2.1 Derived categories

We write $\mathbf{D}(\mathcal{A})$ to denote the derived category of an exact category $\mathcal{A}$ ; see [Reference KrauseKra22, § 4.1]. The main examples for us are the exact category $\operatorname{Mod}(A,R)$ and its subcategory $\operatorname{mod}(A,R)$ .

with kernel given by the complexes in $\operatorname{Mod}(A,R)$ that are acyclic in $\operatorname{Mod} A$ .

In what follows, by a perfect complex over a ring $\Lambda$ we mean a complex that is quasi-isomorphic to a bounded complex of finitely generated projective $\Lambda$ -modules.

Proposition 2.4. The inclusions $\operatorname{mod}(A,R)\subseteq \operatorname{Mod}(A,R)\subseteq \operatorname{Mod} A$ of exact categories induce functors

\[\mathbf{D}^b(\operatorname{mod}(A,R))\longrightarrow\mathbf{D}(\operatorname{Mod}(A,R)) \longrightarrow\mathbf{D}(\operatorname{Mod} A), \]

where the first one and the composite are fully faithful, identifying $\mathbf{D}^b(\operatorname{mod}(A,R))$ with the subcategory of complexes in $\mathbf{D}(\operatorname{Mod} A)$ that are perfect over R.

Proof. The composite of functors in question and also the first one are fully faithful, because $\operatorname{mod}(A,R)$ is cofinal in $\operatorname{Mod} A$ . This means that for any exact sequence

\[0\longrightarrow X\longrightarrow Y\longrightarrow Z\longrightarrow 0\quad \text{with } Z\in\operatorname{mod}(A,R) \]

there is an epimorphism $A\otimes_R Z\to Z$ in $\operatorname{mod}(A,R)$ that factors through $Y\to Z$ ; see [Reference KrauseKra22, Proposition 4.2.15]. It is clear that the essential image consists of complexes in $\mathbf{D}(\operatorname{Mod} A)$ that are perfect over R. As for the converse claim, let X be a complex of A-modules that is perfect over R. We claim that X is quasi-isomorphic to a bounded complex consisting of A-modules that are finite and projective over R; this would complete the argument, for the latter is evidently in the image of $\mathbf{D}^b(\operatorname{mod}(A,R))$ .

Indeed, since X is perfect over R its homology modules $H_i(X)$ are finitely generated as R-modules, and so also as A-modules, and are zero for $|i|\gg 0$ . Without loss of generality we can assume $H_i(X)=0$ for $i<0$ . Thus X has a resolution P over A where each $P_i$ is finitely generated projective and zero for $i<0$ . For each integer n, set $\Omega_n := \operatorname{Coker}(P_{n+1}\to P_n)$ . Then X is quasi-isomorphic to the complex

\[X(n)\,{:}\, \quad \cdots\longrightarrow 0 \longrightarrow \Omega_{n} \longrightarrow P_{n-1}\longrightarrow \cdots\longrightarrow P_0\longrightarrow 0 \longrightarrow\cdots\ \]

for each $n\geqslant s:=\max\{i\mid H_i(X)\ne0\}$ . The A-module $\Omega_s$ is perfect over R. Let p denote its projective dimension. Then the module $\Omega_{s+p}$ is projective over R, and therefore the complex $X(s+p)$ is quasi-isomorphic to X and in $\operatorname{mod}(A,R)$ .

The following example has been suggested by Greg Stevenson in order to illustrate the difference between the derived categories of $\operatorname{Mod}(A,R)$ and $\operatorname{Mod} A$ .

Example 2.5. Let k be a field and choose $A:= k[t]/(t^2)=: R$ . Then $\mathbf{D}(\operatorname{Mod}(A,R))$ equals the homotopy category $\mathbf{K}(\operatorname{Proj} A)$ of complexes of projective A-modules, and the kernel of the canonical functor $\mathbf{D}(\operatorname{Mod}(A,R))\to\mathbf{D}(\operatorname{Mod} A)$ identifies with the stable module category of A. In particular, a complex is in the kernel if and only if it is a direct sum of copies of the complex

\[\cdots\xrightarrow{\ t\ }A\xrightarrow{\ t\ }A\xrightarrow{\ t\ }A\xrightarrow{\ t\ }\cdots. \]

2.2 Frobenius algebras

We are interested in the case when the exact category $\operatorname{Mod}(A,R)$ is Frobenius, that is to say, it has enough projective and enough injective objects, and these coincide.

(1) $\Leftrightarrow$ (3) follows from Lemma 2.2.

A ring A is a Frobenius R-algebra if it is a finite projective R-algebra satisfying the conditions in the result above; see the work of Scheja and Storch [Reference Scheja and StorchSS74, § 14], where this notion is introduced via condition (1). When R is noetherian, this condition can be detected in terms of the fibres of A over R; see Lemma 3.2. Evidently A is a Frobenius R-algebra if and only if the same is true for $A^\mathrm{op}$ , because of the duality (2.1).

2.3 The stable module category

For any Frobenius category $\mathcal{A}$ we write $\operatorname{St}\mathcal{A}$ for the stable category obtained from $\mathcal{A}$ by annihilating the projective and injective objects; this carries a natural triangulated structure [Reference KrauseKra22, § 3.3]. We denote by $\Sigma$ the suspension in $\operatorname{St}\mathcal{A}$ , which for any $M\in\mathcal{A}$ is given by an exact sequence

\[0\longrightarrow M\longrightarrow Q\longrightarrow \Sigma M\longrightarrow 0 \]

such that Q is an injective, and so also a projective, object. For M,N in $\mathcal{A}$ we set

\[\underline{\operatorname{Hom}}_{\mathcal{A}}(M,N):= \operatorname{Hom}_{\mathcal{A}}(M,N)/\mathrm{PHom}_{\mathcal{A}}(M,N)\,, \]

where $\mathrm{PHom}_{\mathcal{A}}(M,N)$ denotes the subgroup of morphisms $M\to N$ that factor through a projective object.

Let A be a Frobenius R-algebra. We set

\[\operatorname{stmod}(A,R):=\operatorname{St}(\operatorname{mod}(A,R))\quad\text{and}\quad \operatorname{StMod}(A,R):=\operatorname{St}(\operatorname{Mod}(A,R))\,, \]

and consider these categories with their natural triangulated structure.

2.4 The homotopy category of projectives

Let $\Lambda$ be a ring and $\mathbf{K}(\operatorname{Proj} \Lambda)$ its homotopy category of (unbounded) complexes of projective $\Lambda$ -modules. Consider the canonical functor $\mathbf{q}\,{:}\, \mathbf{K}(\operatorname{Proj} \Lambda)\to\mathbf{D} (\operatorname{Mod} \Lambda)$ which inverts the quasi-isomorphisms. Taking projective resolutions yields a left adjoint $X\mapsto \mathbf{p} X$ that identifies the derived category $\mathbf{D} (\operatorname{Mod} \Lambda)$ with the full subcategory of K-projective complexes in $\mathbf{K}(\operatorname{Proj} \Lambda)$ .

Let A be a Frobenius R-algebra. We write $\mathbf{K}(\operatorname{Proj} A,R)$ for the full subcategory of complexes in $\mathbf{K}(\operatorname{Proj} A)$ that are K-projective when restricted to R; this is a localising subcategory. Let $\mathbf{Ac}(\operatorname{Proj} A,R)$ denote the full subcategory of complexes in $\mathbf{K}(\operatorname{Proj} A,R)$ that are acyclic; this is again a localising subcategory. The term ‘acyclic’ refers to the exact structure of the ambient exact category, which is either $\operatorname{Mod} A$ or $\operatorname{Mod}(A,R)$ . In both cases we obtain the same class of acyclic complexes. It is clear that a complex is acyclic in $\operatorname{Mod} A$ if it is acyclic in $\operatorname{Mod}(A,R)$ . On the other hand, if a complex is acyclic in $\operatorname{Mod} A$ and also K-projective over R, then it is contractible over R and therefore acyclic in $\operatorname{Mod}(A,R)$ .

Recall that each M in $\operatorname{Mod}(A,R)$ has an injective and a projective resolution, denoted by $\mathbf{i} M$ and $\mathbf{p} M$ ; see (2.2). A complete resolution of an M in $\operatorname{Mod}(A,R)$ is an acyclic complex $\mathbf{t} M$ of projective A-modules with $ \operatorname{Coker}(d^{-1}_{\mathbf{t} M})=M$ ; here, $d^{-1}_{\mathbf{t} M}$ denotes the $(-1)$ st differential in the complex $\mathbf{t} M$ . Then $\mathbf{i} M$ and $\mathbf{t} M$ are K-projective over R, since A is a Frobenius R-algebra; the complex $\mathbf{p} M$ is always K-projective over R.

