Triangulated categories with central action

Let $\mathsf{T}$ be a triangulated category with suspension $\unicode[STIX]{x1D6F4}$ . For objects $X$ and $Y$ in $\mathsf{T}$ set

$$\begin{eqnarray}\operatorname{Hom}_{\mathsf{T}}^{\ast }(X,Y):=\bigoplus _{i\in \mathbb{Z}}\operatorname{Hom}_{\mathsf{T}}(X,\unicode[STIX]{x1D6F4}^{i}Y)\quad \text{and}\quad \operatorname{End}_{\mathsf{T}}^{\ast }(X):=\operatorname{Hom}_{\mathsf{T}}^{\ast }(X,X).\end{eqnarray}$$

Composition makes $\operatorname{End}_{\mathsf{T}}^{\ast }(X)$ a graded ring and $\operatorname{Hom}_{\mathsf{T}}^{\ast }(X,Y)$ a left- $\operatorname{End}_{\mathsf{T}}^{\ast }(Y)$ right- $\operatorname{End}_{\mathsf{T}}^{\ast }(X)$ module.

Let $R$ be a graded commutative noetherian ring. In what follows we will only be concerned with homogeneous elements and ideals in  $R$ . In this spirit, ‘localisation’ will mean homogeneous localisation, and $\operatorname{Spec}R$ will denote the set of homogeneous prime ideals in  $R$ .

We say that a triangulated category $\mathsf{T}$ is $R$ -linear if for each $X$ in $\mathsf{T}$ there is a homomorphism of graded rings $\unicode[STIX]{x1D719}_{X}:R\rightarrow \operatorname{End}_{\mathsf{T}}^{\ast }(X)$ such that the induced left and right actions of $R$ on $\operatorname{Hom}_{\mathsf{T}}^{\ast }(X,Y)$ are compatible in the following sense: for any $r\in R$ and $\unicode[STIX]{x1D6FC}\in \operatorname{Hom}_{\mathsf{T}}^{\ast }(X,Y)$ , one has

$$\begin{eqnarray}\unicode[STIX]{x1D719}_{Y}(r)\unicode[STIX]{x1D6FC}=(-1)^{|r|\,|\unicode[STIX]{x1D6FC}|}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D719}_{X}(r).\end{eqnarray}$$

An exact functor $F:\mathsf{T}\rightarrow \mathsf{U}$ between $R$ -linear triangulated categories is $R$ -linear if the induced map

$$\begin{eqnarray}\operatorname{Hom}_{\mathsf{T}}^{\ast }(X,Y)\longrightarrow \operatorname{Hom}_{\mathsf{U}}^{\ast }(FX,FY)\end{eqnarray}$$

of graded abelian groups is $R$ -linear for all objects $X,Y$ in  $\mathsf{T}$ .

In what follows, we fix a compactly generated $R$ -linear triangulated category $\mathsf{T}$ and write $\mathsf{T}^{\mathsf{c}}$ for its full subcategory of compact objects.

Localisation

Fix an ideal $\mathfrak{a}$ in $R$ . An $R$ -module $M$ is $\mathfrak{a}$ -torsion if $M_{\mathfrak{q}}=0$ for all $\mathfrak{q}$ in $\operatorname{Spec}R$ with $\mathfrak{a}\not \subseteq \mathfrak{q}$ . Analogously, an object $X$ in $\mathsf{T}$ is $\mathfrak{a}$ -torsion if the $R$ -module $\operatorname{Hom}_{\mathsf{T}}^{\ast }(C,X)$ is $\mathfrak{a}$ -torsion for all $C\in \mathsf{T}^{\mathsf{c}}$ . The full subcategory of $\mathfrak{a}$ -torsion objects

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{{\mathcal{V}}(\mathfrak{a})}\mathsf{T}:=\{X\in \mathsf{T}\mid X~\text{is}~\mathfrak{a}\text{-}\text{torsion}\}\end{eqnarray}$$

is localising and the inclusion $\unicode[STIX]{x1D6E4}_{{\mathcal{V}}(\mathfrak{a})}\mathsf{T}\subseteq \mathsf{T}$ admits a right adjoint, denoted $\unicode[STIX]{x1D6E4}_{{\mathcal{V}}(\mathfrak{a})}$ .

Fix a $\mathfrak{p}$ in $\operatorname{Spec}R$ . An $R$ -module $M$ is $\mathfrak{p}$ -local if the localisation map $M\rightarrow M_{\mathfrak{p}}$ is invertible, and an object $X$ in $\mathsf{T}$ is $\mathfrak{p}$ -local if the $R$ -module $\operatorname{Hom}_{\mathsf{T}}^{\ast }(C,X)$ is $\mathfrak{p}$ -local for all $C\in \mathsf{T}^{\mathsf{c}}$ . Consider the full subcategory of $\mathsf{T}$ of $\mathfrak{p}$ -local objects

$$\begin{eqnarray}\mathsf{T}_{\mathfrak{p}}:=\{X\in \mathsf{T}\mid X~\text{is}~\mathfrak{p}\text{-}\text{local}\}\end{eqnarray}$$

and the full subcategory of $\mathfrak{p}$ -local and $\mathfrak{p}$ -torsion objects

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{\mathfrak{p}}\mathsf{T}:=\{X\in \mathsf{T}\mid X~\text{is}~\mathfrak{p}\text{-}\text{local and}~\mathfrak{p}\text{-}\text{torsion}\}.\end{eqnarray}$$

Note that $\unicode[STIX]{x1D6E4}_{\mathfrak{p}}\mathsf{T}\subseteq \mathsf{T}_{\mathfrak{p}}\subseteq \mathsf{T}$ are localising subcategories. The inclusion $\mathsf{T}_{\mathfrak{p}}\rightarrow \mathsf{T}$ admits a left adjoint $X\mapsto X_{\mathfrak{p}}$ while the inclusion $\unicode[STIX]{x1D6E4}_{\mathfrak{p}}\mathsf{T}\rightarrow \mathsf{T}_{\mathfrak{p}}$ admits a right adjoint. We denote by $\unicode[STIX]{x1D6E4}_{\mathfrak{p}}:\mathsf{T}\rightarrow \unicode[STIX]{x1D6E4}_{\mathfrak{p}}\mathsf{T}$ the composition of those adjoints; it is the local cohomology functor with respect to  $\mathfrak{p}$ ; see [Reference Benson, Iyengar and KrauseBIK08, Reference Benson, Iyengar and KrauseBIK11a] for the construction of this functor.

The following observation is clear.

The functor $\unicode[STIX]{x1D6E4}_{{\mathcal{V}}(\mathfrak{a})}$ commutes with exact functors preserving coproducts.

