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 gradedcommutative noetherian ring and \({\textsf{T}}\) a graded Rlinear triangulated category, that is strongly generated, Extfinite, and idempotent complete. Then a graded Rlinear 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
for some object X. The result is proved in Section 2. Without the assumption that R is noetherian and \({\textsf{T}}\) Extfinite, 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 Extfinite 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
When \({\textsf{T}} = {\text {D}}_{}(R)\), the derived category of modules over a ring R, and X and Y are Rmodules viewed as objects in \({\text {D}}_{}(R)\) via the natural embedding, then this coincides with the classical Extgroups.
2.2. Let R be a \(\mathbb {Z}\)graded gradedcommutative ring. This means R decomposes as
and the Koszul sign rule holds
We say \(r \in R_d\) is an homogeneous element of degree d.
2.3. A triangulated category \({\textsf{T}}\) is graded Rlinear if

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

2.
composition is Rbilinear.
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 Raction on \({\text {Ext}}^{*}_{{\textsf{T}}}(X,Y)\) via
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 welldefined 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 Rmodules, and by \({\text {grmod}}(R)\) its full subcategory of finitely generated Rmodules. The nth shift M[n] of a graded Rmodule M is given by \((M[n])_d = M_{n+d}\).
The suspension functor of a graded Rlinear category \({\textsf{T}}\) in the first component of \({\text {Ext}}^{*}_{{\textsf{T}}}(,)\) corresponds to the negative shift in \({\text {grMod}}(R)\):
2.5. A functor \(\textsf{f} :{\textsf{T}}^{op} \rightarrow {\text {grMod}}(R)\) is graded Rlinear 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 Rmodules, 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 Rmodules.
Without explicitly stating, we always assume that a natural transformation between graded Rlinear 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 Rlinear, then any graded representable functor is graded Rlinear.
A graded Rlinear 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 Rlinear, 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 gradedcommutative noetherian ring and \({\textsf{T}}\) a graded Rlinear triangulated category, that is strongly generated, Extfinite, and idempotent complete. Then a graded Rlinear 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 Rlinear triangulated category \({\textsf{T}}\) is Extfinite if for all \(X, Y \in {\textsf{T}}\), the graded Rmodule \({\text {Ext}}^{*}_{{\textsf{T}}}(X,Y)\) 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
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 nonnegative 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 gradedcommutative ring R, a graded Rlinear triangulated category \({\textsf{T}}\), and a graded Rlinear 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
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
given by postcomposition.
2.12. Adapting the definitions in [21, Section 4] a graded Rlinear 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 Extfinite, then any graded representable functor is locally finite. Without the assumption that \({\textsf{T}}\) is Extfinite, 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
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 Rmodule \(\textsf{f}(X)\) 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
whose composition is the identity map on \(\Sigma ^{d_j} X\). Let \(y \in \textsf{f}(Y)\) be the canonical element, for which
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 \({\textsf{T}}\) Extfinite, 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 Extfinite, the Rmodule \(\textsf{g}_Y(X)\) 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
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 Rlinear, the assumption that \({\textsf{S}}\) is closed under suspension is redundant.
Proposition 2.18
Let \({\textsf{T}}\) be a strongly generated, graded Rlinear triangulated category and \(\textsf{f} :{\textsf{T}}^{op} \rightarrow {\text {grMod}}(R)\) a cohomological graded Rlinear 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
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
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
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
Thus by construction of B, there exists a natural transformation \(\zeta _{n+1}\) whose image is \(\zeta _n\).
By this construction, we have
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
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
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
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 Extfinite, any graded representable functor is locally finite. For the converse, we assume \(\textsf{f}\) is locally finite. Since R is noetherian and \({\textsf{T}}\) Extfinite, 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 gradedcommutative ring. A functor \(\textsf{f} :{\textsf{S}} \rightarrow {\textsf{T}}\) between Rlinear graded triangulated categories is graded Rlinear if it is exact and the induced map
is a map of graded Rmodules.