Lemma 2.7. When A is a Frobenius R-algebra the assignment $X\mapsto \operatorname{Coker}(d^{-1}_X)$ induces an R-linear triangle equivalence

\[\mathbf{Ac}(\operatorname{Proj} A,R)\xrightarrow{\scriptstyle{\sim}$}\operatorname{StMod}(A,R). \]

Proposition 2.8. When A is a Frobenius R-algebra the canonical functor

\[ \mathbf{K}(\operatorname{Proj} A,R)\longrightarrow\mathbf{D} (\operatorname{Mod} A) \]

induces a localisation sequence

Proof. The left adjoint $\mathbf{p} \,{:}\, \mathbf{D}(\operatorname{Mod} A) \to \mathbf{K}(\operatorname{Proj} A)$ clearly lands in $\mathbf{K}(\operatorname{Proj} A,R)$ . For any $X\in \mathbf{K}(\operatorname{Proj} A)$ , completing the counit $\mathbf{p} X\to X$ to an exact triangle

\[\mathbf{p} X\longrightarrow X\longrightarrow \mathbf{t} X\longrightarrow \]

provides the left adjoint $\mathbf{t}$ for the inclusion of $\mathbf{Ac}(\operatorname{Proj} A,R)$ .

2.5 Compact generation

Let A be a Frobenius R-algebra. We write $\operatorname{Mod}^{\mathrm{lf}}(A,R)$ for the localising subcategory of $\operatorname{Mod}(A,R)$ generated by the modules in $\operatorname{mod}(A,R)$ . By a localising subcategory we mean a full subcategory closed under all coproducts and satisfying the two-out-of-three property: for any exact sequence

\[0\longrightarrow X\longrightarrow Y\longrightarrow Z\longrightarrow 0, \]

if any two of X,Y, or Z is in the subcategory so is the third. The subcategory $ \operatorname{Mod}^{\mathrm{lf}}(A,R)$ contains all the projective objects in $\operatorname{Mod}(A,R)$ , and so is itself a Frobenius category, with induced exact structure. When R is regular, $ \operatorname{Mod}^{\mathrm{lf}}(A,R)= \operatorname{Mod}(A,R)$ , by Proposition 2.14, but not for general R; see § 11. Set

\[\operatorname{StMod}^{\mathrm{lf}}(A,R) := \operatorname{St}(\operatorname{Mod}^{\mathrm{lf}}(A,R)). \]

Here are the key properties of this triangulated category.

Indeed, the local ring $A:= \mathbb{C}[x,y]_{(x,y)}/(x^2-y^2(y+1))$ , viewed as an algebra over $R:= \mathbb{C}[x]_{(x)}$ , is a Frobenius R-algebra; this can be checked easily. Since R is a DVR, a finitely generated module over it is projective (equivalently, free) if and only if it is torsion-free. Thus an A-module is in $\operatorname{mod}(A,R)$ if and only if it has positive depth, that is to say, it is maximal Cohen–Macaulay. Thus $\operatorname{stmod}(A,R)$ is the stable category of maximal Cohen–Macaulay A-modules. This category is not idempotent complete; see [Reference KrauseKra22, Lemma 6.2.12] for a proof.

For an example where A is the group algebra of a finite group, see the forthcoming work of J. Grodal and A. Krause.

Set $\mathcal{A}:= \operatorname{Mod}(A,R)$ . For a full additive subcategory $\mathcal{C}\subseteq\mathcal{A}$ we write $\mathbf{K}(\mathcal{C})$ for the homotopy category of chain complexes. We set

\[\mathbf{K}^{+,b}(\mathcal{C})=\{X\in\mathbf{K}^+(\mathcal{C})\mid H^nX=0 \textrm{ for }|n|\gg 0\} \]

where $\mathbf{K}^+(\mathcal{C})$ denotes the complexes X with $X^n=0$ for $n\ll 0$ , and the condition $H^nX=0$ means that $d_X^{n-1}$ can be written as the composite

\[X^{n-1}\twoheadrightarrow\operatorname{Ker} d^n_X\rightarrowtail X^n \]

of an admissible epimorphism and an admissible monomorphism in $\mathcal{A}$ . In particular, the subcategory $\mathbf{K}^{+,b}(\mathcal{C})$ depends on the ambient category $\mathcal{A}$ , even though it is not part of the notation. The subcategory $\mathbf{K}^{-,b}(\mathcal{C})$ is defined analogously.

1. (1) X is compact in $\mathbf{K}(\operatorname{Proj} A)$ and K-projective over R;

2. (2) X fits into an extension $0\to X'\to X\to X''\to 0$ such that $X'=\mathbf{i} M$ for some $M\in\operatorname{mod}(A,R)$ and X” is perfect; and

3. (3) X is in $\mathbf{K}^{+,b}(\operatorname{proj} A)$ .

Moreover, such an X is perfect over R.

(2) $\Rightarrow$ (1) It suffices to verify that X” and X’ are compact and K-projective over R. Clearly X” has these properties. Also X’ is K-projective over R. For any Y in $\mathbf{K}(\operatorname{Proj} A)$ one has an isomorphism

\[ \operatorname{Hom}_{\mathbf{K}(A)}(\mathbf{i} M,Y)\cong \operatorname{Hom}_{\mathbf{K}(A)}(M,Y)\,, \]

by [Reference KrauseKra05, Lemma 2.1], so X’ is compact.

(2) $\Leftrightarrow$ (3) This is clear.

The last assertion was verified in proving (1) $\Rightarrow$ (2).

We write $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} A,R)$ for the localising subcategory of $\mathbf{K}(\operatorname{Proj} A)$ that is generated by the objects satisfying the equivalent conditions in Lemma 2.11. Thus one has inclusions

\[\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} A,R) \subseteq \mathbf{K}(\operatorname{Proj} A,R) \subseteq \mathbf{K}(\operatorname{Proj} A) \]

which become equalities when the ring R is regular; cf. Proposition 14. In fact, it is not hard to verify that equality holds on the right only if R is regular. Equality holds on the left when, for instance, $A=R$ , and also in some other cases, but not always; see § 11.

Proposition 2.12. Let A be a Frobenius R-algebra. The triangulated category $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} A,R) $ is compactly generated and the canonical functor $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} A,R)\to\mathbf{D} (\operatorname{Mod} A)$ induces a triangle equivalence

\[\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} A,R)^{c}\xrightarrow{\scriptstyle{\sim}}\, \mathbf{D}^b(\operatorname{mod}(A,R)). \]

Proof. The triangulated category $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} A,R)$ is compactly generated by construction. For the description of the compact objects recall that $\operatorname{mod}(A,R)$ is an exact category with enough injective objects. Thus we have a triangle equivalence which restricts to an equivalence

$$\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} A,R)^{c}=\mathbf{K}^{+,b}(\operatorname{proj} A)\xrightarrow{\scriptstyle{\sim}} \mathbf{D}^b(\operatorname{mod}(A,R)).$$

We write $\mathbf{D}^{\mathrm{perf}}(A)$ for the category of perfect complexes, which identifies with the full subcategory of compact objects in $\mathbf{D}(\operatorname{Mod} A)$ .

and the pair of left adjoints $(\mathbf{p} ,\mathbf{t})$ induces (when restricted to compact objects) a triangle equivalence

$$\mathbf{D}^b(\operatorname{mod}(A,R))/\mathbf{D}^{\mathrm{perf}}(A)\xrightarrow{\scriptstyle{\sim}$}\operatorname{stmod}(A,R)\, .$$

Indeed, we have the full subcategory of acyclic complexes in $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} A,R)$ on the left, which identifies with a localising subcategory of $\operatorname{StMod}(A,R)$ by Lemma 7. This subcategory is generated by the modules from $\operatorname{mod}(A,R)$ since $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} A,R)$ is generated by their injective resolutions, and therefore it equals $\operatorname{StMod}^{\mathrm{lf}}(A,R)$ .

As for the description of the compact objects, $\mathbf{q}$ preserves coproducts so the left adjoint $\mathbf{p} $ preserves compactness. The compacts in $\mathbf{D}(\operatorname{Mod} A)$ identify with the category $\mathbf{D}^{\mathrm{perf}}(A)$ of perfect complexes, while the compacts in $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} A,R)$ identify with $\mathbf{D}^b(\operatorname{mod}(A,R))$ by Proposition 2.12. This yields the last assertion, since for any Frobenius category $\mathcal{A}$ with full subcategory of projectives $\mathcal{P}$ we have

\[ \mathbf{D}^b(\mathcal{A})/\mathbf{D}^b(\mathcal{P})\xrightarrow{\scriptstyle{\sim}}\operatorname{St}\mathcal{A}\,, \]

for example by [Reference KrauseKra22, Proposition 4.4.18].

