Brown representability for triangulated categories with a linear action by a graded ring

Abstract

In this paper we give necessary and sufficient conditions for a functor to be representable in a strongly generated triangulated category which has a linear action by a graded ring, and we discuss some applications and examples.

1 Introduction

Every object X in a category induces a contravariant functor into the category of sets that sends an object Y to the set ${\text {Hom}}^{}_{}(Y,X)$. Any functor that is naturally isomorphic to such a functor is called representable. There are a number of results in various settings, called Brown representability, when every ‘reasonable’ functor is representable. The first such result is due to Brown; see [11].

The first Brown representability result for triangulated categories was established by Neeman [19, Theorem 3.1]. The work on hand was motivated by [10, Theorem 1.3] and [21, 4.3].

Theorem

(see Theorem 2.7). Let R be a $\mathbb {Z}$-graded graded-commutative noetherian ring and ${\textsf{T}}$ a graded R-linear triangulated category, that is strongly generated, Ext-finite, and idempotent complete. Then a graded R-linear cohomological functor $\textsf{f} :{\textsf{T}}^{op} \rightarrow {\text {grMod}}(R)$ is graded representable if and only if $\textsf{f}$ only takes values in ${\text {grmod}}(R)$.

In contrast to the previous works, we characterize the graded representable functors; those are the functors naturally isomorphic to

$$\begin{aligned} \coprod _{d \in \mathbb {Z}} {\text {Hom}}^{}_{{\textsf{T}}}(-,\Sigma ^d X) \end{aligned}$$

for some object X. The result is proved in Section 2. Without the assumption that R is noetherian and ${\textsf{T}}$ Ext-finite, we obtain necessary, through not sufficient, conditions for a functor to be graded representable; see Corollary 2.19.

The study of representability is motivated by the fact, that the characterization of representable functors in a triangulated category ${\textsf{T}}$ yields the existence of a right adjoint functor to a functor ${\textsf{S}} \rightarrow {\textsf{T}}$. For a nice discussion on this, see [20, Introduction]. In Section 3, we show the same holds in the graded setting.

Finally we discuss some examples where Theorem 2.7 yields new insight: When G is a finite group and R a commutative noetherian ring, then ${\text {D}}_{b}({\text {mod}}(RG))$ is Ext-finite as a ${{\,\textrm{H}\,}}^*(G,R)$-linear category. In the second example, we consider the action of Hochschild cohomology ${\text {HH}}^{*}(R/Q)$ on ${\text {D}}_{b}({\text {mod}}(R))$, when Q is a regular ring and $R = Q/(\varvec{f})$ a quotient by a regular sequence.

2 Representable functors in the graded setting

Let ${\textsf{T}}$ be a triangulated category with suspension functor $\Sigma$.

2.1. For objects X and Y in ${\textsf{T}}$, we write

(2.1.1)

When ${\textsf{T}} = {\text {D}}_{}(R)$, the derived category of modules over a ring R, and X and Y are R-modules viewed as objects in ${\text {D}}_{}(R)$ via the natural embedding, then this coincides with the classical Ext-groups.

2.2. Let R be a $\mathbb {Z}$-graded graded-commutative ring. This means R decomposes as

$$\begin{aligned} R = \coprod _{d \in \mathbb {Z}} R_d\,, \end{aligned}$$

and the Koszul sign rule holds

$$\begin{aligned} rs = (-1)^{de} sr \quad \text {for } r \in R_d \text { and }s \in R_e\,. \end{aligned}$$

We say $r \in R_d$ is an homogeneous element of degree d.

2.3. A triangulated category ${\textsf{T}}$ is graded R-linear if

• 1. for any objects X and Y in ${\textsf{T}}$, the abelian group ${\text {Ext}}^{*}_{{\textsf{T}}}(X,Y)$ is a graded R-module with the grading given by the coproduct in (2.1.1), and

• 2. composition is R-bilinear.

