# Lattices over finite group schemes and stratification

Tobias Barthel, Dave Benson, Srikanth B. Iyengar,
Henning Krause, and Julia Pevtsova  Tobias Barthel
Max Planck Institute for Mathematics
Vivatsgasse 7
53111 Bonn
Germany  Dave Benson
Institute of Mathematics
University of Aberdeen
King’s College
Aberdeen AB24 3UE
Scotland U.K.  Srikanth B. Iyengar
Department of Mathematics
University of Utah
Salt Lake City, UT 84112
U.S.A.  Henning Krause
Fakultät für Mathematik
Universität Bielefeld
33501 Bielefeld
Germany.  Julia Pevtsova
Department of Mathematics
University of Washington
Seattle, WA 98195
U.S.A.

###### Abstract.

This work concerns representations of a finite flat group scheme $G$, defined
over a noetherian commutative ring $R$. The focus is on lattices, namely,
finitely generated $G$-modules that are projective as $R$-modules, and on the
full subcategory of all $G$-modules projective over $R$ generated by the
lattices. The stable category of such $G$-modules is a rigidly-compactly
generated, tensor triangulated category. The main result is that this stable
category is stratified and costratified by the natural action of the
cohomology ring of $G$. Applications include formulas for computing the
support and cosupport of tensor products and the module of homomorphisms, and
a classification of the thick ideals in the stable category of lattices.

###### Key words and phrases:

Costratification, finite group, finite flat group scheme, integral
representation, lattice, stable module category, stratification

###### 2020 Mathematics Subject Classification:

16G30 (primary); 18G80, 20C10, 20J06 (secondary)

###### Contents

  1. 1 Introduction
  2. 2 Frobenius algebras
  3. 3 Fibrewise detection
  4. 4 Finite flat group schemes
  5. 5 Enhancements
  6. 6 Cohomology and fibres
  7. 7 Local cohomology and support
  8. 8 Building and fibres
  9. 9 Stratification
  10. 10 Costratification and other applications
  11. 11 An example

##  1\. Introduction

In recent years, there has been a surge of interest in the structure and
properties of tensor triangulated categories arising in algebra, algebraic
geometry, and topology. In the algebraic context the starting point is the
groundbreaking contributions by Hopkins [29] and Neeman [37, 39]. Since then,
the subject has expanded into modular representation theory for finite groups,
finite group schemes over fields, Lie algebras, rings of differential
operators, and equivariant homotopy theory, among others. A central problem is
the classification of thick ideals in essentially small tensor triangulated
categories, along with the more challenging task of identifying localising and
colocalising ideals in the associated big tensor triangulated categories.

In this work we provide such classification for the big category of
representations of finite groups and finite group schemes over commutative
noetherian rings. The tensor triangulated category we introduce serves as a
unifying framework for two contexts that have been studied extensively.
Specialised to the trivial group it is the derived category of a commutative
noetherian ring, and the corresponding classification is due to Neeman. On the
other end of the spectrum, when the coefficient ring $R$ is a field, one gets
the stable module category of a finite group or a finite group scheme. Here is
our main theorem.

######  Theorem A (Theorems 9.1, 10.1).

For $G$ a finite flat group scheme over a commutative noetherian ring $R$, the
tensor triangulated category $\operatorname{Rep}(G,R)$ is stratified and
costratified by the canonical action of the cohomology ring $H^{*}(G,R)$.

The category $\operatorname{Rep}(G,R)$ arises as a suitable ind-completion of
the bounded derived category $\operatorname{rep}(G,R)$ of _lattices_ :
representations of $G$ that are finitely generated and projective over $R$. We
provide two complementary constructions of this category: In Sections 2 and 4,
we use Frobenius exact categories to give a homological model of
$\operatorname{Rep}(G,R)$. In Section 5 we study $\operatorname{Rep}(G,R)$ as
the ind-completion in the $\infty$-categorical sense. The ring $H^{*}(G,R)$,
which identifies with $\operatorname{Ext}^{*}_{RG}(R,R)$, is a graded-
commutative noetherian $R$-algebra, by recent work of van der Kallen [44].

Group representations over rings are pervasive, encompassing representations
over rings of integers of algebraic number fields; see for example [22,
Chapter 3]. In this context the representations of interest are the torsion-
free ones, that is to say, lattices, and these have become objects of study in
their own right [40, 42, 45]. Lattices appear already in the work of Brauer
and Nesbitt [20] in the context of integral representations of finite groups
as an intermediary between ordinary and modular representations. While
classification of indecomposable modules in such categories is usually out of
reach, Theorem A provides structural information on the category
$\operatorname{Rep}(G,R)$. Namely, the stratification by $H^{*}(G,R)$ that it
gives can be restated as a natural bijection between the nonzero localising
ideals of $\operatorname{Rep}(G,R)$, the nonzero colocalising ideals of
$\operatorname{Rep}(G,R)$, and subsets of $\operatorname{Spec}H^{*}(G,R)$:

$\left\\{\begin{gathered}\text{localising ideals}\\\ \text{of
$\operatorname{Rep}(G,R)$}\end{gathered}\,\right\\}\;\stackrel{{\scriptstyle\sim}}{{\longleftrightarrow}}\;\left\\{\begin{gathered}\text{subsets
of}\\\
\text{$\operatorname{Spec}H^{*}(G,R)$}\end{gathered}\,\right\\}\,\;\stackrel{{\scriptstyle\sim}}{{\longleftrightarrow}}\;\left\\{\begin{gathered}\text{colocalising
ideals}\\\ \text{of $\operatorname{Rep}(G,R)$}\end{gathered}\,\right\\}\,.$

The general results in [14, 16] allow us to deduce several other desirable
consequences from Theorem A; see Section 10 for details. These include the
calculation of the Balmer spectrum [1] of $\operatorname{rep}(G,R)$:

(1.1) $\operatorname{Spc}(\operatorname{rep}(G,R))\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\operatorname{Spec}H^{*}(G,R)\,.$

This isomorphism encodes a natural bijection between the nonzero thick ideals
of $\operatorname{rep}(G,R)$ and specialisation closed subsets of
$\operatorname{Spec}H^{*}(G,R)$. This is the restriction of the bijection
above for the big category $\operatorname{Rep}(G,R)$ to the small category
$\operatorname{rep}(G,R)$. In view of the fact that classification of
indecomposable lattices is generally a hopeless task, the parametrisation of
thick tensor ideals offers a partial solution; it classifies $G$-lattices over
$R$ up to homological operations: shifts, extensions, retracts, and tensoring.

### Historical remarks and related works

As already mentioned, the case $G=1$ in Theorem A yields the classification
for the big derived category of $R$-modules and is due to Neeman [37]. This
was preceded by the classification of thick subcategories for the bounded
derived category of lattices over $R$ by Hopkins [29], following the work of
Devinatz, Hopkins, and Smith [25] in stable homotopy theory. Studying
representations of a finite group $G$ over a field $k$, Benson, Carlson, and
Rickard [10] classified thick tensor ideals in $\operatorname{stmod}kG$, the
small stable category of $kG$, which is closely related to
$\operatorname{rep}(G,k)$ as we explain in Section 2. Continuing with the
small categories, the isomorphism (1.1) for $G$ a finite group scheme, and
$R=k$ still a field, was established in [26]. The first result on classifying
localising ideals in a big category is due to Benson, Iyengar, and Krause
[14]; their classification recovers the case $\operatorname{Rep}(G,k)$ of
Theorem A where $k$ is a field and $G$ is a finite group. This was generalised
by the last four authors to finite group schemes in [18].

Moving to more recent history, and in fact to the motivating development for
this work, we arrive at a theorem of Lau [33] who established the calculation
(1.1) for $G$ a finite group and $R$ any commutative noetherian ring. Hence,
Theorem A generalises Lau’s work in two directions: from finite groups to
finite group schemes, and from $\operatorname{rep}(G,R)$ to its ind-completion
$\operatorname{Rep}(G,R)$.Keeping in line with earlier work on representations
over a field, both of these passages are far from routine, and require new
ideas and methods.

For a start, while the objects in $\operatorname{rep}(G,R)$ can be described
in simple terms and have been studied for long, as far as we know the category
$\operatorname{Rep}(G,R)$ has not appeared in the literature before, and even
the objects in it are mysterious, when $R$ is not a regular ring. Our first
task is to establish basic structural properties of $\operatorname{Rep}(G,R)$,
and the paragraph below outlines some of our key findings.

### The category $\operatorname{Rep}(G,R)$

Throughout $G$ is a finite flat group scheme over a noetherian commutative
ring $R$. The group algebra $RG$ is a cocommutative Hopf $R$-algebra that is a
finitely generated and projective as an $R$-module. We consider
representations of $G$ over $R$ which can be identified with $RG$-modules.

Let $\operatorname{Mod}(G,R)$ be the category of $RG$-modules that are
projective as $R$-modules, and $\operatorname{mod}(G,R)$ its full subcategory
consisting of the finitely generated modules, in other words, lattices. We
write $\mathbf{K}(\operatorname{Proj}G,R)$ for the subcategory of
$\mathbf{K}(\operatorname{Proj}G)$ consisting of complexes that are
$K$-projective when restricted to $R$. For our purposes the more pertinent
object is a subcategory of $\mathbf{K}(\operatorname{Proj}G,R)$ whose objects
satisfy a local finiteness condition; this is the category
$\operatorname{Rep}(G,R)$. When $R$ is regular, $\operatorname{Rep}(G,R)$
coincides with $\mathbf{K}(\operatorname{Proj}G)$, and this case is discussed
in [5, 17]. However, these categories differ for general rings. In Section 11
we discuss an example that illustrates this, based on the representation
theory of $\mathbb{Z}/2\times\mathbb{Z}/2$.

The key features of these homotopy categories and the associated stable
categories are described in Sections 2–4, including a recollement relating
them; see Corollary 2.15. The constructions apply more generally to any
Frobenius $R$-algebra and the results of Section 2 are stated in that greater
generality.