2.6 Regular base

A commutative ring R is regular if it is noetherian and the local ring $R_\mathfrak{p}$ has finite global dimension for each $\mathfrak{p}$ in $\operatorname{Spec} R$ ; when the Krull dimension of R is finite, R is regular if and only if its global dimension is finite (see [Reference Bruns and HerzogBH98, § 2.2]). In what follows, we often use the following characterisation of regularity: a noetherian commutative ring R is regular if and only if every complex of projective R-modules is K-projective; equivalently, every acyclic complex of projective modules is contractible (see, for instance, [Reference Iacob and IyengarII09, § 3]).

Proposition 2.14. Let R be a regular ring and A a Frobenius R-algebra. There are equalities

\[ \operatorname{Mod}^{\mathrm{lf}}(A,R)=\operatorname{Mod}(A,R)\quad\text{and}\quad \mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} A,R)=\mathbf{K}(\operatorname{Proj} A). \]

Moreover, there is an equality $\mathbf{D}(\operatorname{Mod}(A,R))=\mathbf{D}(\operatorname{Mod} A)$ .

Proof. By [Reference NeemanNee08, Proposition 7.14], the triangulated category $\mathbf{K}(\operatorname{Proj} A)$ is compactly generated. From Lemma 2.11 and the regularity of R, it follows that each compact object belongs to $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} A,R)$ . Thus $ \mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} A,R)=\mathbf{K}(\operatorname{Proj} A)$ . Comparing the diagrams in Proposition 2.8 and Corollary 2.13 and using Lemma 2.7, we conclude that $\operatorname{StMod}^{\mathrm{lf}}(A,R)=\operatorname{StMod}(A,R)$ . Therefore, any $M \in \operatorname{Mod}(A,R)$ may be represented by an object in $\operatorname{Mod}^{\mathrm{lf}}(A,R)$ up to a projective summand. Since projectives are in $\operatorname{Mod}^{\mathrm{lf}}(A,R)$ , it follows that $\operatorname{Mod}^{\mathrm{lf}}(A,R)=\operatorname{Mod}(A,R)$ . The last equality follows from Proposition 2.3 once one observes that, for any complex in $\operatorname{Mod} (A,R)$ , being acyclic in $\operatorname{Mod} A$ implies being contractible in $\operatorname{Mod} R$ and therefore also acyclic in $\operatorname{Mod}(A,R)$ .

4.1 Tensor structure

Given G-modules X and Y, there is a natural diagonal G-module structure on $X\otimes_RY$ , obtained by restricting its $(G\times G)$ -module structure along the coalgebra map $\Delta\,{:}\, RG\to RG\otimes_RRG$ .

Proof. This follows from the standard adjunction isomorphism

$$\operatorname{Hom}_{G}(P\otimes_RQ,-)\cong \operatorname{Hom}_G(P, \operatorname{Hom}_R(Q,-)).$$

The preceding result implies that $\otimes_R$ induces a tensor product on $\mathbf{K}(\operatorname{Proj} G)$ , the homotopy category of projective G-modules.

4.2 Rigidity

The tensor product on $\mathbf{K}(\operatorname{Proj} G)$ induces one on $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} G,R)$ , the subcategory introduced following Lemma 2.11. By construction, it is compactly generated, and our aim is to show that it is also rigidly-compactly generated, in the sense recalled below.

Let $\mathcal{T}$ be a compactly generated tensor triangulated category, with tensor product $\otimes$ and unit $\mathbb 1$ . We assume that $\mathbb 1$ is compact. Being a compactly generated tensor triangulated category, $\mathcal{T}$ has an internal function object, $\operatorname{\mathcal{H}\!\!\;\mathit{om}}(-,-)$ , defined by the property that

\[\operatorname{Hom}_{\mathcal{T}}(X\otimes Y,Z) \cong \operatorname{Hom}_{\mathcal{T}}(X,\operatorname{\mathcal{H}\!\!\;\mathit{om}}(Y,Z)) \]

for X,Y and Z in $\mathcal{T}$ . There is a natural map

(4.1) \begin{equation}\operatorname{\mathcal{H}\!\!\;\mathit{om}}(X,\mathbb 1)\otimes Y \longrightarrow \operatorname{\mathcal{H}\!\!\;\mathit{om}}(X,Y)\end{equation}

and X is rigid if this map is an isomorphism for all Y. Since $\mathbb 1$ is compact, every rigid object is compact, and one says that $\mathcal{T}$ is rigidly-compactly generated when the converse holds: compact objects and rigid objects coincide. It is straightforward to verify that this property holds if and only if:

1. (1) the subcategory of compact objects is closed under $\otimes$ , and

2. (2) the map (4.1) is an isomorphism when X,Y are compact.

We get back to the tensor triangulated category $\mathbf{K}(\operatorname{Proj} G)$ with product $\otimes_R$ , where G is a finite flat group scheme over R.

Proof. We claim that the tensor product on $\mathbf{K}(\operatorname{Proj} G)$ restricts to a product on $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} G, R)$ . Because $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} G, R)$ is compactly generated it suffices to show that for any compact objects X,Y in $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} G, R)$ , the complex $X\otimes_RY$ is again in $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} G, R)$ . For the rigidity we need to verify that $X\otimes_RY$ is also compact and that (4.1) is an isomorphism. We use the identification of the compact objects in $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} G, R)$ with the objects in $\mathbf{D}^b(\operatorname{mod}(G,R))$ ; see Proposition 2.12. For M,N in $\mathbf{D}^b(\operatorname{mod}(G,R))$ it is clear that the complex $M\otimes^{\mathrm L}_RN$ is in $\mathbf{D}^b(\operatorname{mod}(G,R))$ since M and N are perfect complexes over R. Now we can identify $\operatorname{\mathcal{H}\!\!\;\mathit{om}}(M,N)=\operatorname{RHom}_R(M,N)$ and it remains to observe that the map

\[\operatorname{RHom}_R(M,R) \otimes^{\mathrm L}_R N \longrightarrow \operatorname{RHom}_R(M,N) \]

of G-complexes is an isomorphism, because M is perfect over R.

4.3 Cohomology

With G acting trivially on R, we consider the cohomology algebra $S:= \operatorname{Ext}^*_G(R,R)$ of G. Since G is a group scheme, standard arguments yield that S is a graded-commutative R-algebra. For any G-module M, set

\[H^*(G,M) := \operatorname{Ext}_G^*(R,M). \]

This is a graded module over S, under cup-products; see [Reference BensonBen98]. Friedlander and Suslin [Reference Friedlander and SuslinFS97] prove that S is finitely generated—equivalently, noetherian—when R is a field; this result has been extended by van der Kallen [Reference van der KallenvdKal23] to cover any noetherian commutative ring R (see § 6.6).

For any map $R\to R'$ we write $G_{R'}$ for the group scheme over R’ obtained by base change of G along $R\to R'$ . This induces a map of graded-commutative rings

\[H^*(G,R)\longrightarrow H^*(G_{R'},R'). \]

For any R’-module M’ one has an isomorphism of $H^*(G,R)$ -modules

\[H^*(G,M') \cong H^*(G_{R'},M')\,, \]

where M’ is viewed as a G-module via restriction, as usual. This is compatible with the map of graded algebras $H^*(G,R)\to H^*(G_{R'},R')$ .

4.4 Examples

We list examples of finite flat group schemes over rings; the purpose is to indicate that there is a plethora of such things. We are grateful to Sean Cotner for sharing his expertise on this topic.

When R contains $\mathbb F_p$ , the group algebra of such a group scheme is $R[x]/(x^p-\lambda x)$ for $\lambda \in R$ , and two such group schemes are isomorphic if the scalars $\lambda$ differ by a unit.

Example 4.5. To construct examples of higher rank, consider an $\mathbb F_p$ -algebra R and the additive group scheme $\mathbb G_{a,R}$ over R. Its coordinate algebra is R[x]. Let

\[f\,{:}\, \mathbb G_{a,R} \longrightarrow \mathbb G_{a,R} \]

be a map of group schemes defined by the map on coordinate algebras

\[f^*\,{:}\, R[x] \longrightarrow R[x], \quad f^*(x) = x^{p^n} + a_{p^n-1}x^{p^{n-1}} + \cdots + a_1x \]

for some $n \geqslant 1$ , and elements $a_i$ in R. This is a Hopf algebra map since $f^*(x)$ is primitive by construction and hence defines a map of group schemes. Let G be the scheme theoretic kernel of f. Then G is a finite flat group scheme over R with coordinate algebra $R[x]/(f^*(x))$ , of rank $p^n$ .