This data is equivalent to a ring homomorphism $R \rightarrow {{\,\textrm{Z}\,}}({\textsf{T}})$, where

is the graded center of ${\textsf{T}}$. More precisely, a ring homomorphism $\varphi :R \rightarrow {{\,\text {Z}\,}}({\textsf {T}})$ yields an R-action on ${\text {Ext}}^{*}_{{\textsf{T}}}(X,Y)$ via

$$\begin{aligned} \begin{aligned} {} r \cdot - :{\text{ Ext }}^{*}_{{\textsf {T}}}(X,Y)&\rightarrow {\text{ Ext }}^{*}_{{\textsf {T}}}(X,\Sigma ^d Y) = {\text{ Ext }}^{*}_{{\textsf {T}}}(X,Y)[d] \,, \\{} {} f&\mapsto (\Sigma ^d f) \circ \varphi (r)_X = \varphi (r)_Y \circ f \,, \end{aligned} \end{aligned}$$

for any homogeneous element $r \in R$. Conversely, any homogeneous element $r \in R$ yields a natural transformation $\eta$ given by

for any $X \in {\textsf{T}}$. It is straightforward to check that these identifications are well-defined and mutually inverse. The graded center has been studied in a number of works; for example [8, 13].

2.4. We denote by ${\text {grMod}}(R)$ the category of graded R-modules, and by ${\text {grmod}}(R)$ its full subcategory of finitely generated R-modules. The nth shift M[n] of a graded R-module M is given by $(M[n])_d = M_{n+d}$.

The suspension functor of a graded R-linear category ${\textsf{T}}$ in the first component of ${\text {Ext}}^{*}_{{\textsf{T}}}(-,-)$ corresponds to the negative shift in ${\text {grMod}}(R)$:

$$\begin{aligned} {\text {Ext}}^{*}_{{\textsf{T}}}(\Sigma ^n X,Y) \cong {\text {Ext}}^{*}_{{\textsf{T}}}(X,Y)[-n]\,. \end{aligned}$$

2.5. A functor $\textsf{f} :{\textsf{T}}^{op} \rightarrow {\text {grMod}}(R)$ is graded R-linear if

• 1. the induced map ${\text {Ext}}^{*}_{{\textsf{T}}}(X,Y) \rightarrow {\text {Ext}}^{*}_{R}(\textsf{f}(Y),\textsf{f}(X))$ is a map of graded R-modules, and

• 2. the suspension becomes the negative shift under $\textsf{f}$, that is

  $$\begin{aligned} \textsf{f}(\Sigma ^n X) = \textsf{f}(X)[-n]\,. \end{aligned}$$

The functor $\textsf{f}$ is cohomological if $\textsf{f}$ applied to any exact triangle yields a long exact sequence of graded R-modules.

Without explicitly stating, we always assume that a natural transformation between graded R-linear functors respects this structure.

Definition 2.6

A functor ${\textsf{T}}^{op} \rightarrow {\text {grMod}}(R)$ is graded representable if it is naturally isomorphic to

for some object X in ${\textsf{T}}$.

When ${\textsf{T}}$ is graded R-linear, then any graded representable functor is graded R-linear.

A graded R-linear functor $\textsf{f} :{\textsf{T}}^{op} \rightarrow {\text {grMod}}(R)$ is graded representable if and only if $\textsf{f}_d :{\textsf{T}}^{op} \rightarrow {\text {Mod}}(R_0)$ is representable for an(y) arbitrary integer d. The functors $\textsf{f}_d$ are the degree d part of $\textsf{f}$, that is . Since $\textsf{f}$ is graded R-linear, the degree d part $\textsf{f}_d$ for an integer d encodes all the information of $\textsf{f}$, that is

Theorem 2.7

Let R be a $\mathbb {Z}$-graded graded-commutative noetherian ring and ${\textsf{T}}$ a graded R-linear triangulated category, that is strongly generated, Ext-finite, and idempotent complete. Then a graded R-linear cohomological functor $\textsf{f} :{\textsf{T}}^{op} \rightarrow {\text {grMod}}(R)$ is graded representable if and only if $\textsf{f}$ is locally finite.

Before we give a proof, we recall some definitions and properties:

2.8. A $\mathbb {Z}$-graded ring R is noetherian if and only if $R_0$ is noetherian and R is finitely generated as an $R_0$-algebra; see for example [15, Corollaire (2.1.5)] or [12, Theorem 1.5.5]. In particular, such a ring is bounded below.

