## 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 .

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. 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. 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. 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. 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.  .

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. 1.

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

2. 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 XY. 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 XY 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.