For example, take $R = k[[t]]$ , with k a field of positive characteristic p, and $f^*(x) = x^p - t^{p-1}x$ . The fibres of $G:=\operatorname{Ker} f$ over R are different group schemes. The special fibre, at $t=0$ , has the coordinate algebra $k[x]/(x^p)$ , so $G_k \cong \mathbb G_{a(1)}$ , the first Frobenius kernel of $\mathbb G_a$ . On the other hand, the generic fibre $G_{k((t))}$ has coordinate algebra $k((t))[x]/(x^p - t^{p-1}x)$ , which splits into p copies of k((t)). Hence it is the constant finite group scheme $\mathbb Z/p$ .

Another family of commutative finite flat group schemes is given by the (scheme-theoretic) p-torsion subgroups of supersingular abelian varieties defined over $\mathbb F_p$ -algebras; see [Reference Li and OortLO98, Chapter 13]. The next family is again based on $\mathbb G_a$ but consists of noncommutative group schemes.

Example 4.6. Set $R=k[t]/(t^p)$ where k is a field of characteristic p. Consider the cyclic group $\mathbb Z/p$ with a generator $\sigma$ acting on $\mathbb G_{a,R}$ via multiplication by $(t+1)$ :

\[\sigma \circ s \mapsto (t+1)s \]

for any $s \in \mathbb G_{a,R}$ . On the level of the coordinate algebra $R[\mathbb G_{a}] \cong R[x]$ , the R-algebra automorphism $\sigma$ is defined by $\sigma(x) = (t+1)x$ . This action commutes with the Frobenius homomorphism on $\mathbb G_{a,R}$ and, hence, restricts to the Frobenius kernels $\mathbb G_{a(r),R}$ . We consider finite flat group schemes over R which are semi-direct products with respect to this action: $G_r = \mathbb G_{a(r),R} \rtimes \mathbb Z/p$ .

Explicitly, the group algebra $RG_r$ is the smash product: $RG_r \cong R\mathbb G_{a(r)} \# R\mathbb Z/p$ . The coproduct is inherited from that on $R\mathbb G_{a(r)} \cong R[x_1, \ldots, x_r]/(x_1^p, \ldots, x_r^p)$ and $R\mathbb Z/p \cong R[\sigma]/(\sigma^p-1)$ and is cocommutative. The product is twisted with the action of $\sigma$ :

\begin{align*}(1 \# \sigma)(x_1 \# 1) & = (t+1)x_1\#\sigma; \\[6pt](1 \# \sigma)(x_i \# 1) & = x_i\#\sigma, \quad 1 < i \leqslant r.\end{align*}

These are noncommutative and finite flat over R, with $RG_r$ a free R-module of rank $p^{r+1}$ . The smallest one, $G_1$ , cannot be obtained via base change from a group scheme defined over any integral domain. A proof of this claim was communicated to us by Sean Cotner.

6.1 Universal homeomorphisms

A map $\varphi\,{:}\, A\to B$ of (possibly graded) commutative rings is a universal homeomorphism if for each map $A\to A'$ of rings the map on spectra induced by the base change map $A'\to A'\otimes_AB$ , is a homeomorphism:

\[\operatorname{Spec} (A'\otimes_AB) \xrightarrow{\scriptstyle{\sim}} \operatorname{Spec} A'. \]

In particular $\operatorname{Spec} B\to \operatorname{Spec} A$ is a homeomorphism. A universal homeomorphism is universally bijective (in the obvious sense), and the converse holds if $\varphi$ is finite, but not in general: consider the map $\mathbb Z_{(p)}\to \mathbb F_p\times\mathbb Q$ for some prime number p.

Universal bijectivity is easier to track. Indeed, when $\operatorname{Spec} B\to \operatorname{Spec} A$ is surjective, so is the map induced by base change. So the critical point is that the one-to-one property is preserved under base change. It is not hard to verify that the following conditions are equivalent:

1. (i) $\varphi$ is universally bijective;

2. (ii) $\varphi\otimes_A k(\mathfrak{p})\,{:}\, k(\mathfrak{p})\to B\otimes_A k(\mathfrak{p})$ is universally bijective for $\mathfrak{p} \in \operatorname{Spec} A$ ; and

3. (iii) $\operatorname{Spec}(B\otimes_A k)$ is a point for each map $A\to k$ of rings, where k is a field.

This observation is used repeatedly in what follows, as is the following criterion to detect universal homeomorphisms; see [Sta18, Tag 0BRA] for a proof.

1. (i) B is generated as an A-algebra by elements b such that there exists an $n\geqslant 1$ for which $p^n\cdot b$ and $b^{p^n}$ are in $\mathrm{Image}(\varphi)$ ; and

2. (ii) each $a\in \operatorname{Ker}(\varphi)$ is nilpotent.

Then $\varphi$ is a universal homeomorphism. In particular, if A contains a field of positive characteristic p and $\varphi$ is an F-isomorphism, it is a universal homeomorphism.

Remark 6.3. Let G be a finite flat group scheme over R. Any map $f\,{:}\, R'\to R''$ of R-algebras induces a map

\[H^*(f)\,{:}\, H^*(G,R')\otimes_{R'}R'' \longrightarrow H^*(G,R'') \]

of graded R”-algebras. A caveat: usually $H^*(f)$ would stand for the map of rings $H^*(G,R')\to H^*(G,R'')$ , but in what follows the map above, which is obtained from the latter by base change along $R'\to R''$ , is the relevant one.

Consider the collection

$$U(G) := \{f\, \textrm{a map of} \, R-\textrm{algebras} \mid H^*(f) \, \textrm{is a universal homeomorphism}\}.$$

The following statements are easy to verify, given the characterisations of universal homeomorphisms and universal bijective maps.

1. (1) f is in U(G); and

2. (2) $f\otimes_{K}K_\mathfrak{p}$ is in U(G) for each $\mathfrak{p}$ in $\operatorname{Spec} K$ .

Here is a variation of the result above concerning universal bijectivity.

1. (1) $H^*(f)$ is universally bijective; and

2. (2) $H^*(\overline{f}_\mathfrak{q})$ is universally bijective for each $\mathfrak{q}\in\operatorname{Spec} R''$ .

1. (i) There exists an integer $d\geqslant 1$ with

   \[d\cdot H^{\geqslant 1}(G,-)=0 \quad\text{on $\operatorname{Mod} R$}. \]

2. (ii) The R-algebra $H^*(G,R)$ is finitely generated, and hence noetherian, and the $H^*(G,R)$ -module $H^*(G,M)$ is finitely generated whenever M is a finitely generated G-module.

These results are crucial to all that follows. When G is a finite group, one can take d to be the order of the group. The statement is also obvious when $d\cdot R=0$ , so the essential case is when the structure map $\mathbb Z\to R$ is one-to-one.

In the former case $H^{\geqslant 1}(G,-)=0$ on $\operatorname{Mod} R'$ , and there is nothing left to verify; this is by Remark 6.6(i). We may therefore assume that R’ is p-local. Then, again by Remark 6.6(i), one gets

\[p^n\cdot H^{\geqslant 1}(G,-)=0\quad\text{on $\operatorname{Mod} R'$ and some $n\geqslant 1$.} \]

Set $I:= \operatorname{Ker}(f)$ ; the hypothesis is that $I^s=0$ for some $s\geqslant 1$ . It suffices to verify the statement when $s=2$ , and to that end we verify that $H^*(f)$ satisfies the conditions in Remark 6.2. Condition (2) clearly holds as I is nilpotent. The exact sequence

\[0\longrightarrow I\longrightarrow R'\longrightarrow R''\longrightarrow 0 \]

of R’-modules induces an exact sequence

\[H^*(G,R')\otimes_{R'}R'' \xrightarrow{\ H^*(f)\ } H^*(G,R'')\xrightarrow{\ \delta\ } H^{*+1}(G,I). \]

Arguing as in the proof of [Reference Benson, Iyengar, Krause and PevtsovaBIKP24, Lemma 5.2] or [Reference LauLau23, Lemma 8.17] yields that the connecting map $\delta$ is a derivation: For any x,y in $H^*(G,R'')$ , one has

\[\delta(x\cup y) = \delta(x)\cup y + (-1)^{|x|} x\cup \delta(y). \]

Fix $x\in H^{\geqslant 1}(G,R')$ . When $|x|$ and p are both odd, $x^2=0$ by the graded-commutativity of $H^*(G,R')$ , so we can assume $|x|$ is even or $p=2$ . Either way, it follows from the fact that $\delta\,{:}\, H^*(G,R'')\to H^*(G,R'')\to H^{*+1}(G,I)$ is a derivation that $\delta(x^i) = ix^{i-1}\delta(x)$ . Since $p^n\cdot H^{\geqslant 1}(G,-)=0$ , we deduce that $\delta(x^{p^n})=0$ and hence $x^{p^n}$ is in the image of $H^*(f)$ . Thus condition (i) of Remark 6.2 is also satisfied.