2.9. A graded R-linear triangulated category ${\textsf{T}}$ is Ext-finite if for all $X, Y \in {\textsf{T}}$, the graded R-module ${\text {Ext}}^{*}_{{\textsf{T}}}(X,Y)$ is finitely generated.

A triangulated category ${\textsf{T}}$ is idempotent complete if for every object X in ${\textsf{T}}$ and every idempotent $e \in {{\,\textrm{End}\,}}_{\textsf{T}}(X)$, that is $e^2 = e$, there exists an object Y and maps

$$\begin{aligned} i :Y \rightarrow X \quad \text {and}\quad p :X \rightarrow Y \end{aligned}$$

such that $p \circ i = {{\,\textrm{id}\,}}_Y$ and $i \circ p = e$.

2.10. A subcategory ${\textsf{S}} \subseteq {\textsf{T}}$ is thick if it is triangulated and closed under retracts. Since the intersection of thick subcategories is thick, there exists a smallest thick subcategory of ${\textsf{T}}$ containing an object G, which we denote by ${{\,\textrm{thick}\,}}(G)$. We say G finitely builds an object X in ${\textsf{T}}$ when $X \in {{\,\textrm{thick}\,}}(G)$.

There is an exhaustive filtration of ${{\,\textrm{thick}\,}}_R(G)$: Let ${{\,\textrm{thick}\,}}^1(G)$ be the smallest full subcategory containing G that is closed under finite coproducts, retracts, and suspension. Then

These are full subcategories and form an exhaustive filtration of ${{\,\textrm{thick}\,}}(G)$; cf. [3, 10]. In particular, if X lies in ${{\,\textrm{thick}\,}}(G)$, then there exists an integer n such that $X \in {{\,\textrm{thick}\,}}^n(G)$.

A triangulated category ${\textsf{T}}$ is strongly generated if there exists an object G in ${\textsf{T}}$ and a non-negative integer n such that ${\textsf{T}} = {{\,\textrm{thick}\,}}^n(G)$. The object X is a strong generator of ${\textsf{T}}$; cf.  [21].

In the remainder of this section, we give a proof of Theorem 2.7. We fix a $\mathbb {Z}$-graded graded-commutative ring R, a graded R-linear triangulated category ${\textsf{T}}$, and a graded R-linear cohomological functor $\textsf{f} :{\textsf{T}}^{op} \rightarrow {\text {grMod}}(R)$.

Lemma 2.11

(Graded version of Yoneda’s lemma). For any $X \in {\textsf{T}}$, the map

$$\begin{aligned} {{\,\textrm{Nat}\,}}(\textsf{g}_X,\textsf{f}) \rightarrow \textsf{f}_0(X) \quad \text {given by}\quad \eta \mapsto \eta (X)({{\,\textrm{id}\,}}_X) \end{aligned}$$

is an isomorphism of abelian groups.

Proof

For $u \in \textsf{f}_0(X)$, we define a natural transformation

where $Y \in {\textsf{T}}$ and $f \in {\text {Ext}}^{*}_{{\textsf{T}}}(Y,X)$. Since u is a degree zero element, the map $\eta _u(Y)$ is homogeneous. It is straightforward to verify that this is the inverse of the map in the claim and both are maps of abelian groups. $\square$

In particular, any morphism $f :X \rightarrow Y$ corresponds to a natural transformation

$$\begin{aligned} f_* :\textsf{g}_X \rightarrow \textsf{g}_Y \end{aligned}$$

given by post-composition.

2.12. Adapting the definitions in [21, Section 4] a graded R-linear functor $\textsf{f} :{\textsf{T}}^{op} \rightarrow {\text {grMod}}(R)$ is

• locally finitely generated if for every X in ${\textsf{T}}$, there exists Y in ${\textsf{T}}$ and a natural transformation $\zeta :\textsf{g}_Y \rightarrow \textsf{f}$ such that $\zeta (X)$ is surjective,

• locally finitely presented if it is locally finitely generated and the kernel of any natural transformation $\textsf{g}_Y \rightarrow \textsf{f}$ is locally finitely generated, and

• locally finite if $\textsf{f}$ only takes values in ${\text {grmod}}(R)$.