Lemma 2.2. Let $F:\mathsf{T}\rightarrow \mathsf{U}$ be an exact functor between $R$ -linear compactly generated triangulated categories such that $F$ is $R$ -linear and preserves coproducts. Suppose that the action of $R$ on $\mathsf{U}$ factors through a homomorphism $f:R\rightarrow S$ of graded commutative rings. For any ideal $\mathfrak{a}$ of $R$ there is a natural isomorphism

$$\begin{eqnarray}F\circ \unicode[STIX]{x1D6E4}_{{\mathcal{V}}(\mathfrak{a})}\cong \unicode[STIX]{x1D6E4}_{{\mathcal{V}}(\mathfrak{a}S)}\circ F\end{eqnarray}$$

of functors $\mathsf{T}\rightarrow \mathsf{U}$ , where $\mathfrak{a}S$ denotes the ideal of $S$ that is generated by  $f(\mathfrak{a})$ .

Injective cohomology objects

Given an object $C$ in $\mathsf{T}^{\mathsf{c}}$ and an injective $R$ -module  $I$ , Brown representability yields an object $T(C,I)$ in $\mathsf{T}$ such that

(2.1) $$\begin{eqnarray}\operatorname{Hom}_{R}(\operatorname{Hom}_{\mathsf{T}}^{\ast }(C,-),I)\cong \operatorname{Hom}_{\mathsf{T}}(-,T(C,I)).\end{eqnarray}$$

This yields a functor

$$\begin{eqnarray}T:\mathsf{T}^{\mathsf{c}}\times \mathsf{Inj}\,R\longrightarrow \mathsf{T}.\end{eqnarray}$$

For each $\mathfrak{p}$ in $\operatorname{Spec}R$ , we write $I(\mathfrak{p})$ for the injective hull of $R/\mathfrak{p}$ and set

$$\begin{eqnarray}T_{\mathfrak{p}}:=T(-,I(\mathfrak{p})),\end{eqnarray}$$

viewed as a functor $\mathsf{T}^{\mathsf{c}}\rightarrow \mathsf{T}$ .

Tensor triangulated categories

Let $\mathsf{T}=(\mathsf{T},\otimes ,\unicode[STIX]{x1D7D9})$ be a tensor triangulated category such that $R$ acts on $\mathsf{T}$ via a homomorphism of graded rings $R\rightarrow \operatorname{End}_{\mathsf{T}}^{\ast }(\unicode[STIX]{x1D7D9})$ . Brown representability yields functions objects ${\mathcal{H}}om(X,Y)$ satisfying an adjunction isomorphism

$$\begin{eqnarray}\operatorname{Hom}_{\mathsf{T}}(X\otimes Y,Z)\cong \operatorname{Hom}_{\mathsf{T}}(X,{\mathcal{H}}om(Y,Z))\quad \text{for all}~X,Y,Z~\text{in}~\mathsf{T}.\end{eqnarray}$$

Set $X^{\vee }:={\mathcal{H}}om(X,\unicode[STIX]{x1D7D9})$ for each $X$ in $\mathsf{T}$ . It is part of our definition of a tensor triangulated category that the unit, $\unicode[STIX]{x1D7D9}$ , is compact, and that compact objects are rigid: for all $C,X$ in $\mathsf{T}$ with $C$ compact the natural map

$$\begin{eqnarray}C^{\vee }\otimes X\longrightarrow {\mathcal{H}}om(C,X)\end{eqnarray}$$

is an isomorphism; see [Reference Benson, Iyengar and KrauseBIK08, § 8] for details.

The functors $\unicode[STIX]{x1D6E4}_{\mathfrak{p}}$ and $T_{\mathfrak{p}}$ can be computed as follows:

(2.2) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{\mathfrak{p}}\cong \unicode[STIX]{x1D6E4}_{\mathfrak{p}}(\unicode[STIX]{x1D7D9})\otimes -\quad \text{and}\quad T_{\mathfrak{p}}\cong T_{\mathfrak{p}}(\unicode[STIX]{x1D7D9})\otimes -.\end{eqnarray}$$

Indeed, the first isomorphism is from [Reference Benson, Iyengar and KrauseBIK08, Corollary 8.3], while the second one holds because for each $X\in \mathsf{T}$ and compact object $C$ there are isomorphisms

$$\begin{eqnarray}\displaystyle \operatorname{Hom}_{\mathsf{T}}(X,T_{\mathfrak{p}}(\unicode[STIX]{x1D7D9})\otimes C) & \cong & \displaystyle \operatorname{Hom}_{\mathsf{T}}(X,{\mathcal{H}}om(C^{\vee },T_{\mathfrak{p}}(\unicode[STIX]{x1D7D9})))\nonumber\\ \displaystyle & \cong & \displaystyle \operatorname{Hom}_{\mathsf{T}}(X\otimes C^{\vee },T_{\mathfrak{p}}(\unicode[STIX]{x1D7D9}))\nonumber\\ \displaystyle & \cong & \displaystyle \operatorname{Hom}_{R}(\operatorname{Hom}_{\mathsf{T}}^{\ast }(\unicode[STIX]{x1D7D9},X\otimes C^{\vee }),I(\mathfrak{p}))\nonumber\\ \displaystyle & \cong & \displaystyle \operatorname{Hom}_{R}(\operatorname{Hom}_{\mathsf{T}}^{\ast }(\unicode[STIX]{x1D7D9},{\mathcal{H}}om(C,X),I(\mathfrak{p})))\nonumber\\ \displaystyle & \cong & \displaystyle \operatorname{Hom}_{R}(\operatorname{Hom}_{\mathsf{T}}^{\ast }(C,X),I(\mathfrak{p})).\nonumber\end{eqnarray}$$

The first and the fourth isomorphisms above hold because $C$ is rigid; the second and the last one are adjunction isomorphisms; the third one is by the defining isomorphism (2.1).

We turn now to modules over finite group schemes, following the notation and terminology from [Reference Benson, Iyengar, Krause and PevtsovaBIKP18].

The stable module category

Let $G$ be a finite group scheme over a field $k$ of positive characteristic. The coordinate ring and the group algebra of $G$ are denoted $k[G]$ and $kG$ , respectively. These are finite-dimensional Hopf algebras over $k$ that are dual to each other. We write $\mathsf{Mod}\,G$ for the category of $G$ -modules and $\mathsf{mod}\,G$ for its full subcategory consisting of finite-dimensional $G$ -modules. We often identity $\mathsf{Mod}\,G$ with the category of $kG$ -modules, which is justified by [Reference JantzenJan03, I.8.6].

We write $H^{\ast }(G,k)$ for the cohomology algebra, $\operatorname{Ext}_{G}^{\ast }(k,k)$ , of  $G$ . This is a graded commutative $k$ -algebra, because $kG$ is a Hopf algebra, and acts on $\operatorname{Ext}_{G}^{\ast }(M,N)$ , for any $G$ -modules $M,N$ . Moreover, the $k$ -algebra $H^{\ast }(G,k)$ is finitely generated, and, when $M,N$ are finite dimensional, $\operatorname{Ext}_{G}^{\ast }(M,N)$ is finitely generated over it; this is by a theorem of Friedlander and Suslin [Reference Friedlander and SuslinFS97].