A surjective map $R'\to R''$ is a complete intersection if its kernel can be generated by a regular sequence.

For each integer $i\geqslant 1$ set

\[J_i:= \operatorname{Ker} (H^*(G,R')\xrightarrow{\ t^i\ } H^*(G,R')). \]

The ring $H^*(G,R')$ is noetherian, by van der Kallen’s result (Remark 6.6), so the chain of ideals $J_1\subseteq J_2\subseteq \cdots$ stabilises. Fix an integer s such that $J_i=J_{i+1}$ for $i\geqslant s$ . Since the map $R'/t^s\to R'/t$ is in U(G), by Lemma 6.7, it suffices to verify that the map $R'\to R'/t^s$ is in U(G). Thus, replacing t by $t^s$ , we can assume $J_1=J_i$ for $i\geqslant 1$ . The sequence of G-modules

\[0\longrightarrow R'\xrightarrow{\ t\ } R'\xrightarrow{\ f\ } R'/t\to 0 \]

yields the exact sequence of $H^*(G,R')$ -modules

\[0\longrightarrow R'/t\otimes_{R'} H^*(G,R') \xrightarrow{\ H^*(f)\ } H^*(G,R'/t) \xrightarrow{\ \delta\ } J_1\longrightarrow 0. \]

It suffices to verify $\delta(x^{p^n})=0$ for $x\in H^{\geqslant 1}(G,R'/t)$ ; see Remark 6.2.

Let P be a projective resolution of R’ as a $G_{R'}$ -module. As t is not a zero-divisor on R’, the complex $P/tP$ is a projective resolution of $R'/t$ as a $G_{R'/t}$ -module, and the exact sequence above is induced by the exact sequence

\[0\longrightarrow \operatorname{End}_G(P) \longrightarrow \operatorname{End}_G(P) \longrightarrow \operatorname{End}_G(P/tP)\longrightarrow 0. \]

The product on $H^*(G,R')$ and $H^*(G,R'/t)$ is induced by the composition product on the differential graded algebras $\operatorname{End}_G(P)$ and $\operatorname{End}_G(P/tP)$ , respectively, and the map $H^*(f)$ is induced by the map $\operatorname{End}_G(P)\to \operatorname{End}_G(P/tP)$ of dg algebras. Fix a cycle z in $\operatorname{End}_G(P/tP)$ and a pre-image $\tilde{z}$ in $\operatorname{End}_G(P)$ ; then $\delta([z]) = [d(\tilde{z})/t]$ . The graded-commutativity of the ring $H^*(G,R)$ and the fact that $p^n H^{\geqslant 1}(G,R')=0$ yields $\delta([z]^{p^n})= [tw]$ for some $w\in\operatorname{End}_G(P)$ ; see [Reference LauLau23, Lemma 8.21] for details. Since $td(w) = d(tw) =0$ in $\operatorname{End}_G(P)$ and t is not a zero-divisor, $d(w)=0$ . Then

\[t^2[w] = t[tw] = t \delta([z]^{p^n}) =0 \]

since $t\cdot \operatorname{End}_G(P/tP)=0$ . So [w] is in $J_2$ . Since $J_2=J_1$ we deduce that $t[w]=0$ , that is to say, that $\delta([z]^{p^n})=0$ , as desired.

A map $R\to R'$ of commutative rings is said to be essentially of finite type if R’ is a localisation of a finitely generated R-algebra.

1. (1) $H^*(f)$ is universally bijective; and

2. (2) $H^*(f)$ is a universal homeomorphism if f is essentially of finite type.

Proof. (1) Given Lemma 6.5 it suffices to consider the case when $R'=k$ , a field. We claim that then the map $H^*(f)$ is even a universal homeomorphism. To see this, set $\mathfrak{p}:= \operatorname{Ker}(f)$ , a prime ideal in R. The map $f\,{:}\, R\to k$ factors as

\[R\longrightarrow R_\mathfrak{p} \longrightarrow k(\mathfrak{p})\longrightarrow k\,, \]

where the first two maps are the natural ones. Since both maps $R\to R_\mathfrak{p}$ and $k(\mathfrak{p})\to k$ are flat and hence in U(G), it suffices to verify that the map $R_\mathfrak{p} \to k(\mathfrak{p})$ is in U(G). We can thus assume R is local and $f\,{:}\, R\to k$ is the quotient map, with k the residue field of R.

We induct on $\dim R$ to deduce that f is in U(G), and in particular universally bijective. The base case $\dim R=0$ is covered by Lemma 6.7, for then the ideal $\operatorname{Ker}(f)$ is nilpotent.

Suppose $\dim R\geqslant 1$ and set $I:= \sqrt{(0)}$ the nilradical of R; it is a nilpotent ideal, since R is noetherian. Thus the map $R\to R/I$ is in U(G). Since the map $R\to k$ factors as $R\to R/I\to k$ , it suffices to verify that $R/I\to k$ is in U(G). So we can replace R by $R/I$ and assume it is reduced. Then, since $\dim R\geqslant 1$ , the maximal ideal of R is not an associated prime and hence contains an element t that is not a zero-divisor. The map $R\to k$ factors as

\[R\longrightarrow R/tR\longrightarrow k. \]

The map on the left is in U(G), by Lemma 6.8, and the one on the right is in U(G) by the induction hypothesis, since $\dim(R/tR)=\dim R-1$ . We conclude that f is in U(G).

(2) Since f is essentially of finite type, it factors as $R\to R''\to R'$ where $R\to R''$ is the localisation of a finitely generated polynomial ring over R, and $R''\to R'$ is surjective. Since $R\to R''$ is flat it is in U(G). We may thus replace R by R” and suppose f is surjective, and hence a finite map. It follows that the map $H^*(f)$ is also finite, by van der Kallen’s theorem (Remark 6.6(ii)), and hence the induced map $\operatorname{Spec} H^*(f)$ is universally closed. Since $H^*(f)$ is universally bijective, by part (1), we conclude that it is a universal homeomorphism.

7.1 Localisation and local cohomology

Let $\mathcal{T}$ be a triangulated category with suspension $\Sigma$ . Given objects X and Y in $\mathcal{T}$ , consider the graded abelian groups

\[ \operatorname{Hom}_{\mathcal{T}}^*(X,Y)=\bigoplus_{i\in\mathbb Z}\operatorname{Hom}_{\mathcal{T}}(X,\Sigma^i Y) \quad\text{and}\quad\operatorname{End}_{\mathcal{T}}^{*}(X)= \operatorname{Hom}_{\mathcal{T}}^{*}(X,X). \]

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

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

We say that a triangulated category $\mathcal{T}$ is S-linear if for each X in $\mathcal{T}$ there is a homomorphism of graded rings $\phi_X\,{:}\, S\to \operatorname{End}_{\mathcal{T}}^{*}(X)$ such that the induced left and right actions of S on $\operatorname{Hom}_{\mathcal{T}}^{*}(X,Y)$ are compatible in the following sense: for any $r\in S$ and $\alpha\in\operatorname{Hom}^*_\mathcal{T}(X,Y)$ , one has

\[ \phi_Y(r)\alpha=(-1)^{|r||\alpha|}\alpha\phi_X(r). \]

An exact functor $F\,{:}\, \mathcal{T}\to\mathcal{U}$ between S-linear triangulated categories is S-linear if the induced map

\[ \operatorname{Hom}_{\mathcal{T}}^*(X,Y)\longrightarrow \operatorname{Hom}_{\mathcal{U}}^*(FX,FY) \]

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

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

Let $V\subseteq \operatorname{Spec} S$ be a specialisation closed subset. An S-module M is V-torsion if $M_\mathfrak{p}=0$ for all $\mathfrak{p}$ in $\operatorname{Spec} S\setminus V$ . Analogously, an object X in $\mathcal{T}$ is V-torsion if the S-module $\operatorname{Hom}_\mathcal{T}^*(C,X)$ is V-torsion for all $C\in\mathcal{T}^c$ . For an ideal $I\subseteq S$ we use the more common term I-torsion instead of V(I)-torsion. The full subcategory of V-torsion objects

$$\varGamma_{V}\mathcal{T}:=\{X\in\mathcal{T}\mid X \text{ is }\textit{V}-\text{torsion} \}$$

is localising, and the inclusion $\varGamma_{V}\mathcal{T}\subseteq \mathcal{T}$ admits a right adjoint, denoted $\varGamma_{V}$ . Then each object $X\in\mathcal{T}$ fits into a functorial exact triangle

\[\varGamma_V X\longrightarrow X\longrightarrow L_V X\longrightarrow \]

and $L_V$ denotes the corresponding localisation functor.