Lemma 3.1
Let R be a \(\mathbb {Z}\)graded gradedcommutative ring, and \({\textsf{S}}\), \({\textsf{T}}\) graded Rlinear triangulated categories. Suppose \({\textsf{T}}\) is Extfinite and every cohomological graded Rlinear functor \({\textsf{S}}^{op} \rightarrow {\text {grMod}}(R)\), that is locally finite, is graded representable. Then every graded Rlinear 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 Rlinear functor. Since \({\textsf{T}}\) is Extfinite, 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
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
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 gradedcommutative noetherian ring and \({\textsf{S}}\), \({\textsf{T}}\) Extfinite graded Rlinear triangulated categories. Suppose \({\textsf{S}}\) is strongly generated and idempotent complete. Then every graded Rlinear 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 Ralgebra that is finitely generated as an Rmodule. 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 Rlinear category, and the Rmodule \({\text {Hom}}^{}_{{\text {D}}_{b}({\text {mod}}(A))}(X,Y) = {\text {Ext}}^{0}_{R}(X,Y)\) is finitely generated for any X, Y. In general, the category \({\text {D}}_{b}({\text {mod}}(A))\) need not be Extfinite as an Rlinear 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 Extfinite 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 RGcomplex M is
When \(M = R\), this is a \(\mathbb {Z}\)graded gradedcommutative ring, and every \({{\,\textrm{H}\,}}^*(G,M)\) is a graded \({{\,\textrm{H}\,}}^*(G,R)\)module. In particular, for RGcomplexes X, Y, the identification
holds and the cohomology ring \({{\,\textrm{H}\,}}^*(G,R)\) acts on any Extmodule; see for example [9, Proposition 3.1.8]. So the bounded derived category of finitely generated RGmodules \({\text {D}}_{b}({\text {mod}}(RG))\) 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 RGmodule M. In particular, the derived category \({\text {D}}_{b}({\text {mod}}(RG))\) is Extfinite 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
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 Extfinite over R if and only if the Extmodules \({\text {Ext}}^{*}_{R}(X,Y)\) 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 Extmodules are bounded by definition. For the converse, for every X in \({\text {D}}_{b}({\text {mod}}(R))\), the Extmodule \({\text {Ext}}^{*}_{R}(X,R/\mathfrak {p})\) 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 Extfinite.
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 Extmodules 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.
References
Aihara, T., Takahashi, R.: Generators and dimensions of derived categories of modules. Comm. Algebra 43(11), 5003–5029 (2015)
Avramov, L.L., Buchweitz, R.O.: Homological algebra modulo a regular sequence with special attention to codimension two. J. Algebra 230(1), 24–67 (2000)
Avramov, L.L., Buchweitz, R.O., Iyengar, S.B., Miller, C.: Homology of perfect complexes. Adv. Math. 223(5), 1731–1781 (2010)
Avramov, L.L., Gasharov, V.N., Peeva, I.V.: Complete intersection dimension. Inst. Hautes Études Sci. Publ. Math. 86, 67–114 (1997)
Avramov, L.L., Iyengar, S.B., Lipman, J.: Reflexivity and rigidity for complexes, I. Commutative rings. Algebra Number Theory 4(1), 47–86 (2010)
Balmer, P., Schlichting, M.: Idempotent completion of triangulated categories. J. Algebra 236(2), 819–834 (2001)
Bass, H., Murthy, M.P.: Grothendieck groups and Picard groups of abelian group rings. Ann. Math. (2) 86, 16–73 (1967)
Benson, D., Iyengar, S.B., Krause, H.: Local cohomology and support for triangulated categories. Ann. Sci. Éc. Norm. Supér. (4) 41(4), 573–619 (2008)
Benson, D.J.: Representations and Cohomology. I. Basic Representation Theory of Finite Groups and Associative Algebras. Second edition. Cambridge Studies in Advanced Mathematics, 30. Cambridge University Press, Cambridge (1998)
Bondal, A.I., van den Bergh, M.: Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J. 3(1), 1–36 (2003)
Brown, E.H.: Cohomology theories. Ann. Math. (2) 2(75), 467–484 (1962). (Correction: Ann. of Math. (2), 78:201 (1963))
Bruns, W., Herzog, J.: CohenMacaulay Rings, revised ed., Cambridge Stud. Adv. Math. Cambridge University Press, Cambridge (1998)
Buchweitz, R.O., Flenner, H.: Global Hochschild (co)homology of singular spaces. Adv. Math. 217(1), 205–242 (2008)
Evens, L.: The cohomology ring of a finite group. Trans. Amer. Math. Soc. 101, 224–239 (1961)
Grothendieck, A.: Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes. Inst. Hautes Études Sci. Publ. Math. 8, 222pp. (1961)
Iyengar, S.B., Takahashi, R.: Annihilation of cohomology and strong generation of module categories. Int. Math. Res. Not. IMRN 2, 499–535 (2016)
Letz, J.C.: Local to global principles for generation time over commutative Noetherian rings. Homol. Homotopy Appl. 23(2), 165–182 (2021)
Matsumura, H.: Commutative Ring Theory. Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge (1989)
Neeman, A.: The Grothendieck duality theorem via Bousfield’s techniques and Brown representability. J. Amer. Math. Soc. 9(1), 205–236 (1996)
Neeman, A.: Triangulated Categories. Annals of Mathematics Studies, 148. Princeton University Press, Princeton, NJ ( 2001)
Rouquier, R.: Dimensions of triangulated categories. J. KTheory 1(2), 193–256 (2008)
Venkov, B.B.: Cohomology algebras for some classifying spaces (Russian). Dokl. Akad. Nauk SSSR 127, 943–944 (1959)
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Partly supported by the NSF grant DMS1700985, and the Alexander von Humboldt Foundation in the framework of an Alexander von Humboldt Professorship endowed by the German Federal Ministry of Education and Research.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Letz, J.C. Brown representability for triangulated categories with a linear action by a graded ring. Arch. Math. 120, 135–146 (2023). https://doi.org/10.1007/s00013022018007
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013022018007
Keywords
 Brown representability
 Representable functor
 Triangulated category
 Extfinite
 Graded ring
Mathematics Subject Classification
 18G80