The stable module category $\mathsf{StMod}\,G$ is obtained from $\mathsf{Mod}\,G$ by identifying two morphisms between $G$ -modules when they factor through a projective $G$ -module. An isomorphism in $\mathsf{StMod}\,G$ will be called a stable isomorphism, to distinguish it from an isomorphism in $\mathsf{Mod}\,G$ . In the same vein, $G$ -modules $M$ and $N$ are said to be stably isomorphic if they are isomorphic in $\mathsf{StMod}\,G$ ; this is equivalent to the condition that $M$ are $N$ are isomorphic in $\mathsf{Mod}\,G$ , up to projective summands.

The tensor product over $k$ of $G$ -modules passes to $\mathsf{StMod}\,G$ and yields a tensor triangulated category with unit $k$ and suspension $\unicode[STIX]{x1D6FA}^{-1}$ , the inverse of the syzygy functor. The category $\mathsf{StMod}\,G$ is compactly generated and the subcategory of compact objects identifies with $\mathsf{stmod}\,G$ , the stable module category of finite-dimensional $G$ -modules. See Carlson [Reference CarlsonCar96, § 5] and Happel [Reference HappelHap88, ch. I] for details.

We use the notation $\text{}\underline{\operatorname{Hom}}_{G}(M,N)$ for the $\operatorname{Hom}$ -sets in $\mathsf{StMod}\,G$ . The cohomology algebra $H^{\ast }(G,k)$ acts on $\mathsf{StMod}\,G$ via a homomorphism of $k$ -algebras

$$\begin{eqnarray}-\otimes _{k}M:H^{\ast }(G,k)=\operatorname{Ext}_{G}^{\ast }(k,k)\longrightarrow \text{}\underline{\operatorname{Hom}}_{G}^{\ast }(M,M).\end{eqnarray}$$

Thus, the preceding discussion on localisation functors on triangulated categories applies to the $H^{\ast }(G,k)$ -linear category $\mathsf{StMod}\,G$ .

Koszul objects

Each $b$ in $H^{d}(G,k)$ corresponds to a morphism $k\rightarrow \unicode[STIX]{x1D6FA}^{-d}k$ in $\mathsf{StMod}\,G$ ; let $k/\!\!/b$ denote its mapping cone. This gives a morphism $k\rightarrow \unicode[STIX]{x1D6FA}^{d}(k/\!\!/b)$ . For a sequence of elements $\boldsymbol{b}:=b_{1},\ldots ,b_{n}$ in $H^{\ast }(G,k)$ and a $G$ -module  $M$ , we set

$$\begin{eqnarray}k/\!\!/\boldsymbol{b}:=(k/\!\!/b_{1})\otimes _{k}\cdots \otimes _{k}(k/\!\!/b_{n})\quad \text{and}\quad M/\!\!/\boldsymbol{b}:=M\otimes _{k}k/\!\!/\boldsymbol{b}.\end{eqnarray}$$

It is easy to check that for a $G$ -module $N$ and $s=\sum _{i}|b_{i}|$ , there is an isomorphism

(2.3) $$\begin{eqnarray}\text{}\underline{\operatorname{Hom}}_{G}(M,N/\!\!/\boldsymbol{b})\cong \text{}\underline{\operatorname{Hom}}_{G}(\unicode[STIX]{x1D6FA}^{n+s}M/\!\!/\boldsymbol{b},N).\end{eqnarray}$$

Let $\mathfrak{b}=(\boldsymbol{b})$ be the ideal of $H^{\ast }(G,k)$ generated by  $\boldsymbol{b}$ . By abuse of notation we set $M/\!\!/\mathfrak{b}:=M/\!\!/\boldsymbol{b}$ . If $\boldsymbol{b}^{\prime }$ is a finite set of elements in $H^{\ast }(G,k)$ such that $\sqrt{(\boldsymbol{b}^{\prime })}=\sqrt{(\boldsymbol{b})}$ , then, by [Reference Benson, Iyengar and KrauseBIK15, Proposition 3.10], for any $M$ in $\mathsf{StMod}\,G$ there is an equality

(2.4) $$\begin{eqnarray}\mathsf{Thick}(M/\!\!/\boldsymbol{b})=\mathsf{Thick}(M/\!\!/\boldsymbol{b}^{\prime }).\end{eqnarray}$$

Fix $\mathfrak{p}$ in $\operatorname{Spec}H^{\ast }(G,k)$ . We will repeatedly use the fact that $\unicode[STIX]{x1D6E4}_{\mathfrak{p}}(\mathsf{StMod}\,G)^{\mathsf{c}}$ is generated as a triangulated category by the family of objects $(M/\!\!/\mathfrak{p})_{\mathfrak{p}}$ with $M$ in $\mathsf{stmod}\,G$ ; see [Reference Benson, Iyengar and KrauseBIK11a, Proposition 3.9]. In fact, if $S$ denotes the direct sum of a representative set of simple $G$ -modules, then there is an equality

(2.5) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{\mathfrak{p}}(\mathsf{StMod}\,G)^{\mathsf{c}}=\mathsf{Thick}((S/\!\!/\mathfrak{p})_{\mathfrak{p}}).\end{eqnarray}$$

It turns out that one has $\unicode[STIX]{x1D6E4}_{\mathfrak{m}}(\mathsf{StMod}\,G)=\{0\}$ where $\mathfrak{m}$ denotes $H^{{\geqslant}1}(G,k)$ , the ideal of elements of positive degree; see Lemma 2.5 below. For this reason, it is customary to focus on $\operatorname{Proj}H^{\ast }(G,k)$ , the set of homogeneous prime ideals not containing $\mathfrak{m}$ , when dealing with $\mathsf{StMod}\,G$ .

Tate cohomology

By construction, the action of $H^{\ast }(G,k)$ on $\mathsf{StMod}\,G$ factors through $\text{}\underline{\operatorname{Hom}}_{G}^{\ast }(k,k)$ , the graded ring of endomorphisms of the identity. The latter ring is not noetherian in general, which is one reason to work with $H^{\ast }(G,k)$ . In any case, there is little difference, vis-à-vis their action on $\mathsf{StMod}\,G$ , as the next remarks should make clear.

Remark 2.3. Let $M$ and $N$ be $G$ -modules. The map $\operatorname{Hom}_{G}(M,N)\rightarrow \text{}\underline{\operatorname{Hom}}_{G}(M,N)$ induces a map $\operatorname{Ext}_{G}^{\ast }(M,N)\rightarrow \text{}\underline{\operatorname{Hom}}_{G}^{\ast }(M,N)$ of $H^{\ast }(G,k)$ -modules. This map is surjective in degree zero, with kernel $\operatorname{PHom}_{G}(M,N)$ , the maps from $M$ to $N$ that factor through a projective $G$ -module. It is bijective in positive degrees and hence one gets an exact sequence of graded $H^{\ast }(G,k)$ -modules

(2.6) $$\begin{eqnarray}0\longrightarrow \operatorname{PHom}_{G}(M,N)\longrightarrow \operatorname{Ext}_{G}^{\ast }(M,N)\longrightarrow \text{}\underline{\operatorname{Hom}}_{G}^{\ast }(M,N)\longrightarrow X\longrightarrow 0\end{eqnarray}$$

with $X^{i}=0$ for $i\geqslant 0$ . For degree reasons, the $H^{\ast }(G,k)$ -modules $\operatorname{PHom}_{G}(M,N)$ and $X$ are $\mathfrak{m}$ -torsion. Consequently, for $\mathfrak{p}$ in $\operatorname{Proj}H^{\ast }(G,k)$ the induced localised map is an isomorphism:

