k} k → X ⊗_{k} ik induces for all Y in K(Inj A) an isomorphism

Hom_{K(A)}(X ⊗_{k} ik, Y) ⟶ Hom_{K(A)}(X, Y).

This follows from Lemma 2.1, using in addition the formula (8.1).

(2) Fix an object X in K(Inj A). The map X ≅ X ⊗_{k} ik → X ⊗_{k} tk induces for all Y in S(A) an isomorphism

Hom_{K(A)}(X ⊗_{k} tk, Y) ⟶ Hom_{K(A)}(X, Y).

To see this, consider the exact triangle

X ⊗_{k} pk ⟶ X ⊗_{k} ik ⟶ X ⊗_{k} tk ⟶ Σ(X ⊗_{k} pk).

Now use that

Hom_{K(A)}(X ⊗_{k} pk, Y) ≅ Hom_{K(A)}(pk, Hom_{k}(X, Y)) = 0

since Hom_{k}(X, Y) is acyclic.

(3) We apply Proposition 7.13. First observe that every object in A is Gorenstein injective. The functor

A ⟶^{can} K(A) ⟶^{−⊗tk} S(A)

is naturally isomorphic to the stabilization functor S: A → S(A). This follows from the fact that A ⊗_{k} ik is an injective resolution in A for each object A. Thus

SA ≅ (A ⊗_{k} ik) ⊗_{k} tk ≅ A ⊗_{k} tk.

In Proposition 7.13, it is shown that S induces an equivalence \underline{A} → S(A), with quasiinverse Z^{0}. □

Remark 8.3. The unit of the product in K(Inj A) is ik, and its graded endomorphism ring is the cohomology ring H^{∗}(Λ, k). The unit of the product in S(A) is tk, and its graded endomorphism ring is the Tate cohomology ring Ĥ^{∗}(Λ, k).

Appendix A. The DG category of noetherian objects

Let A be a locally noetherian Grothendieck category. We give an alternative description of the homotopy category K(Inj A) as the derived category of some DG category. Here, we follow closely Keller’s exposition in [22].

Let C be a small DG category. We recall the definition of the derived category D_{dg}(C) of C. The category C_{dg}(C) of cochain complexes has by definition as objects all DG C-modules. A map in C_{dg}(C) is a map of DG C-modules which is homogeneous of degree zero and commutes with the differential. The homotopy category K_{dg}(C) is obtained from C_{dg}(C) by identifying homotopy equivalent maps, where f, g: X → Y are homotopy equivalent if there exists a map s: X → Y of graded modules which is homogeneous of degree −1 and satisfies

(f − g)^{n} = s^{n+1} ^{∘} d + d^{∘} s^{n} for all n ∈ ℤ.

Finally, the derived category of C is obtained from K_{dg}(C) as the localization

D_{dg}(C) = K_{dg}(C)[Q^{−1}]

with respect to the class Q of all maps f which induce an isomorphism H^{∗}f.

Given two cochain complexes X and Y in A, we define the cochain complex Hom_{A}(X, Y). The nth component is

∏_{p∈ℤ}Hom_{A}(X^{p}, Y^{n+p})

and the differential is given by

d(f^{p}) = d^{∘} f^{p} − (−1)^{n}f^{p+1} ^{∘} d.

Now fix a class C of objects in A. We obtain a DG category C̄ by taking as objects for each A in C an injective resolution Ā, and as maps

HomC̄(Ā, B̄) = Hom_{A}(Ā, B̄).

Proposition A.1. Let A be a locally noetherian Grothendieck category, and let C be a class of noetherian objects which generate D^{b}(noeth A), that is, there is no proper thick subcategory containing C. Then the functor

K(Inj A) ⟶ D_{dg}(C̄), X ↦ Hom_{A}(−, X)|C̄,

is an equivalence of triangulated categories.

Proof. The functor is exact. To see that it preserves coproducts, fix an object A in C and a family of objects X_{i} in K(Inj A). Then we have for every n ∈ ℤ

H^{n} ∐_{i} Hom_{A}(Ā, X_{i}) ≅ ∐_{i} H^{n} Hom_{A}(Ā, X_{i}) ≅ ∐_{i} Hom_{K(Inj A)}(Σ^{−n}Ā, X_{i})
≅ Hom_{K(Inj A)}(Σ^{−n}Ā, ∐_{i} X_{i}) ≅ H^{n} Hom_{A}(Ā, ∐_{i} X_{i})

since Ā is compact in K(Inj A) by Lemma 2.1. Thus the canonical map

∐_{i} Hom_{A}(−, X_{i})|C̄ ⟶ Hom_{A}(−, ∐_{i} X_{i})|C̄

is an isomorphism. Furthermore, the functor induces for objects A and B in C bijections

Hom_{K(Inj A)}(Ā, Σ^{n}B̄) ≅ H^{n} Hom_{A}(Ā, B̄) ≅ H^{n} HomC̄(Ā, B̄) ≅ Hom_{D_{dg}(C̄)}(Ā^{∧}, Σ^{n}B̄^{∧}), where Ā^{∧} denotes the free module HomC̄(−, Ā). Using infinite dévissage, we conclude that the functor is fully faithful since C generates K(Inj A). The functor is, up to isomorphism, surjective on objects since the image contains the free C̄-modules which generate D_{dg}(C̄). □

Corollary A.2. Viewing noeth A as DG category, we have an equivalence

K(Inj A) ⟶^{∼} D_{dg}(noeth A).

We remark that the proof of Proposition A.1 works for any homotopy category. To be precise, let X be an additive category with arbitrary coproducts and let C be a set of objects in K(X) which are compact (when viewed as objects in the localizing subcategory generated by C). Define C̄ as before by

HomC̄(A, B) = Hom_{X} (A, B)

for A and B in C. Then the functor

K(X) ⟶ D_{dg}(C̄), X ↦ Hom_{X} (−, X)|C̄,

induces an equivalence between the localizing subcategory which is generated by C, and D_{dg}(C̄).

Acknowledgements. It is a pleasure to thank a number of colleagues for their interest in the topic of this paper: Øyvind Solberg helped to prove the crucial Proposition 2.3. Peter Jørgensen pointed out the relevance of the stable derived category in algebraic geometry. Amnon Neeman communicated an alternative proof of Corollary 4.4 in the algebraic geometric context. Ragnar Buchweitz explained the connection with his earlier unpublished work. Finally, Apostolos Beligiannis provided numerous comments on a preliminary version of this manuscript.

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

   E-mail address: hkrause@math.uni-paderborn.de