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

\[\mathcal{T}_\mathfrak{p}:=\{X\in\mathcal{T}\mid X \text{ is $\mathfrak{p}$-local}\} \]

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

\[\varGamma_\mathfrak{p}\mathcal{T}:=\{X\in\mathcal{T}\mid X \text{ is $\mathfrak{p}$-local and $\mathfrak{p}$-torsion} \}. \]

Note that $\varGamma_\mathfrak{p}\mathcal{T}\subseteq\mathcal{T}_\mathfrak{p}\subseteq\mathcal{T}$ are localising subcategories. The inclusion $\mathcal{T}_\mathfrak{p}\to\mathcal{T}$ admits a left adjoint $X\mapsto X_\mathfrak{p}$ , which identifies with the localisation functor with respect to the specialisation closed set $\{\mathfrak{q}\in\operatorname{Spec} S\mid \mathfrak{q}\not\subseteq \mathfrak{p}\}$ , while the inclusion $\varGamma_\mathfrak{p}\mathcal{T}\to\mathcal{T}_\mathfrak{p}$ admits the right adjoint $X\mapsto \varGamma_{V(\mathfrak{p})}X$ . We denote by $\varGamma_\mathfrak{p}\,{:}\,\mathcal{T}\to\varGamma_\mathfrak{p}\mathcal{T}$ the composite of those adjoints; it is the local cohomology functor with respect to $\mathfrak{p}$ .

7.2 Support

Let $\mathcal{T}$ and S be as above. The support of an object $X\in\mathcal{T}$ is

\[\operatorname{supp}_S X:= \{\mathfrak{p}\in\operatorname{Spec} S\mid \varGamma_{\mathfrak{p}}X\ne 0\}. \]

The next result is an important input in § 9.

Lemma 7.1. Let $\mathcal{T}$ and $\mathcal{U}$ be compactly generated S-linear triangulated categories. Let $E\,{:}\, \mathcal{T}\to \mathcal{U}$ be an S-linear exact functor that preserves arbitrary coproducts and compact objects. With F a right adjoint of E, for each $X\in \mathcal{T}$ one has

\[\operatorname{supp}_S E(X) = \operatorname{supp}_S FE(X). \]

Proof. Since E preserves compact objects, its right adjoint F preserves coproducts. It thus follows from [Reference Benson, Iyengar and KrauseBIK12, Proposition 7.2] that the local cohomology functor $\varGamma_{\mathfrak{p}}$ commutes with E and F, for $\mathfrak{p}$ in $\operatorname{Spec} S$ . In particular $\varGamma_\mathfrak{p} E(X)$ is isomorphic to $E(\varGamma_\mathfrak{p} X)$ , and hence in the essential image of E. As F is a right adjoint of E it is conservative on the essential image of E, so one gets the first equivalence below:

\[\varGamma_\mathfrak{p} E(X)=0 \iff F(\varGamma_\mathfrak{p} E(X))=0 \iff \varGamma_\mathfrak{p} FE(X) =0. \]

The second equivalence holds by the observation above. This is as desired.

7.3 Tensor triangulated categories

From now on $\mathcal{T}$ is a rigidly-compactly generated triangulated category, with product $\otimes$ and unit $\mathbb 1 = \mathbb 1_{\mathcal{T}}$ . We assume that there is a central action of S via a ring homomorphism $S\to\operatorname{Hom}^*_{\mathcal{T}}(\mathbb 1,\mathbb 1)$ . By [Reference Benson, Iyengar and KrauseBIK08, Theorem 8.2], this implies that for each specialisation closed subset V of $\operatorname{Spec} S$ there are natural isomorphisms

\[\varGamma_VX \cong X\otimes \varGamma_V\mathbb 1 \quad\text{and}\quad L_VX \cong X\otimes L_V\mathbb 1. \]

In particular $\varGamma_{\mathfrak{p}} X\cong X\otimes \varGamma_{\mathfrak{p}}\mathbb 1$ for each $\mathfrak{p}\in\operatorname{Spec} S$ .

For any class of objects $\mathcal{X}$ in $\mathcal{T}$ we write $\operatorname{Loc}^\otimes(\mathcal{X})$ for the smallest localising tensor ideal of $\mathcal{T}$ containing $\mathcal{X}$ . The following local-to-global result for tensor triangulated categories is [Reference Benson, Iyengar and KrauseBIK11b, Theorem 3.6].

Theorem 7.2. For each object X in $\mathcal{T}$ we have

\[\operatorname{Loc}^\otimes(X)=\operatorname{Loc}^\otimes(\{\varGamma_{\mathfrak{p}}X\mid \mathfrak{p}\in\operatorname{Spec} S\}). \]

8.1 Reduction to the local case

Fix $\mathfrak{p}\in \operatorname{Spec} R$ . For any G-complex X, set

\[\lambda(X):= X\otimes_R R_\mathfrak{p} \]

viewed as a complex of $G_\mathfrak{p}$ -modules, where $G_\mathfrak{p}$ is shorthand for the group scheme $G_{R_\mathfrak{p}}$ over $R_\mathfrak{p}$ . As $\lambda$ preserves coproducts and compact objects it has a right adjoint, by Brown representability, and we get an adjoint pair of coproduct-preserving functors

Proof. The functor $\lambda$ identifies with the localisation functor

\[ -\otimes_R R_\mathfrak{p}\,{:}\, \operatorname{rep}(G,R) \longrightarrow \operatorname{rep}(G_\mathfrak{p},R_\mathfrak{p}) \]

when restricted to compact objects; see Proposition 2.12. Thus for compact objects X,Y in $\operatorname{Rep}(G, R)$ the unit $Y\to \lambda_r\lambda(Y)$ induces an isomorphism

\[ \operatorname{Hom}_{\mathbf{K}(G)}(X,Y)_\mathfrak{p} \cong\operatorname{Hom}_{\mathbf{K}(G_\mathfrak{p})}(\lambda X,\lambda Y)\cong \operatorname{Hom}_{\mathbf{K}(G)}( X,\lambda_r\lambda (Y)). \]

This yields the first assertion since $\lambda$ and $\lambda_r$ preserve coproducts. The second assertion is an immediate consequence.

8.2 The local case

For the proof of Theorem 8.1 the key is to analyse the situation where $(R,\mathfrak{m},k)$ is a noetherian commutative local ring, with maximal ideal $\mathfrak{m}$ and residue field k. One has an adjoint pair $(\pi,\pi_r)$ of functors

(8.1)

where $\pi(X):= X\otimes_Rk$ . The right adjoint exists because $\pi$ preserves coproducts; since $\pi$ preserves compact objects, it follows that $\pi_r$ also preserves coproducts.

Observe that $\pi$ is a tensor functor. Because of rigidity this yields the following projection formula. For X in $\operatorname{Rep}(G, R)$ and Y in $\operatorname{Rep}(G_k,k)$ , one has

\[X\otimes_R \pi_r (Y) \cong \pi_r(\pi(X)\otimes_k Y). \]

One can verify this directly, but this is a general fact about tensor functors between rigidly-compactly generated tensor triangulated categories; see [Reference Balmer, Dell’Ambrogio and SandersBDS16, Theorem 1.3].