(2.7) $$\begin{eqnarray}\operatorname{Ext}_{G}^{\ast }(M,N)_{\mathfrak{ p}}\xrightarrow[{}]{~\cong ~}\text{}\underline{\operatorname{Hom}}_{G}^{\ast }(M,N)_{\mathfrak{ p}}.\end{eqnarray}$$

More generally, for each $r$ in $\mathfrak{m}$ localisation induces an isomorphism

$$\begin{eqnarray}\operatorname{Ext}_{G}^{\ast }(M,N)_{r}\xrightarrow[{}]{~\cong ~}\text{}\underline{\operatorname{Hom}}_{G}^{\ast }(M,N)_{r}\end{eqnarray}$$

of $H^{\ast }(G,k)_{r}$ -modules. This means that $\operatorname{Proj}H^{\ast }(G,k)$ has a finite cover by affine open sets on which ordinary cohomology and stable cohomology agree.

Given the finite generation result due to Friedlander and Suslin mentioned earlier, the next remark can be deduced from the exact sequence (2.6).

Remark 2.4. When $M,N$ are finite-dimensional $G$ -modules, $\text{}\underline{\operatorname{Hom}}_{G}^{{\geqslant}s}(M,N)$ is a finitely generated $H^{\ast }(G,k)$ -module for any $s\in \mathbb{Z}$ . Moreover the $H^{\ast }(G,k)_{\mathfrak{p}}$ -module

$$\begin{eqnarray}\text{}\underline{\operatorname{Hom}}_{G}^{\ast }(M_{\mathfrak{ p}},N_{\mathfrak{p}})\cong \text{}\underline{\operatorname{Hom}}_{G}^{\ast }(M,N)_{\mathfrak{ p}}\end{eqnarray}$$

is finitely generated for each $\mathfrak{p}$ in $\operatorname{Proj}H^{\ast }(G,k)$ .

To gain a better understanding of the discussion above, it helps to consider the homotopy category of $\mathsf{Inj}\,G$ , the injective $G$ -modules.

The homotopy category of injectives

Let $\mathsf{K}(\mathsf{Inj}\,G)$ and $\mathsf{D}(\mathsf{Mod}\,G)$ denote the homotopy category of $\mathsf{Inj}\,G$ and the derived category of $\mathsf{Mod}\,G$ , respectively. These are also $H^{\ast }(G,k)$ -linear compactly generated tensor triangulated categories, with the tensor product over  $k$ . The unit of the tensor product on $\mathsf{K}(\mathsf{Inj}\,G)$ is an injective resolution of the trivial $G$ -module  $k$ , while that of $\mathsf{D}(\mathsf{Mod}\,G)$ is  $k$ . The canonical quotient functor $\mathsf{K}(\mathsf{Inj}\,G)\rightarrow \mathsf{D}(\mathsf{Mod}\,G)$ induces an equivalence of triangulated category $\mathsf{K}(\mathsf{Inj}\,G)^{\mathsf{c}}\xrightarrow[{}]{{\sim}}\mathsf{D}^{b}(\mathsf{mod}\,G)$ , where the target is the bounded derived category of $\mathsf{mod}\,G$ ; see [Reference KrauseKra05, Proposition 2.3].

Taking Tate resolutions identifies $\mathsf{StMod}\,G$ with $\mathsf{K}_{\text{ac}}(\mathsf{Inj}\,G)$ , the full subcategory of acyclic complexes in $\mathsf{K}(\mathsf{Inj}\,G)$ . In detail, let $\mathsf{p}k$ and $\mathsf{i}k$ be a projective and an injective resolution of the trivial $G$ -module  $k$ , respectively, and let $\mathsf{t}k$ be the mapping cone of the composed morphism $\mathsf{p}k\rightarrow k\rightarrow \mathsf{i}k$ ; this is a Tate resolution of  $k$ . Since projective and injective $G$ -modules coincide, one gets the exact triangle

(2.8) $$\begin{eqnarray}\mathsf{p}k\longrightarrow \mathsf{i}k\longrightarrow \mathsf{t}k\longrightarrow\end{eqnarray}$$

in $\mathsf{K}(\mathsf{Inj}\,G)$ . For a $G$ -module $M$ , the complex $M\otimes _{k}\mathsf{t}k$ is a Tate resolution of $M$ and the assignment $M\mapsto M\otimes _{k}\mathsf{t}k$ induces an equivalence of categories

$$\begin{eqnarray}\mathsf{StMod}\,G\xrightarrow[{}]{{\sim}}\mathsf{K}_{\text{ac}}(\mathsf{Inj}\,G),\end{eqnarray}$$

with quasi-inverse $X\mapsto Z^{0}(X)$ , the submodule of cycles in degree 0. Assigning $X$ in $\mathsf{K}(\mathsf{Inj}\,G)$ to $X\otimes _{k}\mathsf{t}k$ is a left adjoint of the inclusion $\mathsf{K}_{\text{ac}}(\mathsf{Inj}\,G)\rightarrow \mathsf{K}(\mathsf{Inj}\,G)$ . These results are contained in [Reference KrauseKra05, Theorem 8.2]. Consider the composed functor

$$\begin{eqnarray}\unicode[STIX]{x1D70B}:\mathsf{K}(\mathsf{Inj}\,G)\xrightarrow[{}]{-\otimes _{k}\mathsf{t}k}\mathsf{K}_{\text{ac}}(\mathsf{Inj}\,G)\xrightarrow[{}]{~\sim ~}\mathsf{StMod}\,G.\end{eqnarray}$$

A straightforward verification yields that these functors are $H^{\ast }(G,k)$ -linear. The result below is the categorical underpinning of Remark 2.3 and Lemma 2.5.

Lemma 2.6. There is a natural isomorphism $\unicode[STIX]{x1D6E4}_{\mathfrak{m}}X\cong X\otimes _{k}\mathsf{p}k$ for $X\in \mathsf{K}(\mathsf{Inj}\,G)$ . For each $\mathfrak{p}$ in $\operatorname{Proj}H^{\ast }(G,k)$ , the functor $\unicode[STIX]{x1D70B}$ induces triangle equivalences

$$\begin{eqnarray}\mathsf{K}(\mathsf{Inj}\,G)_{\mathfrak{p}}\xrightarrow[{}]{{\sim}}(\mathsf{StMod}\,G)_{\mathfrak{p}}\quad \text{and}\quad \unicode[STIX]{x1D6E4}_{\mathfrak{p}}(\mathsf{K}(\mathsf{Inj}\,G))\xrightarrow[{}]{{\sim}}\unicode[STIX]{x1D6E4}_{\mathfrak{p}}(\mathsf{StMod}\,G).\end{eqnarray}$$