By way of motivation, we state fundamental properties satisfied by
$\operatorname{Rep}(G,R)$ that make it a suitable homotopy category to
consider:

  1. (1)

$\operatorname{Rep}(G,R)$ is a rigidly-compactly generated symmetric tensor
triangulated category with product $\otimes_{R}$, unit $R$, and
$\operatorname{End}^{*}(R)\cong H^{*}(G,R)$.

  2. (2)

The compact objects in $\operatorname{Rep}(G,R)$ identify with

$\operatorname{rep}(G,R)=\mathbf{D}^{b}(\operatorname{mod}(G,R))\,,$

the bounded derived category of lattices.

Inspired by work in [33], Barthel [4] proposed a homotopy theoretic model for
$\operatorname{Rep}(G,R)$, proved that it satisfies the properties above, and
that it is stratified by $H^{*}(G,R)$, when $R$ is a Dedekind domain.
Subsequently, this result was generalised to all regular rings $R$, from a
homotopical [5] and a homological [17] perspective, but the methodology of
neither work readily generalised to arbitrary noetherian coefficients. The
current work provides a model that unifies all these frameworks.

In Section 5 we describe how one can enhance the structure on
$\operatorname{Rep}(G,R)$ so it becomes a symmetric monoidal stable
$\infty$-category. This allows us to compare the model of this paper to the
$\infty$-category of derived representations of a finite group $G$ studied in
[4, 5]. The latter is a natural candidate suggested by the $\infty$-category
context and that extends well to other ring spectra; see [6]. We prove that
the two categories are equivalent in the strongest possible sense, thereby
reconciling the homological with the homotopical approach; see Proposition
5.8. This section serves as motivation and as a bridge from modular
representation theory to higher algebra; the rest of the paper is independent
of it.

Next we describe other key ingredients that go into the proof of Theorem A.

### Reduction to fibres

As noted above, $\operatorname{Rep}(G,R)$ is a rigidly-compactly generated
symmetric tensor triangulated category. Heuristically, it can be viewed as a
sheaf of categories over $\operatorname{Spec}R$ with fibres
$\operatorname{Rep}(G_{k(\mathfrak{p})},k(\mathfrak{p}))$, for $\mathfrak{p}$
ranging over the prime ideals of $R$ and $G_{k(\mathfrak{p})}$ denoting the
group scheme base-changed along $R\to k(\mathfrak{p})$. This perspective is
crucial for Lau’s approach to computing the Balmer spectrum of
$\mathbf{D}^{b}(\operatorname{mod}(G,R))$ for a finite group $G$, and the
approach to stratification in [17, 28].

Let $X_{k(\mathfrak{p})}\colonequals X\otimes_{R}k(\mathfrak{p})$ denote the
base change of an object $X\in\operatorname{Rep}(G,R)$ along $R\to
k(\mathfrak{p})$. The results below illustrate how the global structure of
$\operatorname{Rep}(G,R)$ is controlled by its fibres.

######  Theorem B (Theorem 8.1).

Let $G$ be a finite flat group scheme over a noetherian commutative ring $R$.
The following statements hold.

  1. (1)

(Detection) For any $X\in\operatorname{Rep}(G,R)$ one has

$X=0\quad\iff\quad\text{$X_{k(\mathfrak{p})}=0$ for all
$\mathfrak{p}\in\operatorname{Spec}R$.}$

  2. (2)

(Building) The following conditions are equivalent for
$X,Y\in\operatorname{Rep}(G,R)$:

    1. (a)

$X\in\operatorname{Loc}^{\otimes}(Y)$ in $\operatorname{Rep}(G,R)$;

    2. (b)

${X}_{k(\mathfrak{p})}\in\operatorname{Loc}^{\otimes}({Y}_{k(\mathfrak{p})})$
in $\operatorname{Rep}(G_{k(\mathfrak{p})},k(\mathfrak{p}))$ for each
$\mathfrak{p}\in\operatorname{Spec}R$.

As we explain in Section 3, ‘detection’ holds for any Frobenius algebra; see
Proposition 3.6. Part (2), given in Theorem 8.1 is a generalisation of [17]
from regular to arbitrary noetherian coefficients; it immediately implies (1),
but from a conceptual point of view it is useful to separate these two
conditions.

Another crucial result is that the spectrum of the cohomology ring
$H^{*}(G,R)$ is also controlled by the fibres.

######  Theorem C (Theorem 6.3).

For each $\mathfrak{p}$ in $\operatorname{Spec}R$ base change along $R\to
k(\mathfrak{p})$ induces a homeomorphism

$\operatorname{Spec}H^{*}(G,k(\mathfrak{p}))\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\operatorname{Spec}(H^{*}(G,R)\otimes_{R}k(\mathfrak{p}))\,.$

The proof of this theorem occupies Section 6. The ideas that go into it are
rather different from those involved in proving Theorem B. They build on
earlier work of Benson and Habegger [12], and Lau [33] that establish the
result for finite groups. The critical new inputs are recent results of van
der Kallen [44], concerning torsion and finite generation of the cohomology
ring $H^{*}(G,R)$ for a finite flat group scheme $G$ over $R$; see Section
6.8.

Theorems B and C allow us to employ the general theory of [13, 14] for
describing localising ideals in this context; we recall the relevant notions
in Section 7. One of the key insights of [17] is that stratification for
regular rings may be checked fibrewise, thereby reducing the problem to the
case of field coefficients. After the preparation we have done so far, this
now extends verbatim to arbitrary noetherian rings and finite group schemes,
leading to the stratification part of Theorem A. The costratification is then
a formal consequence, by combining the abstract bootstrap theorem of [7] with
fibrewise detection; see Sections 9 and 10. The results of van der Kallen
mentioned above are again vital inputs in the arguments.

######  Acknowledgements.

It is a pleasure to thank Greg Stevenson for helpful comments on a preliminary
version of this paper. Part of this work was done during the Trimester Program
“Spectral Methods in Algebra, Geometry, and Topology” at the Hausdorff
Institute in Bonn. It is a pleasure to thank HIM for hospitality and for
funding by the Deutsche Forschungsgemeinschaft under Excellence Strategy
EXC-2047/1-390685813. The last four authors thank the American Institute for
Mathematics for its hospitality and the support through the SQuaRE program. TB
is supported by the European Research Council (ERC) under Horizon Europe
(grant No. 101042990) and the Max Planck Institute for Mathematics, SBI was
partly supported by NSF grant DMS-2001368, HK was partly supported by the
Deutsche Forschungsgemeinschaft (SFB-TRR 358/1 2023 - 491392403), and JP was
partly supported by NSF grants DMS-1901854, 2200832 and the Brian and Tiffinie
Pang faculty fellowship.

##  2\. Frobenius algebras

In this section we introduce the various module categories and homotopy
categories of complexes that arise in our work. While our focus is on finite
group schemes, some basic constructions and results apply more generally to
Frobenius algebras. This is the content of this section. The methods build on
those in [31]; we take this work, and also [32], as our standard references
for the material below.

Throughout $R$ is a commutative ring (not necessarily noetherian) and $A$ a
_finite projective $R$-algebra_; that is to say, an $R$-algebra that is
finitely generated and projective as an $R$-module. As usual
$\operatorname{Mod}A$ stands for the category of all left $A$-modules, but we
drop the term ‘left’ in the sequel. We write $\operatorname{Proj}A$ for the
full subcategory of projective $A$-modules. Our focus is on the subcategory of
$\operatorname{Mod}A$ consisting of modules that are projective as
$R$-modules; we denote it $\operatorname{Mod}(A,R)$. We write
$\operatorname{mod}(A,R)$ for the full subcategory of
$\operatorname{Mod}(A,R)$ consisting of modules that are finitely generated
over $R$.

######  Lemma 2.1.

In $\operatorname{Mod}A$ the subcategory $\operatorname{Mod}(A,R)$ is
extension closed and so carries a natural exact structure, given by the short
exact sequences of $A$-modules.∎

Let $\operatorname{Proj}(\operatorname{Mod}(A,R))$ be the full subcategory of
projective objects in $\operatorname{Mod}(A,R)$, and
$\operatorname{Inj}(\operatorname{Mod}(A,R))$ the full subcategory of
injective objects. The duality functor is an equivalence:

(2.2)
$D\colonequals\operatorname{Hom}_{R}(-,R)\colon(\operatorname{mod}(A,R))^{\mathrm{op}}\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\operatorname{mod}(A^{\mathrm{op}},R)\,.$

For all $M\in\operatorname{mod}(A,R)$ and
$N\in\operatorname{mod}(A^{\mathrm{op}},R)$ one has

$\operatorname{Hom}_{A}(M,DN)\cong\operatorname{Hom}_{A^{\mathrm{op}}}(N,DM)\,.$

It follows that the exact category $\operatorname{mod}(A,R)$ has enough
projective and enough injective objects. We set

$\omega_{A/R}\coloneqq\operatorname{Hom}_{R}(A,R)\,;$

this is the _dualising bimodule_ of the $R$-algebra $A$.

For any $A$-module $M$ let $\operatorname{add}M$ denote the full subcategory
of $\operatorname{Mod}A$ consisting of direct summands of finite direct sums
of copies of $M$, and $\mathrm{Add}\,M$ denotes the corresponding subcategory
where all direct sums are permitted. When $M$ is in $\operatorname{Mod}(A,R)$
so is every module in $\mathrm{Add}\,M$.