Lemma 8.3. For any X in $\operatorname{Rep}(G, R)$ there is an equality

\[\operatorname{Loc}^{\otimes}(\pi_r\pi(X)) = \operatorname{Loc}^{\otimes}(\varGamma_{\mathfrak{m}}X). \]

Proof. Let $\mathbf{i}_{G_k} k$ denote the injective resolution of the trivial $G_k$ -module k, which is the unit of the tensor triangulated category $\mathbf{K}(\operatorname{Proj} G_k)$ . We use the fact that $\pi$ is a tensor functor, and that, in particular,

\[\pi(\mathbf{i} R) = (\mathbf{i} R)\otimes_R k \cong \mathbf{i}_{G_k}k. \]

Because $\operatorname{Rep}(G, R)$ is rigid with unit $\mathbf{i} R$ , one has

\[\varGamma_{\mathfrak{m}}X \cong X\otimes_R \varGamma_{\mathfrak{m}}(\mathbf{i} R). \]

The projection formula for the pair $(\pi,\pi_r)$ and the fact that $\mathbf{i}_{G_k}k$ is the unit of $\operatorname{Rep}(G_k,k)$ yields isomorphisms

\[ \pi_r\pi(X) \cong \pi_r(\pi(X)\otimes_k \mathbf{i}_{G_k}k) \cong X\otimes_R \pi_r(\mathbf{i}_{G_k}k). \]

Thus it suffices to verify that

\[\operatorname{Loc}^{\otimes}(\pi_r(\mathbf{i}_{G_k}k)) = \operatorname{Loc}^{\otimes}(\varGamma_{\mathfrak{m}}(\mathbf{i} R)). \]

One can easily verify that $\pi_r(\mathbf{i}_{G_k}k)$ is $\mathfrak{m}$ -torsion, so one has

\[\pi_r(\mathbf{i}_{G_k}k) \cong \varGamma_{\mathfrak{m}} \pi_r(\mathbf{i}_{G_k}k) \cong \pi_r(\mathbf{i}_{G_k}k)\otimes_R \varGamma_{\mathfrak{m}}(\mathbf{i} R). \]

Thus $\pi_r(\mathbf{i}_{G_k}k)$ is in the tensor ideal localising subcategory generated by $\varGamma_{\mathfrak{m}}(\mathbf{i} R)$ .

To complete the proof, it suffices to check that $\varGamma_{\mathfrak{m}}(\mathbf{i} R)$ is in the localising (and in particular the tensor ideal localising) subcategory generated by $\pi_r(\mathbf{i}_{G_k}k)$ . From [Reference Benson, Iyengar and KrauseBIK11a, Proposition 2.11], one gets that

\[\varGamma_{\mathfrak{m}}(\mathbf{i} R)\in\operatorname{Loc}({(\mathbf{i} R)}/\!\!/{\mathfrak{m}}). \]

Here ${(\mathbf{i} R)}/\!\!/{\mathfrak{m}}$ denotes a Koszul object of $\mathfrak{m}$ on $\mathbf{i} R$ ; see [Reference Benson, Iyengar and KrauseBIK11a, § 2.5]. Thus it suffices to verify that

\[{(\mathbf{i} R)}/\!\!/{\mathfrak{m}}\in\operatorname{Thick}(\pi_r(\mathbf{i}_{G_k}k)). \]

The key is that this can be checked in the derived category of G. To that end consider the diagram of pairs of adjoint functors.

It is straightforward to see that the composition of left adjoints from the top left to the bottom right commute:

\[\mathbf{q}_{G_k}\circ \pi \cong \mathbf{q}(-)\otimes^{\mathrm L}_R k. \]

It follows that the composition of right adjoints going the other way commute as well, which allows us to deduce that

\[\pi_r(\mathbf{i}_{G_k}k) \cong \mathbf{i} k. \]

Moreover, as $\mathbf{i}$ is an R-linear functor, one has

\[{(\mathbf{i} R)}/\!\!/{\mathfrak{m}} \cong \mathbf{i} ({R}/\!\!/{\mathfrak{m}}). \]

In $\mathbf{D}(\operatorname{Mod} R)$ the object k finitely builds ${R}/\!\!/{\mathfrak{m}}$ since the cohomology of ${R}/\!\!/{\mathfrak{m}}$ has finite length. The action of G on R and k factors through the augmentation $RG\to R$ . Thus the object k finitely builds ${R}/\!\!/{\mathfrak{m}}$ in $\mathbf{D}(\operatorname{Mod} G)$ . We conclude that

\[{(\mathbf{i} R)}/\!\!/{\mathfrak{m}} \cong \mathbf{i} ({R}/\!\!/{\mathfrak{m}})\in\operatorname{Thick}(\mathbf{i} k)=\operatorname{Thick}(\pi_r(\mathbf{i}_{G_k}k)). \]

This completes the proof that $\pi_r\pi(X)$ and $\varGamma_{\mathfrak{m}}X$ generate the same tensor ideal localising subcategories.

Proof of Theorem 8.1. (1) $\Rightarrow$ (2) This implication is straightforward to verify once one observes that the functor $X\mapsto X\otimes_R k(\mathfrak{p})$ respects coproducts, and the tensor product. For X,Y in $\operatorname{Rep}(G, R)$ there is a natural isomorphism

\[(X\otimes_RY)_{k(\mathfrak{p})} \cong X_{k(\mathfrak{p})} \otimes_{k(\mathfrak{p})} Y_{k(\mathfrak{p})} \quad\text{in}\quad \operatorname{Rep}(G_{k(\mathfrak{p})},k(\mathfrak{p})). \]

(2) $\Rightarrow$ (1) Assume first that R is local and consider the adjoint pair (8.1). Suppose that $\pi(X)\in\operatorname{Loc}^\otimes(\pi(Y))$ . It follows that $\pi_r\pi(X)$ is in $\operatorname{Loc}^\otimes(Y)$ . Indeed, one has $\operatorname{Loc}^\otimes(\pi(Y)) = \operatorname{Loc}(\operatorname{rep}(G_{k(\mathfrak{p})},k(\mathfrak{p})) \otimes \pi(Y))$ , so the projection formula implies

\begin{align*}\pi_r\pi(X) & \in \operatorname{Loc}(\pi_r(\operatorname{rep}(G_{k(\mathfrak{p})},k(\mathfrak{p})) \otimes \pi(Y))) \\& = \operatorname{Loc}(\pi_r(\operatorname{rep}(G_{k(\mathfrak{p})},k(\mathfrak{p}))) \otimes Y) \\& \subseteq \operatorname{Loc}^{\otimes}(Y).\end{align*}

Then Lemma 8.3 implies $\varGamma_\mathfrak{m} X\in\operatorname{Loc}^\otimes(Y)$ . For general R and $\mathfrak{p}$ in $\operatorname{Spec} R$ we have $X_{k(p)}=(X_\mathfrak{p})_{k(p)}$ and therefore $X_{k(\mathfrak{p})}\in\operatorname{Loc}^\otimes(Y_{k(\mathfrak{p})})$ implies

\[\varGamma_\mathfrak{p} X=\varGamma_\mathfrak{p}(X_\mathfrak{p})\in\operatorname{Loc}^\otimes(Y_\mathfrak{p})\subseteq\operatorname{Loc}^\otimes(Y). \]

Here we use Lemma 8.2 to reduce to the local case, where $\mathfrak{p}$ identifies with the maximal ideal of $R_\mathfrak{p}$ . Using Theorem 7.2 we conclude that X is in $\operatorname{Loc}^\otimes(Y)$ .

9.1 Support and fibres

As we will explain in the proof of Theorem 9.1 below, Theorem 8.1 combined with the main theorem of [Reference Benson, Iyengar, Krause and PevtsovaBIKP18] yields a stratification of $\operatorname{Rep}(G, R)$ in terms of subsets of the space

\[\bigsqcup_{\mathfrak{p}\in\operatorname{Spec} R} \operatorname{Spec} S(\mathfrak{p})\,, \]

where to each X in $\operatorname{Rep}(G, R)$ we associate the subset

\[\bigsqcup_{\mathfrak{p}\in\operatorname{Spec} R} \operatorname{supp}_{S(\mathfrak{p})} {X}_{k(\mathfrak{p})}. \]

The task is to relate this to $\operatorname{supp}_S X$ , viewed as a subset of $\operatorname{Spec} S$ .

To that end consider the structure map $\eta\,{:}\, R\to S$ , which induces a map

\[\eta^{a}\,{:}\, \operatorname{Spec} S \longrightarrow \operatorname{Spec} R. \]

The fibre of this map over $\mathfrak{p}$ is

\[(\eta^{a})^{-1}(\mathfrak{p}) = \operatorname{Spec}(S\otimes_R k(\mathfrak{p}))\,, \]

which we identify with a subset of $\operatorname{Spec} S$ in the usual way. In the following we use the map $\kappa_{\mathfrak{p}}\,{:}\, S\otimes_R k(\mathfrak{p})\to S(\mathfrak{p})$ which induces the map

\begin{equation*}\kappa^a_{\mathfrak{p}}\,{:}\, \operatorname{Spec} S(\mathfrak{p}) \longrightarrow \operatorname{Spec} (S\otimes_R k(\mathfrak{p})).\end{equation*}

Theorem 6.1 yields that this map is a bijection; this fact is critical to all that follows. The result below tracks the behavior of supports as we pass to the fibres.