Proof. We identify $\mathsf{StMod}\,G$ with $\mathsf{K}_{\text{ac}}(\mathsf{Inj}\,G)$ . This entails $\unicode[STIX]{x1D6E4}_{\mathfrak{m}}(\mathsf{K}_{\text{ac}}(\mathsf{Inj}\,G))=\{0\}$ , by Lemma 2.5. It is easy to check that $kG$ is $\mathfrak{m}$ -torsion, and hence so is $\mathsf{p}k$ , for it is in the localising subcategory generated by $kG$ , and the class of $\mathfrak{m}$ -torsion objects in $\mathsf{K}(\mathsf{Inj}\,G)$ is a tensor ideal localising subcategory; see, for instance, [Reference Benson, Iyengar and KrauseBIK08, § 8]. Thus, applying $\unicode[STIX]{x1D6E4}_{\mathfrak{m}}(-)$ to the exact triangle (2.8) yields $\mathsf{p}k\cong \unicode[STIX]{x1D6E4}_{\mathfrak{m}}(\mathsf{i}k)$ . It then follows from (2.2) that $X\otimes _{k}\mathsf{p}k\cong \unicode[STIX]{x1D6E4}_{\mathfrak{m}}X$ for any $X$ in $\mathsf{K}(\mathsf{Inj}\,G)$ .

From the construction of $\unicode[STIX]{x1D70B}$ and (2.8), the kernel of $\unicode[STIX]{x1D70B}$ is the subcategory

$$\begin{eqnarray}\{X\in \mathsf{K}(\mathsf{Inj}\,G)\mid X\otimes _{k}\mathsf{p}k\cong X\}.\end{eqnarray}$$

These are precisely the $\mathfrak{m}$ -torsion objects in $\mathsf{K}(\mathsf{Inj}\,G)$ , by the already established part of the result. Said otherwise, $X\in \mathsf{K}(\mathsf{Inj}\,G)$ is acyclic if and only if $\unicode[STIX]{x1D6E4}_{\mathfrak{m}}X=0$ . It follows that $\mathsf{K}_{\text{ac}}(\mathsf{Inj}\,G)$ contains the subcategory $\mathsf{K}(\mathsf{Inj}\,G)_{\mathfrak{p}}$ of $\mathfrak{p}$ -local objects, for each $\mathfrak{p}$ in $\operatorname{Proj}H^{\ast }(G,k)$ . On the other hand, the inclusion $\mathsf{K}_{\text{ac}}(\mathsf{Inj}\,G)\subseteq \mathsf{K}(\mathsf{Inj}\,G)$ preserves coproducts, so its left adjoint $\unicode[STIX]{x1D70B}$ preserves compactness of objects and all compacts of $\mathsf{K}_{\text{ac}}(\mathsf{Inj}\,G)$ are in the image of  $\unicode[STIX]{x1D70B}$ . Given this a simple calculation shows that $\mathsf{K}(\mathsf{Inj}\,G)_{\mathfrak{p}}$ contains ${\mathsf{K}_{\text{ac}}(\mathsf{Inj}\,G)}_{\mathfrak{p}}$ . Thus ${\mathsf{K}_{\text{ac}}(\mathsf{Inj}\,G)}_{\mathfrak{p}}=\mathsf{K}(\mathsf{Inj}\,G)_{\mathfrak{p}}$ .◻

Endofiniteness

Following Crawley-Boevey [Reference Crawley-BoeveyCra91, Reference Crawley-BoeveyCra92], a module $X$ over an associative ring $A$ is endofinite if $X$ has finite length as a module over $\operatorname{End}_{A}(X)$ .

An object $X$ of a compactly generated triangulated category $\mathsf{T}$ is endofinite if the $\operatorname{End}_{\mathsf{T}}(X)$ -module $\operatorname{Hom}_{\mathsf{T}}(C,X)$ has finite length for all $C\in \mathsf{T}^{\mathsf{c}}$ ; see [Reference Krause and ReichenbachKR00].

Let $A$ be a self-injective algebra, finite dimensional over some field. Then an $A$ -module is endofinite if and only if it is endofinite as an object of $\mathsf{StMod}\,A$ . This follows from the fact $X$ is an endofinite $A$ -module if and only if the $\operatorname{End}_{A}(X)$ -module $\operatorname{Hom}_{A}(C,X)$ has finite length for every finite-dimensional $A$ -module  $C$ .

Proof. By Brown representability, $F$ has a left adjoint, say $F^{\prime }$ . It preserves compactness, as $F$ preserves coproducts. For $X\in \mathsf{T}$ and $C\in \mathsf{U}^{\mathsf{c}}$ , there is an isomorphism

$$\begin{eqnarray}\operatorname{Hom}_{\mathsf{U}}(C,FX)\cong \operatorname{Hom}_{\mathsf{T}}(F^{\prime }C,X)\end{eqnarray}$$

of $\operatorname{End}_{\mathsf{T}}(X)$ -modules. Thus if $X$ is endofinite, then $\operatorname{Hom}_{\mathsf{U}}(C,FX)$ is a module of finite length over $\operatorname{End}_{\mathsf{T}}(X)$ , and therefore also over $\operatorname{End}_{\mathsf{U}}(FX)$ . For the converse, observe that each compact object in $\mathsf{T}$ is isomorphic to a direct summand of an object of the form $F^{\prime }C$ for some $C\in \mathsf{U}^{\mathsf{c}}$ .◻

Transpose and dual

Let $G^{\text{op}}$ be the opposite group scheme of $G$ ; it can be realised as the group scheme associated to the cocommutative Hopf algebra $(kG)^{\text{op}}$ . Since $kG$ is a $G$ -bimodule, the assignment $M\mapsto \operatorname{Hom}_{G}(M,kG)$ defines a functor

$$\begin{eqnarray}(-)^{t}:\mathsf{Mod}\,G\longrightarrow \mathsf{Mod}\,G^{\text{op}}.\end{eqnarray}$$

Let $M$ be a finite-dimensional $G$ -module and $P_{1}\xrightarrow[{}]{f}P_{0}\rightarrow M$ a minimal projective presentation. The transpose of $M$ is the $G^{\text{op}}$ -module $\operatorname{Tr}M:=\operatorname{Coker}(f^{t})$ . By construction, there is an exact sequence of $G^{\text{op}}$ -modules:

$$\begin{eqnarray}0\longrightarrow M^{t}\longrightarrow P_{0}^{t}\xrightarrow[{}]{~f^{t}~}P_{1}^{t}\longrightarrow \operatorname{Tr}M\longrightarrow 0.\end{eqnarray}$$

The $P_{i}^{t}$ are projective $G^{\text{op}}$ -modules, so this yields an isomorphism of $G^{\text{op}}$ -modules

$$\begin{eqnarray}M^{t}\cong \unicode[STIX]{x1D6FA}^{2}\operatorname{Tr}M.\end{eqnarray}$$

Given a $G^{\text{op}}$ -module $N$ , the $k$ -vector space $\operatorname{Hom}_{k}(N,k)$ has a natural structure of a $G$ -module, and the assignment $N\mapsto \operatorname{Hom}_{k}(N,k)$ yields a functor

$$\begin{eqnarray}D:=\operatorname{Hom}_{k}(-,k):\mathsf{stmod}\,G^{\text{op}}\longrightarrow \mathsf{stmod}\,G.\end{eqnarray}$$