######  Lemma 2.3.

The exact category $\operatorname{Mod}(A,R)$ has enough projective and enough
injective objects, with

$\operatorname{Proj}(\operatorname{Mod}(A,R))=\mathrm{Add}\,A=\operatorname{Proj}A\quad\text{and}\quad\operatorname{Inj}(\operatorname{Mod}(A,R))=\mathrm{Add}\,\omega_{A/R}\,.$

###### Proof.

We view $\operatorname{Proj}R$ as an exact category; in it all objects are
projective and injective. The forgetful functor
$u\colon\operatorname{Mod}(A,R)\to\operatorname{Proj}R$ is exact. Thus its
left adjoint $u_{\lambda}=A\otimes_{R}-$ preserves projectivity, while its
right adjoint $u_{\rho}=\operatorname{Hom}_{R}(A,-)$ preserves injectivity.
For $M\in\operatorname{Mod}(A,R)$ the counit $\varepsilon_{M}\colon
u_{\lambda}u(M)\to M$ provides an epimorphism such that $u_{\lambda}u(M)$ is
projective, while the unit $\eta_{M}\colon M\to u_{\rho}u(M)$ provides a
monomorphism such that $u_{\rho}u(M)$ is injective. The map
$u(\varepsilon_{M})$ is a split epimorphism, while $u(\eta_{M})$ is a split
monomorphism. Thus $\varepsilon_{M}$ and $\eta_{M}$ provide exact sequences in
$\operatorname{Mod}(A,R)$.

The equality
$\operatorname{Proj}(\operatorname{Mod}(A,R))=\operatorname{Proj}A$ is clear
since $u_{\lambda}$ maps the category of free $R$-modules onto the category of
free $A$-modules.

Given that $u_{\rho}$ preserves injectives and that it also preserves all
direct sums, since $A$ is finitely generated as an $R$-module, it follows that
the objects in $\operatorname{Inj}(\operatorname{Mod}(A,R))$ are precisely the
direct summands of direct sums of copies of $\omega_{A/R}$. ∎

We obtain a pair of functors
$\Sigma,\Omega\colon\operatorname{Mod}(A,R)\to\operatorname{Mod}(A,R)$ by
completing, for any module $M$ in $\operatorname{Mod}(A,R)$, the morphisms
$\eta_{M}$ and $\varepsilon_{M}$ to exact sequences

(2.4) $\begin{gathered}0\longrightarrow
M\xrightarrow{\eta_{M}}u_{\rho}u(M)\longrightarrow\Sigma M\longrightarrow 0\\\
0\longrightarrow\Omega M\longrightarrow
u_{\lambda}u(M)\xrightarrow{\varepsilon_{M}}M\longrightarrow
0\,.\end{gathered}$

Iterating this process yields an injective and projective resolution of $M$,
which we denote $\mathbf{i}M$ and $\mathbf{p}M$, respectively.

### Derived categories

We write $\mathbf{D}(\mathcal{A})$ to denote the derived category of an exact
category $\mathcal{A}$; see [32, Section 4.1]. The main examples for us are
the exact category $\operatorname{Mod}(A,R)$ and its subcategory
$\operatorname{mod}(A,R)$.

######  Proposition 2.5.

The inclusion $\operatorname{Mod}(A,R)\subseteq\operatorname{Mod}A$ of exact
categories induces a quotient functor

${\mathbf{D}(\operatorname{Mod}(A,R))}$${\mathbf{D}(\operatorname{Mod}A)}$

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

###### Proof.

Consider the localising subcategory of $\mathbf{D}(\operatorname{Mod}(A,R))$
generated by $A$. Its right orthogonal in
$\mathbf{D}(\operatorname{Mod}(A,R))$ is the full subcategory of complexes
that are acyclic in $\operatorname{Mod}A$. It follows that the right adjoint
of the inclusion is a quotient functor that identifies with the functor
$\mathbf{D}(\operatorname{Mod}(A,R))\to\mathbf{D}(\operatorname{Mod}A)$
induced by the inclusion $\operatorname{Mod}(A,R)\to\operatorname{Mod}A$. ∎

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 2.6.

The inclusions
$\operatorname{mod}(A,R)\subseteq\operatorname{Mod}(A,R)\subseteq\operatorname{Mod}A$
of exact categories induces 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 for any exact sequence

$0\longrightarrow X\longrightarrow Y\longrightarrow Z\longrightarrow
0\qquad\text{with}\qquad Z\in\operatorname{mod}(A,R)$

there is an epimorphism $A\otimes_{R}Z\to Z$ in $\operatorname{mod}(A,R)$ that
factors through $Y\to Z$; see [32, 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 to 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 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
$\Omega_{n}\colonequals\operatorname{Coker}(P_{n+1}\to P_{n})$. Then $X$ is
quasi-isomorphic to the complex

$X(n)\colon\quad\cdots\longrightarrow
0\longrightarrow\Omega_{n}\longrightarrow
P_{n-1}\longrightarrow\cdots\longrightarrow P_{0}\longrightarrow
0\longrightarrow\cdots\ $

for each $n\geq s\colonequals\max\\{i\mid H_{i}(X)\neq 0\\}$. The $A$-module
$\Omega_{s}$ is perfect over $R$ and 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
illustrates the difference between the derived categories of
$\operatorname{Mod}(A,R)$ and $\operatorname{Mod}A$.

######  Example 2.7.

Let $k$ be a field and choose $A\colonequals k[t]/(t^{2})\equalscolon R$. 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

$\cdots\xrightarrow{\ t\ }A\xrightarrow{\ t\ }A\xrightarrow{\ t\
}A\xrightarrow{\ t\ }\cdots\,.$

### 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.

######  Lemma 2.8.

Let $R$ be a commutative ring and $A$ a finite projective $R$-algebra. The
following conditions are equivalent.

  1. (1)

There is an equality $\operatorname{add}\omega_{A/R}=\operatorname{add}A$.

  2. (2)

The exact category $\operatorname{mod}(A,R)$ is a Frobenius category.

  3. (3)

The exact category $\operatorname{Mod}(A,R)$ is a Frobenius category.

###### Proof.

The equivalence (1) $\Leftrightarrow$ (2) is clear since
$\operatorname{add}\omega_{A/R}$ equals the category of injective objects in
$\operatorname{mod}(A,R)$, while $\operatorname{add}A$ equals the category of
projectives.

(1) $\Leftrightarrow$ (3) follows from Lemma 2.3. ∎

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 [41, §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 3.3. Evidently $A$ is a Frobenius
$R$-algebra if and only if so is $A^{\mathrm{op}}$, because of the duality
(2.2).

### 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 [32, Section 3.3]. We denote $\Sigma$
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

$\underline{\operatorname{Hom}}_{\mathcal{A}}(M,N)\colonequals\operatorname{Hom}_{\mathcal{A}}(M,N)/\mathrm{PHom}_{\mathcal{A}}(M,N)\,,$

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

$\operatorname{stmod}(A,R)\colonequals\operatorname{St}(\operatorname{mod}(A,R))\quad\text{and}\quad\operatorname{StMod}(A,R)\colonequals\operatorname{St}(\operatorname{Mod}(A,R))\,,$

and consider these categories with their natural triangulated structure.

### 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
$\mathbf{q}\colon\mathbf{K}(\operatorname{Proj}\Lambda)\to\mathbf{D}(\operatorname{Mod}\Lambda)$
which inverts the quasi-isomorphisms. Taking projective resolutions yields a
left adjoint 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 $\mathbf{i}M$ and $\mathbf{p}M$; see (2.4). 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 2.9.

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

$\mathbf{Ac}(\operatorname{Proj}A,R)\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\operatorname{StMod}(A,R)\,.$

###### Proof.

Each $X$ in $\mathbf{Ac}(\operatorname{Proj}A,R)$ is contractible when
restricted to $R$. Thus the assignment
$X\mapsto\operatorname{Coker}(d^{-1}_{X})$ is well-defined. The quasi-inverse
takes $M\in\operatorname{Mod}(A,R)$ to a complete resolution $\mathbf{t}M$ in
$\operatorname{Proj}A$. ∎

######  Proposition 2.10.

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

${\mathbf{Ac}(\operatorname{Proj}A,R)}$${\mathbf{K}(\operatorname{Proj}A,R)}$${\mathbf{D}(\operatorname{Mod}A)}$$\scriptstyle{\mathbf{t}}$$\scriptstyle{\mathbf{p}}$

###### Proof.

The left adjoint
$\mathbf{p}\colon\mathbf{D}(\operatorname{Mod}A)\to\mathbf{K}(\operatorname{Proj}A)$
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
$\mathbf{Ac}(\operatorname{Proj}A,R)$. ∎

### 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 Lemma
2.16, but not always; see Section 11. Set

$\operatorname{StMod}^{\mathrm{lf}}(A,R)\colonequals\operatorname{St}(\operatorname{Mod}^{\mathrm{lf}}(A,R))\,.$

Here are the key properties of this triangulated category.

######  Proposition 2.11.

Let $A$ be a Frobenius $R$-algebra. The triangulated category
$\operatorname{StMod}^{\mathrm{lf}}(A,R)$ is compactly generated and its
subcategory of compact objects identifies with the idempotent completion of
$\operatorname{stmod}(A,R)$.

###### Proof.

The canonical functor
$\operatorname{Mod}^{\mathrm{lf}}(A,R)\to\operatorname{StMod}^{\mathrm{lf}}(A,R)$
preserves coproducts and hence $\operatorname{stmod}(A,R)$ consists of compact
objects. They generate $\operatorname{StMod}^{\mathrm{lf}}(A,R)$ as a
triangulated category by definition so an arbitrary compact object is a direct
summand of an object in $\operatorname{StMod}^{\mathrm{lf}}(A,R)$; see [32,
Proposition 3.4.15]. ∎

######  2.12.

The category $\operatorname{stmod}(A,R)$ is idempotent complete when $R$ is a
complete local ring. This need not be the case for a general $R$.

Indeed, the local ring
$A\colonequals\mathbb{C}[x,y]_{(x,y)}/(x^{2}-y^{2}(y+1))$, viewed as an
algebra over $R\colonequals\mathbb{C}[x]_{(x)}$, 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 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 [32, Lemma 6.2.12]
for a proof.

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

Set $\mathcal{A}\colonequals\operatorname{Mod}(A,R)$. 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

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

where $\mathbf{K}^{+}(\mathcal{C})$ denotes the complexes $X$ with $X^{n}=0$
for $n\ll 0$, and the condition $H^{n}X=0$ 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.