When $\textsf{f} :{\textsf{T}}^{op} \rightarrow {\text {grMod}}(R)$ is locally finitely generated or locally finitely presented, then $\textsf{f}_d :{\textsf{T}}^{op} \rightarrow {\text {Mod}}(R_0)$ is locally finitely generated or locally finitely presented in the sense of [21, Section 4], respectively. The same need not hold for locally finite, for examples, see Section 3.

If ${\textsf{T}}$ is Ext-finite, then any graded representable functor is locally finite. Without the assumption that ${\textsf{T}}$ is Ext-finite, we can make the following statement:

Lemma 2.13

Any graded representable functor is locally finitely presented.

Proof

It is clear that a graded representable functor is locally finitely generated. Let $\textsf{g}_X$ be a graded representable functor, and $\textsf{g}_Y \rightarrow \textsf{g}_X$ a natural transformation. By Yoneda’s lemma 2.11, this corresponds to a morphism $Y \rightarrow X$. If we complete this to an exact triangle $Z \rightarrow Y \rightarrow X \rightarrow \Sigma Z$, the sequence

$$\begin{aligned} \textsf{g}_Z \rightarrow \textsf{g}_Y \rightarrow \textsf{g}_X \end{aligned}$$

is exact on ${\textsf{T}}$. In particular, the kernel of $\textsf{g}_Y \rightarrow \textsf{g}_X$ is locally finitely generated. $\square$

Lemma 2.14

If $\textsf{f}$ is locally finite, then $\textsf{f}$ is locally finitely generated.

Proof

Let X be an object in ${\textsf{T}}$. Then the R-module $\textsf{f}(X)$ is finitely generated, and we can choose a finite set of homogeneous generators $x_1, \dots , x_n$ of $\textsf{f}(X)$ in degrees $d_1, \dots , d_n$. Set

For every generator $x_j$, we obtain canonical maps

$$\begin{aligned} \Sigma ^{d_j} X \xrightarrow {i_j} Y \xrightarrow {p_j} \Sigma ^{d_j} X \end{aligned}$$

whose composition is the identity map on $\Sigma ^{d_j} X$. Let $y \in \textsf{f}(Y)$ be the canonical element, for which

$$\begin{aligned} x_j = \textsf{f}(i_j)(y) \quad \text {for } 1 \le j \le n\,. \end{aligned}$$

Because of the suspensions introduced in the definition of Y, the element y is homogeneous of degree 0. By Yoneda’s lemma 2.11, the element y corresponds to the natural transformation $\zeta :\textsf{g}_Y \rightarrow \textsf{f}$ with $\zeta (Y)({{\,\textrm{id}\,}}_Y) = y$. Then $\zeta (y)(i_j) = x_j$, and $\zeta (X)$ is surjective. That is $\textsf{f}$ is locally finitely generated. $\square$

In general a locally finite functor need not be locally finitely presented. This requires further assumptions on R and ${\textsf{T}}$:

Lemma 2.15

If R is noetherian and ${\textsf{T}}$ Ext-finite, then a locally finite functor $\textsf{f}:{\textsf{T}}^{op} \rightarrow {\text {grMod}}(R)$ is locally finitely presented.

Proof

By Lemma 2.14, the functor $\textsf{f}$ is locally finitely generated. Let $\textsf{g}_Y \rightarrow \textsf{f}$ be a natural transformation. We set

Since ${\textsf{T}}$ is Ext-finite, the R-module $\textsf{g}_Y(X)$ is finitely generated. By assumption on $\textsf{f}$, so is $\textsf{f}(X)$. Since R is noetherian, the kernel $\textsf{f}'(X)$ is also finitely generated. Thus $\textsf{f}'$ is a locally finite functor and by Lemma 2.14 it is locally finitely generated. In particular, $\textsf{f}$ is locally finitely presented. $\square$

2.16. Let $(\textsf{f}_i,\eta _i)_{i > 0}$ be a direct system of cohomological functors $\textsf{f}_i :{\textsf{T}} \rightarrow {\textsf{A}}$ where ${\textsf{A}}$ an abelian category and natural transformations $\eta _i :\textsf{f}_i \rightarrow \textsf{f}_{i+1}$. Following [21, 4.2.2], a direct system $(\textsf{f}_i,\eta _i)_{i > 0}$ is almost constant on a subcategory ${\textsf{S}}$ of ${\textsf{T}}$ if for every $X \in {\textsf{S}}$, the sequence

$$\begin{aligned} 0 \rightarrow \ker (\eta _i(X)) \rightarrow \textsf{f}_i(X) \rightarrow {{\,\textrm{colim}\,}}_j \textsf{f}_j(X) \rightarrow 0 \end{aligned}$$

is exact for all positive integers i.