The Auslander–Reiten translate

In what follows we write $\unicode[STIX]{x1D70F}$ for the Auslander–Reiten translate of  $G$ :

$$\begin{eqnarray}\unicode[STIX]{x1D70F}:=D\circ \operatorname{Tr}:\mathsf{stmod}\,G\rightarrow \mathsf{stmod}\,G.\end{eqnarray}$$

Given an extension of fields $K/k$ , for any finite-dimensional $G$ -module $M$ there is a stable isomorphism of $G_{K}$ -modules

$$\begin{eqnarray}(\unicode[STIX]{x1D70F}M)_{K}\cong \unicode[STIX]{x1D70F}(M_{K}).\end{eqnarray}$$

Nakayama functor

The Nakayama functor

$$\begin{eqnarray}\unicode[STIX]{x1D708}:\mathsf{Mod}\,G\xrightarrow[{}]{~\sim ~}\mathsf{Mod}\,G\end{eqnarray}$$

is given by the assignment

$$\begin{eqnarray}M\mapsto D(kG)\otimes _{kG}M\cong \unicode[STIX]{x1D6FF}_{G}\otimes _{k}M,\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}_{G}=\unicode[STIX]{x1D708}(k)$ is the modular character of $G$ ; see [Reference JantzenJan03, I.8.8]. Since the group of characters of $G$ is finite, by [Reference WaterhouseWat79, §§ 2.1 and 2.2], there exists a positive integer $d$ such that $\unicode[STIX]{x1D6FF}_{G}^{\otimes d}\cong k$ and hence as functors on $\mathsf{Mod}\,G$ there is an equality

(4.1) $$\begin{eqnarray}\unicode[STIX]{x1D708}^{d}=\text{id}.\end{eqnarray}$$

When $M$ is a finite-dimensional $G$ -module, there are natural stable isomorphisms

$$\begin{eqnarray}\unicode[STIX]{x1D708}M\cong D(M^{t})\cong \unicode[STIX]{x1D6FA}^{-2}\unicode[STIX]{x1D70F}M.\end{eqnarray}$$

When in addition $M$ is projective, one has

$$\begin{eqnarray}\operatorname{Hom}_{G}(M,-)^{\vee }\cong (M^{t}\otimes _{kG}-)^{\vee }\cong \operatorname{Hom}_{G}(-,\unicode[STIX]{x1D708}M).\end{eqnarray}$$

Let $K/k$ be an extension of fields. For any $G$ -module $M$ there is a natural isomorphism of $G_{K}$ -modules

(4.2) $$\begin{eqnarray}(\unicode[STIX]{x1D708}M)_{K}\cong \unicode[STIX]{x1D708}(M_{K}).\end{eqnarray}$$

This is clear for $M=kG$ since

$$\begin{eqnarray}K\otimes _{k}\operatorname{Hom}_{k}(kG,k)\cong \operatorname{Hom}_{k}(kG,K)\cong \operatorname{Hom}_{K}(K\otimes _{k}kG,K),\end{eqnarray}$$

and the general case follows by taking a free presentation of  $M$ .

Tate duality

For finite groups, the duality theorem below is classical and due to Tate [Reference Cartan and EilenbergCE56, ch. XII, Theorem 6.4]. A proof of the extension to finite group schemes was sketched in [Reference Benson, Iyengar, Krause and PevtsovaBIKP17, § 2], and is reproduced here for readers convenience.

Theorem 4.2. Let $G$ be a finite group scheme over a field $k$ . For any $G$ -modules $M,N$ with $M$ finite dimensional, there are natural isomorphisms

$$\begin{eqnarray}\text{}\underline{\operatorname{Hom}}_{G}(M,N)^{\vee }\cong \text{}\underline{\operatorname{Hom}}_{G}(N,\unicode[STIX]{x1D6FA}^{-1}\unicode[STIX]{x1D70F}M)\cong \text{}\underline{\operatorname{Hom}}_{G}(N,\unicode[STIX]{x1D6FA}\unicode[STIX]{x1D708}M).\end{eqnarray}$$

Proof. A formula of Auslander and Reiten [Reference AuslanderAus78, Proposition I.3.4], see also [Reference KrauseKra03, Corollary p. 269], yields the first isomorphism below

$$\begin{eqnarray}\text{}\underline{\operatorname{Hom}}_{G}(M,N)^{\vee }\cong \operatorname{Ext}_{G}^{1}(N,\unicode[STIX]{x1D70F}M)\cong \text{}\underline{\operatorname{Hom}}_{G}(N,\unicode[STIX]{x1D6FA}^{-1}\unicode[STIX]{x1D70F}M).\end{eqnarray}$$

The second isomorphism is standard. It remains to recall that $\unicode[STIX]{x1D70F}M\cong \unicode[STIX]{x1D6FA}^{2}\unicode[STIX]{x1D708}M$ .◻

Restricted to finite-dimensional $G$ -modules, Tate duality is the statement that the $k$ -linear category $\mathsf{stmod}\,G$ has Serre duality, with Serre functor $\unicode[STIX]{x1D6FA}\unicode[STIX]{x1D708}$ . A refinement of this Serre duality will be proved in § 7.

Small triangulated categories with central action

Let $\mathsf{C}$ be an essentially small $R$ -linear triangulated category. Fix $\mathfrak{p}\in \operatorname{Spec}R$ and let $\mathsf{C}_{\mathfrak{p}}$ denote the triangulated category obtained from $\mathsf{C}$ by keeping the objects of $\mathsf{C}$ and setting

$$\begin{eqnarray}\operatorname{Hom}_{\mathsf{C}_{\mathfrak{ p}}}^{\ast }(X,Y):=\operatorname{Hom}_{C}^{\ast }(X,Y)_{\mathfrak{ p}}.\end{eqnarray}$$

Then $\mathsf{C}_{\mathfrak{p}}$ is an $R_{\mathfrak{p}}$ -linear triangulated category and localising the morphisms induces an exact functor $\mathsf{C}\rightarrow \mathsf{C}_{\mathfrak{p}}$ ; see [Reference BalmerBal10, Theorem 3.6] or [Reference Benson, Iyengar and KrauseBIK15, Lemma 3.5].

Let $\unicode[STIX]{x1D6FE}_{\mathfrak{p}}\mathsf{C}$ be the full subcategory of $\mathfrak{p}$ -torsion objects in $\mathsf{C}_{\mathfrak{p}}$ , namely

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{\mathfrak{p}}\mathsf{C}:=\{X\in \mathsf{C}_{\mathfrak{p}}\mid \operatorname{End}_{\mathsf{C}_{\mathfrak{ p}}}^{\ast }(X)~\text{is}~\mathfrak{p}\text{-}\text{torsion}\}.\end{eqnarray}$$

This is a thick subcategory of $\mathsf{C}_{\mathfrak{p}}$ ; see [Reference Benson, Iyengar and KrauseBIK15, p. 458f]. In [Reference Benson, Iyengar and KrauseBIK15] this category is denoted $\unicode[STIX]{x1D6E4}_{\mathfrak{p}}\mathsf{C}$ . The notation has been changed to avoid confusion.