######  Lemma 2.13.

Let $A$ be a Frobenius $R$-algebra. For an object $X$ in
$\mathbf{K}(\operatorname{Proj}A)$ the following conditions are equivalent.

  1. (1)

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

  2. (2)

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

  3. (3)

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

Moreover, such an $X$ is perfect over $R$.

###### Proof.

(1)$\Rightarrow$(2) Let $X$ be compact and $K$-projective over $R$. The
restriction functor
$\mathbf{K}(\operatorname{Proj}A)\to\mathbf{K}(\operatorname{Proj}R)$
preserves compactness, since its right adjoint $\operatorname{Hom}_{R}(A,-)$
preserves coproducts. Thus $X$ is compact over $R$. Since it is also
$K$-projective over $R$ we deduce that it is perfect over $R$. So $X$ belongs
to $\mathbf{K}^{+}(\operatorname{proj}A)$ by [38, Proposition 7.6], and
$H^{i}(X)=0$ for all $i\geq n$ and some $n\in\mathbb{Z}$. We truncate in
degree $n$ and write $X$ as an extension $0\to X^{\prime}\to X\to
X^{\prime\prime}\to 0$ such that $X^{\prime}$ is concentrated in degrees
$i\geq n$ and $X^{\prime\prime}$ is concentrated in degrees $i<n$. Clearly,
$X^{\prime\prime}$ is perfect and hence $X^{\prime}$ is $K$-projective over
$R$. Splicing $X^{\prime\prime}$ together with a projective resolution of
$M\coloneqq\operatorname{Ker}d^{n}_{X}$, yields a complex over $R$ that is
acyclic and $K$-projective, and hence contractible. Thus
$X^{\prime\prime}=\mathbf{i}M$ with $M\in\operatorname{mod}(A,R)$.

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

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

by [31, Lemma 2.1], so $X^{\prime}$ 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 2.13. 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 2.16. 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 Section 11.

######  Proposition 2.14.

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

$\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj}A,R)^{c}\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\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
$\mathbf{K}^{+}(\operatorname{proj}A)\xrightarrow{\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\mathbf{D}^{+}(\operatorname{mod}(A,R))$
which restricts to an equivalence

$\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj}A,R)^{c}=\mathbf{K}^{+,b}(\operatorname{proj}A)\xrightarrow{\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\mathbf{D}^{b}(\operatorname{mod}(A,R)).\qed$

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

######  Corollary 2.15.

Let $A$ be a Frobenius $R$-algebra. Then the canonical functor
$\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj}A,R)\to\mathbf{D}(\operatorname{Mod}A)$
induces a recollement

${\operatorname{StMod}^{\mathrm{lf}}(A,R)}$${\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj}A,R)}$${\mathbf{D}(\operatorname{Mod}A)}$$\scriptstyle{\mathbf{t}}$$\scriptstyle{\mathbf{p}}$

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

$\mathbf{D}^{b}(\operatorname{mod}(A,R))/\mathbf{D}^{\mathrm{perf}}(A)\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\operatorname{stmod}(A,R)\,.$

###### Proof.

Recall that $\mathbf{p}$ is left adjoint to
$\mathbf{q}\colon\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj}A,R)\to\mathbf{D}(\operatorname{Mod}A)$.
The latter preserves coproducts so it admits a right adjoint, by Brown
representability. This yields the right half of the recollement; the left half
is a consequence.

Indeed, we have the full subcategory of acyclic complexes in
$\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj}A,R)$ on the left, that
identifies with a localising subcategory of $\operatorname{StMod}(A,R)$ by
Lemma 2.9. 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 to 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 2.14. This yields the
last assertion, since for any Frobenius category $\mathcal{A}$ with full
subcategory of projectives $\mathcal{P}$ we have

$\mathbf{D}^{b}(\mathcal{A})/\mathbf{D}^{b}(\mathcal{P})\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\operatorname{St}\mathcal{A}\,,$

for example by [32, Proposition 4.4.18]. ∎

### 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 [21, Section 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, [30, Section 3].

######  Proposition 2.16.

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

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

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

###### Proof.

By [38, Proposition 7.14], the triangulated category
$\mathbf{K}(\operatorname{Proj}A)$ is compactly generated. From Lemma 2.13 and
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 2.10 and Corollary 2.15 and using Lemma
2.9, 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 2.5 once one observes that for any complex
in $\operatorname{Mod}(A,R)$ being acyclic in $\operatorname{Mod}A$ implies
contractible in $\operatorname{Mod}R$ and therefore also acyclic in
$\operatorname{Mod}(A,R)$. ∎

##  3\. Fibrewise detection

Let $R$ be a noetherian commutative ring and $A$ a finite projective
$R$-algebra. In this section we prove that certain homological properties of
$A$-modules can be detected locally. To that end, for each $\mathfrak{p}$ in
$\operatorname{Spec}R$, let $k(\mathfrak{p})$ denote the residue field of $R$
at $\mathfrak{p}$, and for any $A$-complex $X$, set

${X}_{k(\mathfrak{p})}\colonequals X\otimes_{R}k(\mathfrak{p})\,.$

The assignment $X\mapsto{X}_{k(\mathfrak{p})}$ induces exact _fibre functors_

$\displaystyle\operatorname{Mod}(A,R)$
$\displaystyle\longrightarrow\operatorname{Mod}_{k(\mathfrak{p})}A_{k(\mathfrak{p})}=\operatorname{Mod}A_{k(\mathfrak{p})}$
$\displaystyle\mathbf{K}(\operatorname{Proj}A,R)$
$\displaystyle\longrightarrow\mathbf{K}(\operatorname{Proj}A_{k(\mathfrak{p})},k(\mathfrak{p}))=\mathbf{K}(\operatorname{Proj}A_{k(\mathfrak{p})})\,.$

We prove that these functors are _jointly conservative_ when $\mathfrak{p}$
runs through $\operatorname{Spec}R$; see Proposition 3.6. This requires some
preparations.

######  Lemma 3.1.

Let $R$ be a noetherian commutative ring, $A$ a flat $R$-algebra, and $F$ an
$A$-module that is flat over $R$. If for each
$\mathfrak{p}\in\operatorname{Spec}R$ the $A_{k(\mathfrak{p})}$-module
$F_{k(\mathfrak{p})}$ is flat, then the $A$-module $F$ is flat.

###### Proof.

The argument follows Deligne’s proof of [24, Prop. 5.5.23].

We begin with a general observation. For any ideal $I$ of $R$ we write

$V(I)=\\{\mathfrak{p}\in\operatorname{Spec}R\mid I\subseteq\mathfrak{p}\\}$

for the corresponding closed subset. Then for $V=V(I)$ and any $A$-module $M$
its submodule of sections

$\varGamma_{V}M=\\{x\in M\mid I^{n}x=0\text{ for some }n>0\\}$

equals the maximal $A$-submodule $N\subseteq M$ _supported on $V$_, that is
$N_{\mathfrak{p}}=0$ for $\mathfrak{p}\not\in V$. Note that
$M=\operatorname*{colim}_{V}\varGamma_{V}M$ where $V$ runs through all closed
subsets of $\operatorname{Spec}R$.

We prove by noetherian induction on closed subsets $V$ of
$\operatorname{Spec}R$ that for each (right) $A$-module $M$ supported on $V$
one has

$\operatorname{Tor}^{A}_{i}(M,F)=0\quad\text{for $i\geq 1$.}$

Say $V=V(I)$ for some reduced ideal $I$ in $R$. For each integer $n\geq 0$ the
subset

$M_{n}\colonequals\\{x\in M\mid I^{n}x=0\\}$

is an $A$-submodule of $M$, and since $M$ is supported on $V$ we get that
$M=\bigcup_{n\geq 0}M_{n}$. Since
$\operatorname{Tor}^{A}_{i}(M,F)=\operatorname*{colim}_{n}\operatorname{Tor}^{A}_{i}(M_{n},F)$,
it suffices to verify the desired vanishing for $M$ such that $I^{n}M=0$. Then
$M$ has a finite filtration

$0=I^{n}M\subseteq I^{n-1}M\subseteq\cdots\subseteq IM\subseteq M$

by $A$-submodules and each subquotient is annihilated by $I$. Thus we can
replace $M$ by one of these subquotients and assume $IM=0$. Let $\mathfrak{p}$
be a prime ideal in $R$ minimal in $V(I)$, and consider the exact sequence

(3.2) $0\longrightarrow M^{\prime}\longrightarrow M\xrightarrow{\ \iota\
}M_{k(\mathfrak{p})}\longrightarrow M^{\prime\prime}\longrightarrow 0.$

Observe that $\iota$ is $A$-linear so the exact sequence is one of
$A$-modules. Since $I$ is reduced and $\mathfrak{p}$ is minimal over $I$, the
map $\iota_{\mathfrak{p}}$ is bijective, and hence
$M^{\prime}_{\mathfrak{p}}=0=M^{\prime\prime}_{\mathfrak{p}}$. In particular
$M^{\prime}$ and $M^{\prime\prime}$ can be expressed as filtered colimits of
$A$-submodules supported on proper closed subsets $W\subset V$. Thus the
induction hypothesis yields

$\operatorname{Tor}^{A}_{i}(M^{\prime},F)=0=\operatorname{Tor}^{A}_{i}(M^{\prime\prime},F)\quad\text{for
$i\geq 1$.}$

Moreover since $A$ and $F$ are flat over $R$ one has the first isomorphism
below

$\operatorname{Tor}^{A}_{i}(M_{k(\mathfrak{p})},F)\cong\operatorname{Tor}^{A_{k(\mathfrak{p})}}_{i}(M_{k(\mathfrak{p})},F_{k(\mathfrak{p})})=0\quad\text{for
$i\geq 1$.}$

The equality holds by the hypotheses on the flatness of $F_{k(\mathfrak{p})}$.
These computations and (3.2) yield the desired vanishing. ∎

A ring $\Lambda$ is called _self-injective_ if injective and projective
$\Lambda$-modules coincide. Then the opposite ring $\Lambda^{\mathrm{op}}$ is
also self-injective and the ring is artinian; cf. [32, Example 3.3.4] for
details. The importance of the result below is that, for algebras finite
projective over a noetherian commutative ring, the Frobenius property is
detected on fibres. In [17] such an $A$ is said to be _fibrewise self-
injective_ ; in this work, we prefer to call them Frobenius algebras, in view
of Lemma 2.8.