A direct system $(X_i,f_i)_{i>0}$ of objects $X_i$ and morphisms $f_i :X_i \rightarrow X_{i+1}$ in ${\textsf{T}}$ is almost constant on ${\textsf{S}}$ if the induced direct system of functors $(\textsf{g}_{X_i},(f_i)_*)_{i>0}$ is almost constant on ${\textsf{S}}$.

For almost constant direct systems, the following hold; see [21, Proposition 4.13].

Facts 2.17

Let ${\textsf{S}} \subseteq {\textsf{T}}$ be a subcategory closed under suspension, and $(\textsf{f}_i,\eta _i)_{i>0}$ a direct system that is almost constant on ${\textsf{S}}$. Then

• 1. $(\textsf{f}_{ni+r})_{i \geqslant 0}$ is almost constant on ${{\,\textrm{thick}\,}}^n({\textsf{S}})$ for any $r > 0$, and

• 2. $\textsf{f}_{n+1} \rightarrow {{\,\textrm{colim}\,}}\textsf{f}_i$ is split surjective on ${{\,\textrm{thick}\,}}^n({\textsf{S}})$.

If the functors $\textsf{f}_i$ are graded R-linear, the assumption that ${\textsf{S}}$ is closed under suspension is redundant.

Proposition 2.18

Let ${\textsf{T}}$ be a strongly generated, graded R-linear triangulated category and $\textsf{f} :{\textsf{T}}^{op} \rightarrow {\text {grMod}}(R)$ a cohomological graded R-linear functor. Then $\textsf{f}$ is locally finitely presented if and only if $\textsf{f}$ is a retract of a graded representable functor.

Proof

We assume $\textsf{f}$ is locally finitely presented. Let $G \in {\textsf{T}}$ be a strong generator of ${\textsf{T}}$ with ${{\,\textrm{thick}\,}}^d(G) = {\textsf{T}}$. Then there exist $A_1 \in {\textsf{T}}$ and a natural transformation $\zeta _1 :\textsf{g}_{A_1} \rightarrow \textsf{f}$ such that $\zeta _1(G)$ is surjective. Inductively we construct a direct system

$$\begin{aligned} \textsf{g}_{A_1} \rightarrow \textsf{g}_{A_2} \rightarrow \cdots \end{aligned}$$

with compatible natural transformations $\zeta _i :\textsf{g}_{A_i} \rightarrow \textsf{f}$: Assume we have constructed $A_i$ and $\zeta _i$ for $i \le n$. Since $\textsf{f}$ is locally finitely presented, there exists

$$\begin{aligned} \textsf{g}_B \rightarrow \ker (\textsf{g}_{A_n} \rightarrow \textsf{f}) \end{aligned}$$

that is surjective on G. This induces a natural transformation $\textsf{g}_B \rightarrow \textsf{g}_{A_n}$, which by the graded version of Yoneda’s lemma 2.11 corresponds to a morphism $f :B \rightarrow A_n$. We complete this morphism to an exact triangle

$$\begin{aligned} B \rightarrow A_n \rightarrow A_{n+1} \rightarrow \Sigma B \end{aligned}$$

and apply $\textsf{f}_0$, the degree 0 part of $\textsf{f}$. By the graded version of Yoneda’s lemma 2.11, we obtain the exact sequence

$$\begin{aligned} {{\,\textrm{Nat}\,}}(\textsf{g}_B,\textsf{f}) \leftarrow {{\,\textrm{Nat}\,}}(\textsf{g}_{A_n},\textsf{f}) \leftarrow {{\,\textrm{Nat}\,}}(\textsf{g}_{A_{n+1}},\textsf{f})\,. \end{aligned}$$

Thus by construction of B, there exists a natural transformation $\zeta _{n+1}$ whose image is $\zeta _n$.