Remark 7.1. Let $F:\mathsf{C}\rightarrow \mathsf{C}$ be an $R$ -linear equivalence. It is straightforward to check that this induces triangle equivalences $F_{\mathfrak{p}}:\mathsf{C}_{\mathfrak{p}}\xrightarrow[{}]{{\sim}}\mathsf{C}_{\mathfrak{p}}$ and $\unicode[STIX]{x1D6FE}_{\mathfrak{p}}\mathsf{C}\xrightarrow[{}]{{\sim}}\unicode[STIX]{x1D6FE}_{\mathfrak{p}}\mathsf{C}$ making the following diagram commutative.

Remark 7.2. Let $\mathsf{T}$ be a compactly generated $R$ -linear triangulated category. Set $\mathsf{C}:=\mathsf{T}^{\mathsf{c}}$ and fix $\mathfrak{p}\in \operatorname{Spec}R$ . The triangulated categories $\mathsf{T}_{\mathfrak{p}}$ and $\unicode[STIX]{x1D6E4}_{\mathfrak{p}}\mathsf{T}$ are compactly generated. The left adjoint of the inclusion $\mathsf{T}_{\mathfrak{p}}\rightarrow \mathsf{T}$ induces (up to direct summands) a triangle equivalence $\mathsf{C}_{\mathfrak{p}}\xrightarrow[{}]{{\sim}}(\mathsf{T}_{\mathfrak{p}})^{\mathsf{c}}$ and restricts to a triangle equivalence

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{\mathfrak{p}}\mathsf{C}\xrightarrow[{}]{{\sim}}(\unicode[STIX]{x1D6E4}_{\mathfrak{p}}\mathsf{T})^{\mathsf{c}}.\end{eqnarray}$$

This follows from the fact that the localisation functor $\mathsf{T}\rightarrow \mathsf{T}_{\mathfrak{p}}$ taking $X$ to $X_{\mathfrak{p}}$ preserves compactness and that for compact objects $X,Y$ in $\mathsf{T}$

$$\begin{eqnarray}\operatorname{Hom}_{\mathsf{T}}^{\ast }(X,Y)_{\mathfrak{ p}}\xrightarrow[{}]{{\sim}}\operatorname{Hom}_{\mathsf{T}_{\mathfrak{p}}}^{\ast }(X_{\mathfrak{ p}},Y_{\mathfrak{p}}).\end{eqnarray}$$

For details we refer to [Reference Benson, Iyengar and KrauseBIK08].

Local Serre duality

Let $R$ be a graded commutative ring that is local; thus there is a unique homogeneous maximal ideal, say  $\mathfrak{m}$ . Extrapolating from Bondal and Kapranov [Reference Bondal and KapranovBK89, § 3], we call an $R$ -linear triangle equivalence $F:\mathsf{C}\xrightarrow[{}]{{\sim}}\mathsf{C}$ a Serre functor if for all objects $X,Y$ in $\mathsf{C}$ there is a natural isomorphism

(7.1) $$\begin{eqnarray}\operatorname{Hom}_{R}(\operatorname{Hom}_{\mathsf{C}}^{\ast }(X,Y),I(\mathfrak{m}))\xrightarrow[{}]{{\sim}}\operatorname{Hom}_{\mathsf{C}}(Y,FX).\end{eqnarray}$$

The situation when $R$ is a field was the one considered in [Reference Bondal and KapranovBK89]. For a general ring  $R$ , the appearance of $\operatorname{Hom}_{R}^{\ast }(-,I(\mathfrak{m}))$ , which is the Matlis duality functor, is natural for it is an extension of vector-space duality; see also Lemma A.2. The definition proposed above is not the only possible extension to the general context, but it is well suited for our purposes.

For an arbitrary graded commutative ring $R$ , we say that an $R$ -linear triangulated category $\mathsf{C}$ satisfies local Serre duality if there exists an $R$ -linear triangle equivalence $F:\mathsf{C}\xrightarrow[{}]{{\sim}}\mathsf{C}$ such that for every $\mathfrak{p}\in \operatorname{Spec}R$ and some integer $d(\mathfrak{p})$ the induced functor $\unicode[STIX]{x1D6F4}^{-d(\mathfrak{p})}F_{\mathfrak{p}}:\unicode[STIX]{x1D6FE}_{\mathfrak{p}}\mathsf{C}\xrightarrow[{}]{{\sim}}\unicode[STIX]{x1D6FE}_{\mathfrak{p}}\mathsf{C}$ is a Serre functor for the $R_{\mathfrak{p}}$ -linear category $\unicode[STIX]{x1D6FE}_{\mathfrak{p}}\mathsf{C}$ . Thus for all objects $X,Y$ in $\unicode[STIX]{x1D6FE}_{\mathfrak{p}}\mathsf{C}$ there is a natural isomorphism

$$\begin{eqnarray}\operatorname{Hom}_{R}(\operatorname{Hom}_{\mathsf{C}_{\mathfrak{ p}}}^{\ast }(X,Y),I(\mathfrak{p}))\xrightarrow[{}]{{\sim}}\operatorname{Hom}_{\mathsf{C}_{\mathfrak{p}}}(Y,\unicode[STIX]{x1D6F4}^{-d(\mathfrak{p})}F_{\mathfrak{ p}}X).\end{eqnarray}$$

For a compactly generated triangulated category, the Gorenstein property 6.1 is linked to local Serre duality for the subcategory of compact objects.

Proposition 7.3. Let $R$ be a graded commutative noetherian ring and $\mathsf{T}$ a compactly generated $R$ -linear triangulated category. Suppose that $\mathsf{T}$ is Gorenstein, with global Serre functor $F$ and shifts $\{d(\mathfrak{p})\}$ . Then for each $\mathfrak{p}\in \operatorname{supp}_{R}(\mathsf{T})$ , object $X\in (\unicode[STIX]{x1D6E4}_{\mathfrak{p}}\mathsf{T})^{\mathsf{c}}$ and $Y\in \mathsf{T}_{\mathfrak{p}}$ there is a natural isomorphism

$$\begin{eqnarray}\operatorname{Hom}_{R}(\operatorname{Hom}_{\mathsf{T}}^{\ast }(X,Y),I(\mathfrak{p}))\cong \operatorname{Hom}_{\mathsf{T}}(Y,\unicode[STIX]{x1D6F4}^{-d(\mathfrak{p})}F_{\mathfrak{ p}}(X)).\end{eqnarray}$$