######  Lemma 3.3.

Let $R$ be a noetherian commutative ring. A finite projective $R$-algebra $A$
is Frobenius if and only if ${A}_{k(\mathfrak{p})}$ is self-injective for each
$\mathfrak{p}$ in $\operatorname{supp}_{R}A$.

###### Proof.

Since $A$ is finite projective over $R$, for each $\mathfrak{p}$ in
$\operatorname{Spec}R$ one has an isomorphism of
${A}_{k(\mathfrak{p})}$-bimodules:

$\omega_{A/R}\otimes_{R}k(\mathfrak{p})\cong\omega_{{{A}_{k(\mathfrak{p})}}/k(\mathfrak{p})}\,.$

An algebra $\Lambda$ that is finite dimensional over a field $k$ is self-
injective if and only if
$\operatorname{add}\omega_{\Lambda/k}=\operatorname{add}\Lambda$. Thus the
isomorphism above and Lemma 2.8 yield that when $A$ is Frobenius, the ring
${A}_{k(\mathfrak{p})}$ is self-injective.

Suppose ${A}_{k(\mathfrak{p})}$ is self-injective for each $\mathfrak{p}$ in
$\operatorname{Spec}R$. Then the ${A}_{k(\mathfrak{p})}$-module
$\omega_{A/R}\otimes_{R}k(\mathfrak{p})$ is projective for each
$\mathfrak{p}$, by the isomorphism above. For finitely presented modules
flatness and projectivity agree, so Lemma 3.1 yields that $\omega_{A/R}$ is a
projective $A$-module, that is to say, it is in $\operatorname{add}A$. Using
duality, we deduce that $A$ is in $\operatorname{add}\omega_{A/R}$. Thus the
$R$-algebra $A$ is Frobenius, by Lemma 2.8. ∎

######  Lemma 3.4.

Let $R$ be a noetherian commutative ring, $A$ a Frobenius $R$-algebra, and $M$
an $A$-module that is projective over $R$. If $M$ is flat over $A$, then so
are the $A$-modules $\Sigma M$ and $\Omega M$.

###### Proof.

Clearly, when the $A$-module $M$ is finitely generated, if it is projective,
so are $\Sigma M$ and $\Omega M$. A module is flat if and only if it is a
filtered colimit of finitely generated projective modules. Thus the assertion
follows from the fact that the functors $\Sigma$ and $\Omega$ defined via the
exact sequences (2.4) preserve filtered colimits. ∎

######  Proposition 3.5.

Let $R$ be a noetherian commutative ring and $A$ a Frobenius $R$-algebra. For
each $M\in\operatorname{Mod}(A,R)$ the following conditions are equivalent.

  1. (1)

$M$ is a flat $A$-module.

  2. (2)

$M$ is a projective $A$-module.

  3. (3)

$M_{k(\mathfrak{p})}$ is a projective $A_{k(\mathfrak{p})}$-module for each
$\mathfrak{p}\in\operatorname{Spec}R$.

  4. (4)

$M_{k(\mathfrak{p})}$ is a flat $A_{k(\mathfrak{p})}$-module for each
$\mathfrak{p}\in\operatorname{Spec}R$.

###### Proof.

(1) $\Rightarrow$ (2) It follows from Lemma 3.4 that the flat modules in
$\operatorname{Mod}(A,R)$ are closed under the suspension $\Sigma$ and its
stable inverse $\Sigma^{-}\colonequals\Omega$. For each $n\in\mathbb{Z}$ we
obtain an exact sequence $0\to\Sigma^{n}M\to P_{n}\to\Sigma^{n+1}M\to 0$ such
that $P_{n}$ is projective, by setting

$\Sigma^{n}M\colonequals\begin{cases}\Sigma(\Sigma^{n-1}M)&n>0,\\\ M&n=0,\\\
\Sigma^{-}(\Sigma^{n+1}M)&n<0.\end{cases}$

These yield an exact sequence

$0\longrightarrow\bigoplus_{n\in\mathbb{Z}}\Sigma^{n}M\longrightarrow\bigoplus_{n\in\mathbb{Z}}P_{n}\longrightarrow\bigoplus_{n\in\mathbb{Z}}\Sigma^{n}M\longrightarrow
0\,.$

Thus the module $\bar{M}=\bigoplus_{n\in\mathbb{Z}}\Sigma^{n}M$ is flat and
periodic. Then it follows from [11, Theorem 2.5] that $\bar{M}$ projective,
and in particular $M=\Sigma^{0}M$ is projective.

(2) $\Rightarrow$ (3) $\Rightarrow$ (4) Clear.

(4) $\Rightarrow$ (1) This follows from Lemma 3.1. ∎

######  Proposition 3.6.

Let $R$ be a noetherian commutative ring and $A$ a Frobenius $R$-algebra. For
an object $X$ in $\mathbf{K}(\operatorname{Proj}A,R)$ we have $X=0$ if and
only if $X_{k(\mathfrak{p})}=0$ in
$\mathbf{K}(\operatorname{Proj}A_{k(\mathfrak{p})})$ for all
$\mathfrak{p}\in\operatorname{Spec}R$.

###### Proof.

The ‘only if’ direction is clear. So suppose $X_{k(\mathfrak{p})}=0$ for all
$\mathfrak{p}\in\operatorname{Spec}R$. Applying the functor $\mathbf{p}$ from
Proposition 2.10 we obtain an object $\mathbf{p}X$ in the derived category
$\mathbf{D}(\operatorname{Mod}A)$ such that
$\mathbf{p}X\otimes_{R}k(\mathfrak{p})$ is acyclic for all
$\mathfrak{p}\in\operatorname{Spec}R$. Considering $\mathbf{p}X$ as an object
in $\mathbf{D}(\operatorname{Mod}R)$ and applying a result of Neeman [37,
Lemma 2.12] we conclude that $\mathbf{p}(X)$ is acyclic. Hence $X$ is homotopy
equivalent to an object in $\mathbf{Ac}(\operatorname{Proj}A,R)$. Since the
latter is equivalent to $\operatorname{StMod}(A,R)$ by Lemma 2.9, the
statement now follows from Proposition 3.5. ∎

##  4\. Finite flat group schemes

From now on $R$ is a noetherian commutative ring and $G$ a finite flat group
scheme over $R$. The coordinate algebra $R[G]$ is a finitely generated
projective $R$-module and a commutative Hopf $R$-algebra. The group algebra
$RG=\operatorname{Hom}_{R}(R[G],R)$ is then a finitely generated projective
$R$-module and a cocommutative Hopf algebra. Hence, $RG$ is a Frobenius
$R$-algebra, so the results of the previous sections apply. Examples of finite
flat groups schemes are given at the end of this section.

There is an equivalence of categories between $R$-linear representations of
$G$ and $RG$-modules. Since $RG$ is a Hopf algebra there is more structure
associated to its various module categories. This is the topic of this
section. In what follows, by a $G$-module we mean an $RG$-module, and we write
$\operatorname{mod}(G,R)$ for $\operatorname{mod}(RG,R)$,
$\operatorname{Proj}G$ instead of $\operatorname{Proj}RG$, and so forth.

### Tensor structure

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

######  Lemma 4.1.

Let $P,Q$ be $G$-modules. If the $G$-module $P$ is projective and $Q$ is
projective over $R$, then the $G$-module $P\otimes_{R}Q$ is projective.

###### Proof.

This follows from the standard adjunction isomorphism

$\operatorname{Hom}_{G}(P\otimes_{R}Q,-)\cong\operatorname{Hom}_{G}(P,\operatorname{Hom}_{R}(Q,-))\,.\qed$

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

######  Lemma 4.2.

The triangulated category $\mathbf{K}(\operatorname{Proj}G)$ is tensor
triangulated, with product $\otimes_{R}$ and unit the injective resolution
$\mathbf{i}R$ in $\operatorname{mod}(G,R)$.

###### Proof.

We have seen that $\otimes_{R}$ provides a tensor product on
$\mathbf{K}(\operatorname{Proj}G)$. It remains to verify the assertion about
the unit; for this see [19, Proposition 5.3], whose proof extends to finite
flat group schemes, and also [17, Lemma 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 2.13. 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 $\mathds{1}$. We assume that $\mathds{1}$ 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

(4.3) $\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(X,\mathds{1})\otimes
Y\longrightarrow\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(X,Y)$

and $X$ is _rigid_ if this map is an isomorphism for all $Y$. Since
$\mathds{1}$ 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)

(4.3) 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
finite flat group scheme over $R$.

######  Proposition 4.4.

The tensor product $\otimes_{R}$ induces on
$\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj}G,R)$ a tensor-triangulated
structure that is rigidly-compactly generated.

###### 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
$X\otimes_{R}Y$ is again in
$\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj}G,R)$. For the rigidity we need
to verify that $X\otimes_{R}Y$ is also compact and that (4.3) 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 2.14. For $M,N$ in
$\mathbf{D}^{b}(\operatorname{mod}(G,R))$ it is clear that the complex
$M\otimes^{\mathrm{L}}_{R}N$ 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 $R$. ∎

### Cohomology

With $G$ acting trivially on $R$, we consider the cohomology algebra
$S\colonequals\operatorname{Ext}^{*}_{G}(R,R)$ 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

$H^{*}(G,M)\colonequals\operatorname{Ext}_{G}^{*}(R,M)\,.$

This is a graded module over $S$, under cup-products; see [9]. Friedlander and
Suslin [27] prove that $S$ is finitely generated—equivalently, noetherian—when
$R$ is a field; this result has been extended by van der Kallen [44] to cover
any noetherian commutative ring $R$; see 6.8.