Lemma 9.2. For $\mathfrak{p}$ in $\operatorname{Spec} R$ and X in $\operatorname{Rep}(G, R)$ there is an equality

\[\kappa^a_{\mathfrak{p}}(\operatorname{supp}_{S(\mathfrak{p})}{X}_{k(\mathfrak{p})}) = \operatorname{supp}_SX \cap (\eta^{a})^{-1}(\mathfrak{p}). \]

Proof. By Lemma 8.2, we can assume that $(R,\mathfrak{m},k)$ is a local ring and $\mathfrak{p}=\mathfrak{m}$ , the maximal ideal of R. Set $S_k:= S\otimes_Rk =S/\mathfrak{m} S$ . Via the map $\kappa_\mathfrak{m}\,{:}\, S_k\to S(\mathfrak{m})$ , the $S(\mathfrak{m})$ -action on $\mathbf{K}(\operatorname{Proj} G_k)$ induces an $S_k$ -action. Applying [Reference Benson, Iyengar and KrauseBIK12, Corollary 7.8(1)] to the identity functor on $\mathbf{K}(\operatorname{Proj} G_k)$ yields the first equality below:

\begin{align*} \kappa^a_{\mathfrak{m}}(\operatorname{supp}_{S(\mathfrak{m})}X_k) &= \operatorname{supp}_{S_k} X_k \\ &= \operatorname{supp}_{S} X_k \\ &=\operatorname{supp}_{S}(\pi_r\pi X) \\ &=\operatorname{supp}_{S}(\varGamma_\mathfrak{m} X) \\ &= \operatorname{supp}_{S}(X)\cap V(\mathfrak{m} S) \\ &=\operatorname{supp}_{S}(X)\cap (\eta^a)^{-1}(\mathfrak{m}).\end{align*}

The same observation applied to the map $S\to S_k$ yields the second equality. Now consider the adjunction (8.1). Then Lemma 7.1 gives the third equality, and the next one follows from Lemma 8.3. Now observe that $\varGamma_\mathfrak{m}=\varGamma_{V(\mathfrak{m})}=\varGamma_{V(\mathfrak{m} S)}$ by [Reference Benson, Iyengar and KrauseBIK12, Proposition 7.5]. Then [Reference Benson, Iyengar and KrauseBIK08, Theorem 5.6] yields the fifth equality, and the last one is clear since $V(\mathfrak{m} S)= (\eta^a)^{-1}(\mathfrak{m})$ .

Proof of Theorem 9.1. The main task is to verify that when X,Y are objects in $\operatorname{Rep}(G, R)$ with $\operatorname{supp}_SX\subseteq \operatorname{supp}_SY$ , one has $X\in \operatorname{Loc}^{\otimes}(Y)$ . By Theorem 6.1, for each $\mathfrak{p}$ in $\operatorname{Spec} R$ the map $\kappa_\mathfrak{p}^a$ is a bijection, so Lemma 9.2 yields an inclusion

\[\operatorname{supp}_{S(\mathfrak{p})}{X}_{k(\mathfrak{p})}\subseteq \operatorname{supp}_{S(\mathfrak{p})}{Y}_{k(\mathfrak{p})}. \]

Since ${G}_{k(\mathfrak{p})}$ is a finite flat group scheme algebra over $k(\mathfrak{p})$ , the triangulated category $\mathbf{K}(\operatorname{Proj} G_{k(\mathfrak{p})})$ is stratified by the action of its cohomology algebra, $S(\mathfrak{p})$ ; this is the main result of [Reference Benson, Iyengar, Krause and PevtsovaBIKP18]. Thus the inclusion above implies

\[{X}_{k(\mathfrak{p})}\in \operatorname{Loc}^{\otimes}({Y}_{k(\mathfrak{p})}). \]

This holds for each $\mathfrak{p}$ in $\operatorname{Spec} R$ , so we can apply Theorem 8.1 to deduce that X is in $\operatorname{Loc}^{\otimes}(Y)$ as desired.

It remains to observe that $\operatorname{supp}_S \mathbb 1= \operatorname{Spec} S$ . This follows from Lemma 9.2, for ${\mathbb 1}_{k(\mathfrak{p})}$ is the unit of $\mathbf{K}(\operatorname{Proj} G_{k(\mathfrak{p})})$ and its support is $\operatorname{Spec} S(\mathfrak{p})$ .

A standard argument (see [Reference Benson, Iyengar and KrauseBIK11a, Theorem 6.1]) yields the last part of the statement, concerning the Balmer spectrum of $\operatorname{rep}(G,R)$ , for this identifies with the category of compact objects in $\operatorname{Rep}(G,R)$ .

10.1 Applications

We record some consequences of Theorems 9.1 and 10.1. The one below is by [Reference Benson, Iyengar and KrauseBIK12, Corollary 9.9].

Corollary 10.3. Let R and G be as before. The map sending a localising subcategory $\mathsf{S}$ of $\operatorname{Rep}(G, R)$ to its right orthogonal $\mathsf{S}^\perp$ induces a bijection

\[\bigg\{\begin{gathered} \text{tensor closed localising}\\ \text{subcategories of $\operatorname{Rep}(G, R)$}\end{gathered}\;\bigg\} \xrightarrow{\scriptstyle{\sim}} \bigg\{\begin{gathered} \text{Hom closed colocalising}\\ \text{subcategories of $\operatorname{Rep}(G, R)$}\end{gathered}\bigg\}. \]

The inverse map sends a colocalising subcategory $\mathsf{S}$ to its left orthogonal $^\perp\mathsf{S}$ .

One also gets (co)stratification of the big stable category of lattices.

Corollary 10.4. The tensor triangulated category $\operatorname{StMod}^{\mathrm{lf}} (G,R)$ is stratified and costratified by the action of S, and its support equals $\operatorname{Proj} S$ . Consequently

\[\operatorname{Spc}(\operatorname{stmod}(G,R)) \xrightarrow{\scriptstyle{\sim}} \operatorname{Proj} H^*(G,R). \]

Given an object M in $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} G,R)$ we write $\operatorname{supp}_GM$ and $\mathrm{cosupp}_GM$ for its support and cosupport, respectively, computed using the action of $H^*(G,R)$ on $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} G,R)$ ; see § 7. Here is a noteworthy consequence of the fact that $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} G,R)$ is stratified by the action of $H^*(G,R)$ : for M,N in $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} G,R)$ one has equalities

(10.1) \begin{align} \operatorname{supp}_G(M\otimes_RN)&= \operatorname{supp}_G M\cap \operatorname{supp}_G N,\nonumber \\[-10pt]&\\[-6pt] \mathrm{cosupp}_G\operatorname{\mathcal{H}\!\!\;\mathit{om}}(M,N) &= \operatorname{supp}_G M\cap\mathrm{cosupp}_G N.\nonumber\end{align}

The formula for the support of the tensor product can be deduced from Corollary 10.4 as in the proof of [Reference Benson, Iyengar and KrauseBIK11b, Theorem 11.1]. The cosupport of the function objects follows from [Reference Benson, Iyengar and KrauseBIK12, Theorem 9.5].

Given a subgroup scheme $H\leqslant G$ , restriction induces a map of graded R-algebras $H^*(G,R)\to H^*(H,R)$ ; let $\mathrm{res}^*$ denote the induced map on spectra. For any M in $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} G,R)$ there are equalities

(10.2) \begin{align}\operatorname{supp}_H(M\downarrow_H) &=(\mathrm{res}^*_{G,H})^{-1}(\operatorname{supp}_GM), \nonumber \\[-10pt]&\\[-6pt]\mathrm{cosupp}_H(M\downarrow_H) &=(\mathrm{res}^*_{G,H})^{-1}(\mathrm{cosupp}_GM).\nonumber\end{align}

There are further applications, including a proof of the telescope conjecture for $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj} G,R)$ and $\operatorname{StMod}^{\mathrm{lf}}(G,R)$ , and versions of Theorems 9.1 and 10.1 for the Frobenius category $\operatorname{Mod}^{\mathrm{lf}}(G,R)$ . The statements and proofs are analagous to those in [Reference Benson, Iyengar and KrauseBIK11a, Reference Benson, Iyengar and KrauseBIK12], which deal with the case when R is a field.

11.1 Thick and localising ideals

We continue with a description of thick and localising ideals of the stable categories $\operatorname{StMod}^{\mathrm{lf}}(G,R)$ and $\operatorname{StMod}(G,R)$ .

The ideal $(t)\subset RG$ is isomorphic to R as an RG-module, so the exact sequence

\[0\longrightarrow (t) \longrightarrow RG \longrightarrow R\longrightarrow 0 \]

defines an element in $H^1(G,R)=\operatorname{Ext}^1_{RG}(R,R)$ ; we denote it as $\eta$ . Since $H^*(G,R)$ is a commutative R-algebra, one gets a morphism of R-algebras

\[R[\eta]\longrightarrow H^*(G,R). \]

A simple computation yields the following.