Proof. Given Remark 7.2 we can assume $X=C_{\mathfrak{p}}$ for a $\mathfrak{p}$ -torsion compact object $C$ in  $\mathsf{T}$ . The desired isomorphism is a concatenation of the following natural ones:

$$\begin{eqnarray}\displaystyle \operatorname{Hom}_{R}(\operatorname{Hom}_{\mathsf{T}}^{\ast }(C_{\mathfrak{p}},Y),I(\mathfrak{p})) & \cong & \displaystyle \operatorname{Hom}_{R}(\operatorname{Hom}_{\mathsf{T}}^{\ast }(C,Y),I(\mathfrak{p}))\nonumber\\ \displaystyle & \cong & \displaystyle \operatorname{Hom}_{\mathsf{T}}(Y,T_{\mathfrak{p}}(C))\nonumber\\ \displaystyle & \cong & \displaystyle \operatorname{Hom}_{\mathsf{T}}(Y,\unicode[STIX]{x1D6F4}^{-d(\mathfrak{p})}\unicode[STIX]{x1D6E4}_{\mathfrak{p}}F(C))\nonumber\\ \displaystyle & \cong & \displaystyle \operatorname{Hom}_{\mathsf{T}}(Y,\unicode[STIX]{x1D6F4}^{-d(\mathfrak{p})}\unicode[STIX]{x1D6E4}_{{\mathcal{V}}(\mathfrak{p})}F(C)_{\mathfrak{p}})\nonumber\\ \displaystyle & \cong & \displaystyle \operatorname{Hom}_{\mathsf{T}}(Y,\unicode[STIX]{x1D6F4}^{-d(\mathfrak{p})}\unicode[STIX]{x1D6E4}_{{\mathcal{V}}(\mathfrak{p})}F_{\mathfrak{p}}(C_{\mathfrak{p}}))\nonumber\\ \displaystyle & \cong & \displaystyle \operatorname{Hom}_{\mathsf{T}}(Y,\unicode[STIX]{x1D6F4}^{-d(\mathfrak{p})}F_{\mathfrak{p}}(C_{\mathfrak{p}})).\nonumber\end{eqnarray}$$

In this chain, the first map is induced by the localisation $C\mapsto C_{\mathfrak{p}}$ and is an isomorphism because $Y$ is $\mathfrak{p}$ -local. The second one is by the definition of $T_{\mathfrak{p}}(C)$ ; the third is by the Gorenstein property of  $\mathsf{T}$ ; the fourth is by the definition of $\unicode[STIX]{x1D6E4}_{\mathfrak{p}}$ ; the last two are explained by Remark 7.2, where for the last one uses also the fact that $C_{\mathfrak{p}}$ , and hence also $F_{\mathfrak{p}}(C_{\mathfrak{p}})$ , is $\mathfrak{p}$ -torsion.◻

Example 7.5. In the notation of Example 6.3, when $A$ is a (commutative noetherian) Gorenstein ring, local Serre duality reads: For each $\mathfrak{p}\in \operatorname{Spec}A$ and integer $n$ there are natural isomorphisms

$$\begin{eqnarray}\operatorname{Hom}_{A_{\mathfrak{p}}}(\operatorname{Ext}_{A_{\mathfrak{p}}}^{n}(X,Y),I(\mathfrak{p}))\cong \operatorname{Ext}_{A_{\mathfrak{p}}}^{n+\dim A_{\mathfrak{p}}}(Y,X),\end{eqnarray}$$

where $X$ is a perfect complexes of $A_{\mathfrak{p}}$ -modules with finite length cohomology, and $Y$ is a complex of $A_{\mathfrak{p}}$ -modules.

Auslander–Reiten triangles

Let $\mathsf{C}$ be an essentially small triangulated category. Following Happel [Reference HappelHap88], an exact triangle $X\xrightarrow[{}]{\unicode[STIX]{x1D6FC}}Y\xrightarrow[{}]{\unicode[STIX]{x1D6FD}}Z\xrightarrow[{}]{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6F4}X$ in $\mathsf{C}$ is an Auslander–Reiten triangle if:

1. (i) any morphism $X\rightarrow X^{\prime }$ that is not a split monomorphism factors through $\unicode[STIX]{x1D6FC}$ ;

2. (ii) any morphism $Z^{\prime }\rightarrow Z$ that is not a split epimorphism factors through $\unicode[STIX]{x1D6FD}$ ;

3. (iii) $\unicode[STIX]{x1D6FE}\neq 0$ .

In this case, the endomorphism rings of $X$ and $Z$ are local, and in particular the objects are indecomposable. Moreover, each of $X$ and $Z$ determines the AR-triangle up to isomorphism. Assuming conditions (ii) and (iii), the condition (i) is equivalent to the following.

1. (i′) The endomorphism ring of $X$ is local.

See [Reference KrauseKra00, § 2] for details.

Let $\mathsf{C}$ be a Krull–Schmidt category, that is, each object decomposes into a finite direct sum of objects with local endomorphism rings. We say that $\mathsf{C}$ has AR-triangles if for every indecomposable object $X$ there are AR-triangles

$$\begin{eqnarray}V\rightarrow W\rightarrow X\rightarrow \unicode[STIX]{x1D6F4}V\quad \text{and}\quad X\rightarrow Y\rightarrow Z\rightarrow \unicode[STIX]{x1D6F4}X.\end{eqnarray}$$

The next proposition establishes the existence of AR-triangles; it is the analogue of a result of Reiten and Van den Bergh [Reference Reiten and Van den BerghRV02, I.2] for triangulated categories that are Hom-finite over a field.

Proof. Let $F$ be a Serre functor for $\mathsf{C}$ and $X$ an indecomposable object in  $\mathsf{C}$ . The ring $\operatorname{End}_{\mathsf{C}}(X)$ is thus local; let $J$ be its maximal ideal and $I$ the right ideal of $\operatorname{End}_{\mathsf{C}}^{\ast }(X)$ that it generates. One has $I^{0}=J$ , by Remark 7.7, so $\operatorname{End}_{\mathsf{C}}^{\ast }(X)/I$ is nonzero. Choose a nonzero morphism $\operatorname{End}_{\mathsf{C}}^{\ast }(X)/I\rightarrow I(\mathfrak{m})$ and let $\unicode[STIX]{x1D6FE}:X\rightarrow FX$ be the corresponding morphism in $\mathsf{C}$ provided by Serre duality (7.1). We claim that the induced exact triangle

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}^{-1}FX\rightarrow W\rightarrow X\xrightarrow[{}]{\unicode[STIX]{x1D6FE}}FX\end{eqnarray}$$

is an AR-triangle. Indeed, by construction, if $X^{\prime }\rightarrow X$ is not a split epimorphism, then the composition $\operatorname{Hom}_{\mathsf{C}}^{\ast }(X,X^{\prime })\rightarrow \operatorname{End}_{\mathsf{C}}^{\ast }(X)/I\rightarrow I(\mathfrak{m})$ is zero, and therefore the naturality of (7.1) yields that the composition $X^{\prime }\rightarrow X\xrightarrow[{}]{\unicode[STIX]{x1D6FE}}FX$ is zero. Moreover as $X$ is indecomposable so is  $FX$ .

Applying this construction to $F^{-1}\unicode[STIX]{x1D6F4}X$ yields an AR-triangle starting at  $X$ .◻

The following observations about graded rings has been, and will again be, used.

Let $R$ be a graded commutative ring. An $R$ -linear triangulated category $\mathsf{C}$ is noetherian if the $R$ -module $\operatorname{Hom}_{\mathsf{C}}^{\ast }(X,Y)$ is noetherian for all $X,Y$ in  $\mathsf{C}$ .