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

$H^{*}(G,R)\longrightarrow H^{*}(G_{R^{\prime}},R^{\prime})\,.$

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

$H^{*}(G,M^{\prime})\cong H^{*}(G_{R^{\prime}},M^{\prime})\,,$

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

### 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.

######  Example 4.5.

For $R$ a noetherian commutative ring (satisfying mild conditions) Oort and
Tate [43] classified finite group schemes with coordinate algebra
$R[x]/(x^{p})$. When $R$ is a complete local ring with residue field of
characteristic $p$, there is such a group scheme for any factorisation $p=ac$
in $R$. Two such group schemes are isomorphic if the pairs $(a,c)$ and
$(a^{\prime},c^{\prime})$ differ by multiplication by a unit in $R$.

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 $\lambda$’s differ by a unit.

######  Example 4.6.

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

$f\colon\mathbb{G}_{a,R}\longrightarrow\mathbb{G}_{a,R}$

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

$f^{*}\colon R[x]\longrightarrow R[x],\quad
f^{*}(x)=x^{p^{n}}+a_{p^{n}-1}x^{p^{n-1}}+\ldots+a_{1}x$

for some $n\geq 1$, 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
$G\colonequals\operatorname{Ker}f$ 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 [34, Chapter 13]. The next family
is again based on $\mathbb{G}_{a}$ but consists of noncommutative group
schemes.

######  Example 4.7.

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 groups 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 $\sigma$:

$\displaystyle(1\\#\sigma)(x_{1}\\#1)$ $\displaystyle=(t+1)x_{1}\\#\sigma;$
$\displaystyle(1\\#\sigma)(x_{i}\\#1)$ $\displaystyle=x_{i}\\#\sigma,\quad
1<i\leq r.$

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.

##  5\. Enhancements

Let $G$ be a finite flat group scheme over a noetherian commutative ring $R$.
In this section we exhibit an $\infty$-categorical enhancement of the
triangulated category $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj}G,R)$ and,
when $G$ is a finite group, compare it to the derived $\infty$-category
$\operatorname{\mathcal{R}\\!\\!\;\mathit{ep}}(G,R)$ of representations
studied in [4, 5]. While this enhancement is not used in the remainder of this
paper, it plays an important role in related developments; see 5.9. The reader
not familiar with the language of $\infty$-categories may consult Lurie’s
writings [35, 36], or any of the other accounts that have since appeared on
this topic.

We begin with the construction of an enhancement of
$\mathbf{K}(\operatorname{Proj}A)$, for any ring $A$. Let
$\operatorname{Ch}(\operatorname{Proj}A)$ be the category of (unbounded) chain
complexes of projective $A$-modules, viewed as a differential graded category.
Following [36, Construction 1.3.1.6], we define

$\mathcal{K}(\operatorname{Proj}A)\coloneqq\mathrm{N}_{\mathrm{dg}}(\operatorname{Ch}(\operatorname{Proj}A))$

as the differential graded nerve of $\operatorname{Ch}(\operatorname{Proj}A)$.
This is a stable $\infty$-category whose homotopy category identifies with
$\mathbf{K}(\operatorname{Proj}A)$; see [36, §1.3.1].

With $R$ and $G$ as above, the tensor product on
$\operatorname{Ch}(\operatorname{Proj}G)$ induces a symmetric monoidal
structure on $\mathcal{K}(\operatorname{Proj}G)$. It follows as in the proof
of Proposition 4.4 that the localising subcategory of
$\mathcal{K}(\operatorname{Proj}G)$ spanned by those objects which satisfy the
equivalent conditions of Lemma 2.13 inherits a structure of symmetric monoidal
stable $\infty$-category. We denote this $\infty$-category
$\mathcal{K}(\operatorname{Proj}G,R)$; by construction, its homotopy category
identifies with $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj}G,R)$.

Let $\mathcal{D}^{b}(\operatorname{mod}(G,R))$ be the symmetric monoidal
stable $\infty$-category underlying
$\mathbf{D}^{b}(\operatorname{mod}(G,R))$.111As in [33], one observes that
$\mathbf{D}^{b}(\operatorname{mod}(G,R))$ is equivalent to the derived
category of perfect complexes on the quotient stack
$\operatorname{Spec}(R)\mathbin{/\mkern-6.0mu/}G$. Since compact objects,
generators, and rigidity are detected at the level of the homotopy categories,
Propositions 2.14 and 4.4 translate into the statement that
$\mathcal{K}(\operatorname{Proj}G,R)$ is rigidly-compactly generated with full
subcategory of compact objects determined by the equivalence

$\mathcal{K}(\operatorname{Proj}G,R)^{c}\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\,\mathcal{D}^{b}(\operatorname{mod}(G,R))\,,$

induced by the localisation functor that inverts the quasi-isomorphisms.
Equivalently, $\mathcal{K}(\operatorname{Proj}G,R)$ is the ind-completion of
$\mathcal{D}^{b}(\operatorname{mod}(G,R))$, i.e., there is an equivalence

(5.1) $\mathcal{K}(\operatorname{Proj}G,R)\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\,\operatorname{Ind}\mathcal{D}^{b}(\operatorname{mod}(G,R))\,,$

of rigidly-compactly generated symmetric monoidal stable $\infty$-categories.
Passage to the homotopy category thus provides a model for
$\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj}G,R)$. We refer to [35, §5.3.5],
[36, Proposition 1.1.3.6], and [36, Corollary 4.8.1.14] for a discussion of
ind-completions of $\infty$-categories and their monoidal properties.

In the remainder of this section, we specialise to the case that $G$ is a
finite group. In [4], the first author introduced a derived $\infty$-category
of $R$-linear $G$-representation; we recall the definition.

######  5.2.

Write $BG$ for the classifying space of the finite group $G$, viewed as an
$\infty$-groupoid. The derived $\infty$-category of representations is

$\operatorname{\mathcal{R}\\!\\!\;\mathit{ep}}(G,R)\coloneqq\operatorname{Ind}\operatorname{Fun}(BG,\mathcal{D}^{\mathrm{perf}}(R))\,,$

the ind-category of the category of local systems on $BG$ with coefficients in
perfect $R$-modules. Equipped with the pointwise tensor product,
$\operatorname{\mathcal{R}\\!\\!\;\mathit{ep}}(G,R)$ has the structure of a
symmetric monoidal stable $\infty$-category.

The starting point for our comparison between
$\operatorname{\mathcal{R}\\!\\!\;\mathit{ep}}(G,R)$ and
$\mathcal{K}(\operatorname{Proj}G,R)$ is the observation that, by
construction, $\operatorname{\mathcal{R}\\!\\!\;\mathit{ep}}(G,R)$ is rigidly-
compactly generated with full subcategory of compact objects given by
$\operatorname{Fun}(BG,\mathcal{D}^{\mathrm{perf}}(R))$. Consider first the
$\infty$-category $\operatorname{Fun}(BG,\mathcal{D}(R))$ of local systems on
$BG$ with coefficients in $\mathcal{D}(R)$, equipped with the pointwise
symmetric monoidal structure. Let $\mathcal{D}(RG)$ be the derived
$\infty$-category of $RG$-modules with symmetric monoidal structure coming
from the coproduct on the cocommutative Hopf algebra $RG$, as above; its
homotopy category is $\mathbf{D}(\operatorname{Mod}(RG))$. The next result is
folklore, so we only sketch the proof.

######  Lemma 5.3.

Let $G$ be a finite group and $R$ a commutative ring. There is an equivalence
$\operatorname{Fun}(BG,\mathcal{D}(R))\simeq\mathcal{D}(RG)$ of symmetric
monoidal stable $\infty$-categories, which restricts to a symmetric monoidal
equivalence

$\operatorname{Fun}(BG,\mathcal{D}^{\mathrm{perf}}(R))\simeq\mathcal{D}^{b}(\operatorname{mod}(G,R))\,.$

###### Proof.

The $R$-linear dual $A=\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(RG,R)$ of
$RG$ has the structure of a dualisable commutative algebra in
$\mathcal{D}(G,R)$. Induction along the unit map $R\to A$ identifies with the
restriction functor $\mathrm{Res}\colon\mathcal{D}(RG)\to\mathcal{D}(R)$ and
the corresponding cosimplicial descent diagram takes the form

$\Delta\longrightarrow\mathrm{Cat}_{\infty},\qquad\bullet\mapsto\textstyle\prod_{G^{\times\bullet}}\mathcal{D}(R)\coloneqq(\mathcal{D}(R)\rightrightarrows\prod_{G}\mathcal{D}(R)\Rrightarrow\ldots)\,.$

We thus obtain a symmetric monoidal exact comparison functor to the
totalisation,

(5.4)
$\mathcal{D}(RG)\longrightarrow\mathrm{Tot}\textstyle\prod_{G^{\times\bullet}}\mathcal{D}(R)\,.$

On the one hand, since $\mathrm{Res}$ is conservative, it follows from the
Barr–Beck–Lurie theorem [36, Theorem 4.7.3.5] that the functor (5.4) is an
equivalence. On the other hand, the standard simplicial bar resolution of $BG$
implies that the totalisation in (5.4) recovers the category of local systems
on $BG$ with coefficients in $\mathcal{D}(R)$,

$\operatorname{Fun}(BG,\mathcal{D}(R))\simeq\mathrm{Tot}\textstyle\prod_{G^{\times\bullet}}\mathcal{D}(R)\,,$

equipped with its pointwise symmetric monoidal structure. Composing these two
symmetric monoidal equivalences, we obtain the first part of the claim:

(5.5) $\operatorname{Fun}(BG,\mathcal{D}(R))\simeq\mathcal{D}(RG)\,.$

The equivalence of (5.5) restricts to an equivalence on dualisable objects:

(5.6)
$\operatorname{Fun}(BG,\mathcal{D}(R))^{\mathrm{dbl}}\simeq\mathcal{D}(RG)^{\mathrm{dbl}}\,.$

On the one hand, since the tensor product is pointwise, the left side of (5.6)
identifies with
$\operatorname{Fun}(BG,\mathcal{D}(R)^{\mathrm{dbl}})\simeq\operatorname{Fun}(BG,\mathcal{D}(R)^{\mathrm{perf}})$.
On the other hand, the conservative functor $\mathcal{D}(RG)\to\mathcal{D}(R)$
detects dualisable objects, so the right side of (5.6) is naturally equivalent
to $\mathcal{D}^{b}(\operatorname{mod}(G,R))$ by Proposition 2.6, as claimed.
∎

######  5.7.

Let $i^{*}R\in\operatorname{Fun}(BG,\mathcal{D}(R))$ be the local system
arising from pushing $R$ forward along the inclusion of a chosen basepoint
$i\colon\ast\to BG$. Unwinding the construction, the equivalence of Lemma 5.3
is determined by sending $i^{*}R$ to the regular representation
$RG\in\mathcal{D}(RG)$.