By this construction, we have

$$\begin{aligned} \ker (\textsf{g}_{A_n}(G) \rightarrow \textsf{f}(G)) = \ker (\textsf{g}_{A_n}(G) \rightarrow \textsf{g}_{A_{n+1}}(G))\,. \end{aligned}$$

Using this and that $\zeta _1(G)$ is surjective, it is straightforward to verify that the direct system is almost constant on G. Then the induced natural transformation ${{\,\textrm{colim}\,}}_i \textsf{g}_{A_i} \rightarrow \textsf{f}$ is a natural isomorphism. By Facts 2.17, the natural transformation

$$\begin{aligned} \textsf{g}_{A_{d+1}} \rightarrow {{\,\textrm{colim}\,}}_i \textsf{g}_{A_i} \xrightarrow {\sim } \textsf{f} \end{aligned}$$

is split surjective, and thus $\textsf{f}$ is a retract of $\textsf{g}_{A_{d+1}}$ on ${\textsf{T}}$.

For the converse direction, we assume $\textsf{f}$ is the retract of $\textsf{g}_X$ for some object X. Then we have a canonical projection and a canonical injection

$$\begin{aligned} \textsf{g}_X \rightarrow \textsf{f} \quad \text {and}\quad \textsf{f} \rightarrow \textsf{g}_X\,, \end{aligned}$$

respectively. The canonical projection is surjective on ${\textsf{T}}$, the canonical injection is injective. In particular, the canonical projection yields that $\textsf{f}$ is locally finitely generated. Given a natural transformation $\textsf{g}_Y \rightarrow \textsf{f}$, its kernel coincides with the kernel of the composition $\textsf{g}_Y \rightarrow \textsf{f} \rightarrow \textsf{g}_X$. By Lemma 2.13, any representable functor is locally finitely presented, and thus is $\textsf{f}$. $\square$

Corollary 2.19

If ${\textsf{T}}$ is additionally idempotent complete, then every locally finitely presented functor is graded representable.

Proof

Let $\textsf{f}$ be a locally finitely presented functor. By Proposition 2.18, it is a retract of a graded representable functor $\textsf{g}_X$. Then the natural transformation

$$\begin{aligned} \textsf{g}_X \rightarrow \textsf{f} \rightarrow \textsf{g}_X \end{aligned}$$

corresponds to an idempotent $e :X \rightarrow X$. Since ${\textsf{T}}$ is idempotent complete, there exists a retract of Y of X such that e decomposes as the natural inclusion and projection morphism. Then $\textsf{f} \rightarrow \textsf{g}_X \rightarrow \textsf{g}_Y$ is a natural isomorphism, and $\textsf{f}$ is graded representable. $\square$

Proof of Theorem 2.7

Since ${\textsf{T}}$ is Ext-finite, any graded representable functor is locally finite. For the converse, we assume $\textsf{f}$ is locally finite. Since R is noetherian and ${\textsf{T}}$ Ext-finite, we can apply Lemma 2.15 to obtain that $\textsf{f}$ is locally finitely presented. Then $\textsf{f}$ is graded representable by Corollary 2.19. $\square$

3 Applications

Adjoint functors. As explained in [20, Introduction], there is a connection between representable functors and adjoint functors. In our context, we obtain the following:

Let R be a $\mathbb {Z}$-graded graded-commutative ring. A functor $\textsf{f} :{\textsf{S}} \rightarrow {\textsf{T}}$ between R-linear graded triangulated categories is graded R-linear if it is exact and the induced map

$$\begin{aligned} {\text {Ext}}^{*}_{{\textsf{S}}}(X,Y) \rightarrow {\text {Ext}}^{*}_{{\textsf{T}}}(\textsf{f}(X),\textsf{f}(Y)) \end{aligned}$$

is a map of graded R-modules.

Lemma 3.1

Let R be a $\mathbb {Z}$-graded graded-commutative ring, and ${\textsf{S}}$, ${\textsf{T}}$ graded R-linear triangulated categories. Suppose ${\textsf{T}}$ is Ext-finite and every cohomological graded R-linear functor ${\textsf{S}}^{op} \rightarrow {\text {grMod}}(R)$, that is locally finite, is graded representable. Then every graded R-linear functor $\textsf{f} :{\textsf{S}} \rightarrow {\textsf{T}}$ has a right adjoint.