######  Proposition 5.8.

Let $G$ be a finite group and $R$ a noetherian commutative ring. The
equivalence of Lemma 5.3 induces an equivalence

$\operatorname{\mathcal{R}\\!\\!\;\mathit{ep}}(G,R)\simeq\mathcal{K}(\operatorname{Proj}G,R)$

of symmetric monoidal stable $\infty$-categories.

###### Proof.

Recall that both sides of the desired equivalence are rigidly-compactly
generated $\infty$-categories. Combining Lemma 5.3 with (5.1) thus yields
equivalences

$\displaystyle\operatorname{\mathcal{R}\\!\\!\;\mathit{ep}}(G,R)$
$\displaystyle=\operatorname{Ind}\operatorname{Fun}(BG,\mathcal{D}^{\mathrm{perf}}(R))$
$\displaystyle\simeq\operatorname{Ind}\mathcal{D}^{b}(\operatorname{mod}(G,R))$
$\displaystyle\simeq\operatorname{Ind}(\mathcal{K}(\operatorname{Proj}G,R)^{c})$
$\displaystyle\simeq\mathcal{K}(\operatorname{Proj}G,R)$

of symmetric monoidal stable $\infty$-categories. ∎

######  5.9.

One feature of the $\infty$-category
$\operatorname{\mathcal{R}\\!\\!\;\mathit{ep}}(G,R)$ is that its construction
extends to any topological group $G$ and any symmetric monoidal coefficient
$\infty$-category, such as perfect modules over commutative ring spectra. This
allows one to transport questions from modular representation theory to higher
algebra and, vice versa, apply homotopical techniques to problems in
representation theory; see [6].

##  6\. Cohomology and fibres

As in Section 4, we fix a noetherian commutative ring $R$ and a finite flat
group scheme $G$ over $R$. For each map of rings $R\to k$ with $k$ a field,
consider the $k$-algebra

$S(k)\colonequals\operatorname{Ext}_{G_{k}}^{*}(k,k)\,.$

The functor $-\otimes^{\mathrm{L}}_{R}k$ induces a map of graded $R$-algebras
$S\to S(k)$ and hence a map of graded $k$-algebras

(6.1) $\phi_{k}\colon S\otimes_{R}k\longrightarrow S(k)\,.$

The source and target are graded-commutative and finitely generated. Consider
the induced map on spectra:

(6.2)
$\operatorname{Spec}(\phi_{k})\colon\operatorname{Spec}S(k)\longrightarrow\operatorname{Spec}(S\otimes_{R}k)\,.$

In [33] it is proved that this map is a homeomorphism when $G$ is a finite
group. The argument, going back to [12], extends to finite flat group schemes
and yields the result below. The key new input is work of van der Kallen [44],
recalled in 6.8.

######  Theorem 6.3.

The map $\operatorname{Spec}(\phi_{k})$ is a homeomorphism.

When the characteristic of $k$ is zero, the source and target of the map
$\phi_{k}$ are zero in non-negative degrees, by a result of van der Kallen’s
recalled below 6.8, so $\operatorname{Spec}(\phi_{k})$ is trivially a
homeomorphism. The real content of the result is in the case when the
characteristic of $k$ is positive. We deduce it from a more general result,
Theorem 6.11; see 6.12. Its statement and proof are modeled on those in
Benson–Habegger [12] and Lau [33, Section 8], and require some preparation.

### Universal homeomorphisms

A map $\varphi\colon A\to B$ of (possibly graded) commutative rings is a
_universal homeomorphism_ if for each map $A\to A^{\prime}$ of rings the map
on spectra induced by the base change map $A^{\prime}\to
A^{\prime}\otimes_{A}B$, is a homeomorphism:

$\operatorname{Spec}(A^{\prime}\otimes_{A}B)\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\operatorname{Spec}A^{\prime}\,.$

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. (1)

$\varphi$ is universally bijective;

  2. (2)

$\varphi\otimes_{A}k(\mathfrak{p})\colon k(\mathfrak{p})\to
B\otimes_{A}k(\mathfrak{p})$ is universally bijective for
$\mathfrak{p}\in\operatorname{Spec}A$;

  3. (3)

$\operatorname{Spec}(B\otimes_{A}k)$ 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 [23, Lemma 10.46.7] for a
proof.

######  6.4.

Let $\varphi\colon A\to B$ be a map of commutative rings and $p$ be a prime
number. Assume the following conditions hold:

  1. (1)

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

  2. (2)

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.

######  6.5.

Let $G$ be a finite flat group scheme over $R$. Any map $f\colon R^{\prime}\to
R^{\prime\prime}$ of $R$-algebras induces a map

$H^{*}(f)\colon
H^{*}(G,R^{\prime})\otimes_{R^{\prime}}R^{\prime\prime}\longrightarrow
H^{*}(G,R^{\prime\prime})$

of graded $R^{\prime\prime}$-algebras. Consider the collection

$U(G)\colonequals\\{\text{$f$ a map of $R$-algebras}\mid\text{$H^{*}(f)$ is a
universal homeomorphism}\\}.$

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

######  Lemma 6.6.

The collection $U(G)$ is closed under compositions and contains all flat maps.
When $K$ is a commutative ring, $R$ a $K$-algebra, and $f$ a map of
$R$-algebras, the following conditions are equivalent:

  1. (1)

$f$ is in $U(G)$;

  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.

######  Lemma 6.7.

Let $f\colon R^{\prime}\to R^{\prime\prime}$ be a map of commutative rings and
$\overline{f}_{\mathfrak{q}}$ the composite $R^{\prime}\to R^{\prime\prime}\to
k(\mathfrak{q})$, where $\mathfrak{q}\in\operatorname{Spec}R^{\prime\prime}$.
The following conditions are equivalent:

  1. (1)

$H^{*}(f)$ is universally bijective;

  2. (2)

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

######  6.8.

Recall that $R$ is a noetherian commutative ring. Van der Kallen [44, Theorem
1] proves the following results:

  1. (1)

There exists an integer $d\geq 1$ with

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

  2. (2)

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.

######  Lemma 6.9.

Any surjective map $f$ of $R$-algebras with $\operatorname{Ker}(f)$ nilpotent
is in $U(G)$.

###### Proof.

It suffices to check that $f_{\mathfrak{p}}$ is in $U(G)$ for each
$\mathfrak{p}$ in $\operatorname{Spec}R^{\prime}$, by Lemma 6.6, so we may
assume $R^{\prime}$ is a local ring. In particular, it contains $\mathbb{Q}$
or its residue field is of positive characteristic $p$ and $R^{\prime}$ is
$p$-local, that is to say, the natural map
$R^{\prime}\to\mathbb{Z}_{(p)}\otimes_{\mathbb{Z}}R^{\prime}$ is bijective.

In the former case $H^{\geqslant 1}(G,-)=0$ on $\operatorname{Mod}R^{\prime}$,
and there is nothing left to verify; this is by 6.8(1). So we may assume
$R^{\prime}$ is $p$-local. Then, again by 6.8(1), one gets

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

Set $I\colonequals\operatorname{Ker}(f)$; the hypothesis is that $I^{s}=0$ for
some $s\geq 1$. It suffices to verify the statement when $s=2$, and to that
end we verify that $H^{*}(f)$ satisfies the conditions in 6.4. Condition (2)
clearly holds as $I$ is nilpotent. The exact sequence

$0\longrightarrow I\longrightarrow R^{\prime}\longrightarrow
R^{\prime\prime}\longrightarrow 0$

of $R^{\prime}$-modules induces an exact sequence

$H^{*}(G,R^{\prime})\xrightarrow{\ H^{*}(f)\
}H^{*}(G,R^{\prime\prime})\xrightarrow{\ \delta\ }H^{*}(G,I)\,.$

Arguing as in the proof [17, Lemma 5.2], or [33, Lemma 8.17], yields that the
connecting map $\delta$ is a derivation: For any $x,y$ in
$H^{*}(G,R^{\prime\prime})$, one has

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

Fix $x\in H^{\geqslant 1}(G,R^{\prime})$. When $|x|$ and $p$ are both odd,
$x^{2}=0$ by the graded-commutativity of $H^{*}(G,R^{\prime})$. So we can
assume $|x|$ is even or $p=2$. Either way, it follows from the fact that
$\delta\colon H^{*}(G,R^{\prime\prime})\to H^{*+1}(G,I)$ 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 (1) of 6.4 is also satisfied. ∎

A surjective map $R^{\prime}\to R^{\prime\prime}$ is a _complete intersection_
if its kernel can be generated by a regular sequence.

######  Lemma 6.10.

Any surjective complete intersection map $f\colon R^{\prime}\to
R^{\prime\prime}$ of $R$-algebras with $R^{\prime}$ noetherian is in $U(G)$.