Proof

We adapt the proof of [20, Theorem 8.4.4]. Given $Y \in {\textsf{T}}$, we define a functor $\textsf{h} :{\textsf{S}} \rightarrow {\text {grMod}}(R)$ by

This is a graded R-linear functor. Since ${\textsf{T}}$ is Ext-finite, this functor is locally finite. So by assumption, ${\textsf{h}}$ is graded representable, that is there exists an object $\textsf{f}'(Y) \in {\textsf{S}}$ such that

$$\begin{aligned} {\text {Ext}}^{*}_{{\textsf{T}}}(\textsf{f}(-),Y) \cong {\text {Ext}}^{*}_{{\textsf{S}}}(-,\textsf{f}'(Y))\,. \end{aligned}$$

It remains to verify that $\textsf{f}'$ is a functor and this isomorphism is natural in both components. Let $f :Y \rightarrow Z$ be a morphism in ${\textsf{T}}$. Then the induced map

$$\begin{aligned} {\text {Ext}}^{*}_{{\textsf{S}}}(-,\textsf{f}'(Y)) \rightarrow {\text {Ext}}^{*}_{{\textsf{S}}}(-,\textsf{f}'(Z)) \end{aligned}$$

corresponds to a morphism $\textsf{f}'(Y) \rightarrow \textsf{f}'(Z)$ by Yoneda’s lemma 2.11. Thus $\textsf{f}'$ is a functor. The above isomorphism is natural by construction. So $\textsf{f}'$ is a right adjoint of $\textsf{f}$.

Corollary 3.2

Let R be a $\mathbb {Z}$-graded graded-commutative noetherian ring and ${\textsf{S}}$, ${\textsf{T}}$ Ext-finite graded R-linear triangulated categories. Suppose ${\textsf{S}}$ is strongly generated and idempotent complete. Then every graded R-linear functor $\textsf{f} :{\textsf{S}} \rightarrow {\textsf{T}}$ has a right adjoint. $\square$

Derived category. Let R be a commutative noetherian ring and A an R-algebra that is finitely generated as an R-module. Then A is noetherian; see for example [18, Theorem 3.7]. The bounded derived category of finitely generated modules over A, denoted by ${\text {D}}_{b}({\text {mod}}(A))$, has a canonical structure as an R-linear category, and the R-module ${\text {Hom}}^{}_{{\text {D}}_{b}({\text {mod}}(A))}(X,Y) = {\text {Ext}}^{0}_{R}(X,Y)$ is finitely generated for any X, Y. In general, the category ${\text {D}}_{b}({\text {mod}}(A))$ need not be Ext-finite as an R-linear category. By [6, Corollary 2.10], the category ${\text {D}}_{b}({\text {mod}}(A))$ is idempotent complete.

3.3. In general the question whether ${\text {D}}_{b}({\text {mod}}(A))$ is strongly generated is rather difficult. When A is artinian, then ${\text {D}}_{b}({\text {mod}}(A))$ is strongly generated by [21, Proposition 7.37]. When $A = R$ is a commutative notherian ring, then ${\text {D}}_{b}({\text {mod}}(R))$ is strongly generated when R is either essentially of finite type over a field or over an equicharacteristic excellent local ring; see [1, Main Theorem] and [16, Corollary 7.2].

In the following, we discuss two examples in which ${\text {D}}_{b}({\text {mod}}(A))$ is Ext-finite for some cohomology ring connected to A.

Finite croup over a commutative ring. We consider $A = RG$, the group algebra of a finite group G.

3.4. The group cohomology of the group algebra RG with coefficients in an RG-complex M is

When $M = R$, this is a $\mathbb {Z}$-graded graded-commutative ring, and every ${{\,\textrm{H}\,}}^*(G,M)$ is a graded ${{\,\textrm{H}\,}}^*(G,R)$-module. In particular, for RG-complexes X, Y, the identification

$$\begin{aligned} {\text {Ext}}^{*}_{{\text {D}}_{b}({\text {mod}}(RG))}(X,Y) = {\text {Ext}}^{*}_{RG}(X,Y) \cong {{\,\textrm{H}\,}}^*(G,{\text {Hom}}^{}_{R}(X,Y)) \end{aligned}$$

holds and the cohomology ring ${{\,\textrm{H}\,}}^*(G,R)$ acts on any Ext-module; see for example [9, Proposition 3.1.8]. So the bounded derived category of finitely generated RG-modules ${\text {D}}_{b}({\text {mod}}(RG))$ is graded ${{\,\textrm{H}\,}}^*(G,R)$-linear.

3.5. By [14, 22], the group cohomology ring ${{\,\textrm{H}\,}}^*(G,R)$ is noetherian, and ${{\,\textrm{H}\,}}^*(G,M)$ is finitely generated over ${{\,\textrm{H}\,}}^*(G,R)$ for every finitely generated RG-module M. In particular, the derived category ${\text {D}}_{b}({\text {mod}}(RG))$ is Ext-finite as a graded ${{\,\textrm{H}\,}}^*(G,R)$-linear triangulated category.

Corollary 3.6

Let R be a commutative notherian ring and G a finite group. If ${\text {D}}_{b}({\text {mod}}(RG))$ is strongly generated, then a graded ${{\,\textrm{H}\,}}^*(G,R)$-linear functor

$$\begin{aligned} \textsf{f} :{\text {D}}_{b}({\text {mod}}(RG)) \rightarrow {\text {grMod}}({{\,\textrm{H}\,}}^*(G,R)) \end{aligned}$$

is graded representable if and only if $\textsf{f}$ is locally finite. $\square$

Regular ring modulo a regular sequence. We consider $R = A$ a commutative noetherian ring.

3.7. The category ${\text {D}}_{b}({\text {mod}}(R))$ is Ext-finite over R if and only if the Ext-modules ${\text {Ext}}^{*}_{R}(X,Y)$ are bounded for all X and Y in ${\text {D}}_{b}({\text {mod}}(R))$. That is precisely when R is regular: When R is regular, the Ext-modules are bounded by definition. For the converse, for every X in ${\text {D}}_{b}({\text {mod}}(R))$, the Ext-module ${\text {Ext}}^{*}_{R}(X,R/\mathfrak {p})$ is bounded and $X_\mathfrak {p}$ has finite projective dimension for any prime ideal $\mathfrak {p}$ of R. Then X has finite projective dimension; see [7, Lemma 4.5] for modules, and [5, Theorem 4.1] and [17, Theorem 3.6] for complexes.

When R is regular, the bounded derived category ${\text {D}}_{b}({\text {mod}}(R))$ is strongly generated if and only if R is a strong generator. The later holds precisely when R has finite global dimension, that is R has finite Krull dimension. Then Rouquier’s representability theorem [21, Corollary 4.18] applies.

3.8. Suppose $R = Q/(\varvec{f})$ is the quotient of a regular ring Q by a regular sequence $\varvec{f} = f_1, \ldots , f_c$. Then there exist cohomological operators $\varvec{\chi } = \chi _1, \ldots , \chi _c$ in degree 2 such that for X, Y in ${\text {D}}_{b}({\text {mod}}(R))$, the graded modules ${\text {Ext}}^{*}_{R}(X,Y)$ are finitely generated over the noetherian graded ring $R[\varvec{\chi }]$; see [4, Theorem (4.2)]. In particular, the category ${\text {D}}_{b}({\text {mod}}(R))$ is $R[\varvec{\chi }]$-linear and Ext-finite.

The ring of cohomological operators coincides with the Hochschild cohomology

see [2, Section 3].

Corollary 3.9

Let $R = Q/(\varvec{f})$ be the quotient of a regular ring Q by a regular sequence $\varvec{f} = f_1, \ldots , f_c$ with cohomological operators $\varvec{\chi }$. If ${\text {D}}_{b}({\text {mod}}(R))$ is strongly generated, then any graded $R[\varvec{\chi }]$-linear functor $\textsf{f} :{\text {D}}_{b}({\text {mod}}(R)) \rightarrow {\text {grMod}}(R[\varvec{\chi }])$ is graded representable if and only if $\textsf{f}$ is locally finite. $\square$

3.10. For Corollaries 3.6 and 3.9, it is crucial that the ring action on the derived category is graded since the Ext-modules need not be not bounded. In particular, Corollaries 3.6 and 3.9 are not consequences of [21, 4.3], but require Theorem 2.7.