###### Proof.

Arguing as in the proof of Lemma 6.9 we can assume $R^{\prime}$ is local, with
residue field of positive characteristic $p$ and that $p^{n}\cdot H^{\geqslant
1}(G,-)=0$ on $\operatorname{Mod}R^{\prime}$. Moreover, since $U(G)$ is closed
under compositions it suffices to consider the case
$R^{\prime\prime}=R^{\prime}/t$, where $t$ is not a zero-divisor in
$R^{\prime}$. At this point one can argue as in the proof of [33, Lemma 8.21]
to deduce the desired statement. Here is a sketch of the argument.

For each integer $i\geq 1$ set

$J_{i}\colonequals\operatorname{Ker}(H^{*}(G,R^{\prime})\xrightarrow{\ t^{i}\
}H^{*}(G,R^{\prime\prime}))\,.$

The ring $H^{*}(G,R^{\prime})$ is noetherian, by van der Kallen’s result 6.8,
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 $i\geq s$. Since the map
$R^{\prime}/t^{s}\to R^{\prime}/t$ is in $U(G)$, by Lemma 6.9, it suffices to
verify that the map $R^{\prime}\to R^{\prime}/t^{s}$ is in $U(G)$. Thus,
replacing $t$ by $t^{s}$, we can assume $J_{1}=J_{i}$ for $i\geq 1$. The
sequence of $G$-modules

$0\longrightarrow R^{\prime}\xrightarrow{\ t\ }R^{\prime}\xrightarrow{\ f\
}R^{\prime}/t\to 0$

yields the exact sequence of $H^{*}(G,R^{\prime})$-modules

$0\longrightarrow
R^{\prime}/t\otimes_{R^{\prime}}H^{*}(G,R^{\prime})\xrightarrow{\ H^{*}(f)\
}H^{*}(G,R^{\prime}/t)\xrightarrow{\ \delta\ }J_{1}\longrightarrow 0\,.$

It suffices to verify $\delta(x^{p^{n}})=0$ for $x\in H^{\geqslant
1}(G,R^{\prime}/t)=0$; see 6.4.

Let $P$ be a projective resolution of $R^{\prime}$ as a
$G_{R^{\prime}}$-module. As $t$ is not a zero-divisor on $R^{\prime}$, the
complex $P/tP$ is a projective resolution of $R^{\prime}/t$ as a
$G_{R^{\prime}/t}$-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 $H^{*}(G,R^{\prime})$ and $H^{*}(G,R^{\prime}/t)$ 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 $p^{n}H^{\geqslant
1}(G,R^{\prime})=0$ yields $\delta([z]^{p^{n}})=[tw]$ for some
$w\in\operatorname{End}_{G}(P)$; see [33, 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 $R\to R^{\prime}$ of commutative rings is said to be _essentially of
finite type_ if $R^{\prime}$ is a localisation of a finitely generated
$R$-algebra.

######  Theorem 6.11.

Let $R$ be a noetherian commutative ring, $G$ a finite flat group scheme over
$R$, and $f\colon R\to R^{\prime}$ a map of commutative rings. The following
statements hold.

  1. (1)

$H^{*}(f)$ is universally bijective;

  2. (2)

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

###### Proof.

(1) Given Lemma 6.7 it suffices to consider the case when $R^{\prime}=k$, a
field. We claim that then the map $H^{*}(f)$ is even a universal
homeomorphism. To see this, set
$\mathfrak{p}\colonequals\operatorname{Ker}(f)$, a prime ideal in $R$. The map
$f\colon R\to k$ 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 $f\colon R\to k$ 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 6.9 for
then the ideal $\operatorname{Ker}(f)$ is nilpotent.

Suppose $\dim R\geq 1$ and set $I\colonequals\sqrt{(0)}$ 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 $\dim R\geq 1$, 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 6.10, 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 $R\to
R^{\prime\prime}\to R^{\prime}$ where $R\to R^{\prime\prime}$ is the
localisation of a finitely generated polynomial ring over $R$, and
$R^{\prime\prime}\to R^{\prime}$ is surjective. Since $R\to R^{\prime\prime}$
is flat it is in $U(G)$. We may thus replace $R$ by $R^{\prime\prime}$ 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 6.8(2), 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. ∎

######  6.12.

Theorem 6.3 is the special case of Theorem 6.11 where $R^{\prime}$ is a field.
On the other hand, as can be seen from the proof, the latter result is proved
by (a simple) reduction to the case where $R^{\prime}$ is a field.

##  7\. Local cohomology and support

This section is a reminder about local cohomology functors and a general
local-to-global principle for triangulated categories; for details, see [13,
14].

### 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 $\phi_{X}\colon
S\to\operatorname{End}_{\mathcal{T}}^{*}(X)$ 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 $F\colon\mathcal{T}\to\mathcal{U}$ 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}\coloneqq\\{X\in\mathcal{T}\mid X\text{ is
$V$-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}}\coloneqq\\{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}\coloneqq\\{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
$\varGamma_{\mathfrak{p}}\colon\mathcal{T}\to\varGamma_{\mathfrak{p}}\mathcal{T}$
the composite of those adjoints; it is the _local cohomology functor_ with
respect to $\mathfrak{p}$.

### Support

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

$\operatorname{supp}_{S}X\colonequals\\{\mathfrak{p}\in\operatorname{Spec}S\mid\varGamma_{\mathfrak{p}}X\neq
0\\}\,.$

The next result is an important input in the Section 9.

######  Lemma 7.1.

Let $\mathcal{T}$ and $\mathcal{U}$ be compactly generated $S$-linear
triangulated categories. Let $E\colon\mathcal{T}\to\mathcal{U}$ 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 [16, 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. ∎

### Tensor triangulated categories

From now on $\mathcal{T}$ is a rigidly-compactly generated triangulated
category, with product $\otimes$ and unit
$\mathds{1}=\mathds{1}_{\mathcal{T}}$. We assume that there is a central
action of $S$ via a ring homomorphism
$S\to\operatorname{Hom}^{*}_{\mathcal{T}}(\mathds{1},\mathds{1})$. By [13,
Theorem 8.2], this implies that for each specialisation closed subset $V$ of
$\operatorname{Spec}S$ there are natural isomorphisms

$\varGamma_{V}X\cong X\otimes\varGamma_{V}\mathds{1}\quad\text{and}\quad
L_{V}X\cong X\otimes L_{V}\mathds{1}\,.$

In particular $\varGamma_{\mathfrak{p}}X\cong
X\otimes\varGamma_{\mathfrak{p}}\mathds{1}$ 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 [15, Theorem 3.6].

######  Theorem 7.2.

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

$\operatorname{Loc}^{\otimes}(X)=\operatorname{Loc}^{\otimes}(\\{\varGamma_{\mathfrak{p}}X\mid\mathfrak{p}\in\operatorname{Spec}S\\})\,.\qed$

##  8\. Building and fibres

Let $R$ be a noetherian commutative ring and $G$ a finite flat group scheme
over $R$. To lighten the notation we use the abbreviations

$\operatorname{Rep}(G,R)\colonequals\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj}G,R)\quad\text{and}\quad\operatorname{rep}(G,R)\colonequals\mathbf{D}^{b}(\operatorname{mod}(G,R))\,.$

One has $\operatorname{Rep}(G,R)^{c}=\operatorname{rep}(G,R)$, by Proposition
2.14. Moreover, when $R$ is a field, one has
$\operatorname{Rep}(G,R)=\mathbf{K}(\operatorname{Proj}G)$, by Proposition
2.16.

Let $k(\mathfrak{p})$ be the residue field of $R$ at
$\mathfrak{p}\in\operatorname{Spec}R$, and for any $G$-complex $X$ set

${X}_{k(\mathfrak{p})}\colonequals X\otimes_{R}k(\mathfrak{p})\,.$

The assignment $X\mapsto{X}_{k(\mathfrak{p})}$ induces an exact functor

$\pi_{\mathfrak{p}}\colon\operatorname{Rep}(G,R)\longrightarrow\operatorname{Rep}(G_{k(\mathfrak{p})},k(\mathfrak{p}))=\mathbf{K}(\operatorname{Proj}G_{k(\mathfrak{p})})\,.$

Here is the main result in this section.

######  Theorem 8.1.

Let $R$ be a noetherian commutative ring, $G$ a finite flat group scheme over
$R$, and fix $X,Y$ in $\operatorname{Rep}(G,R)$. The following statements are
equivalent.

  1. (1)

$X\in\operatorname{Loc}^{\otimes}(Y)$ in $\operatorname{Rep}(G,R)$;

  2. (2)

${X}_{k(\mathfrak{p})}\in\operatorname{Loc}^{\otimes}({Y}_{k(\mathfrak{p})})$
in $\operatorname{Rep}(G_{k(\mathfrak{p})},k(\mathfrak{p}))$ for each
$\mathfrak{p}\in\operatorname{Spec}R$.

The proof is given towards the end of the section.

### Reduction to the local case

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

$\lambda(X)\colonequals X\otimes_{R}R_{\mathfrak{p}}$

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

${\operatorname{Rep}(G,R)}$${\operatorname{Rep}(G_{\mathfrak{p}},R_{\mathfrak{p}})\,.}$$\scriptstyle{\lambda}$$\scriptstyle{\lambda_{r}}$

######  Lemma 8.2.

The composite $\lambda_{r}\lambda$ is naturally isomorphic to the functor
$X\mapsto X_{\mathfrak{p}}$. Thus $\lambda_{r}$ identifies
$\operatorname{Rep}(G_{\mathfrak{p}},R_{\mathfrak{p}})$ with the subcategory
of $\mathfrak{p}$-local objects in $\operatorname{Rep}(G,R)$.

###### Proof.

The functor $\lambda$ identifies with the localisation functor

$-\otimes_{R}R_{\mathfrak{p}}\colon\operatorname{rep}(G,R)\longrightarrow\operatorname{rep}(G_{\mathfrak{p}},R_{\mathfrak{p}})$

when restricted to compact objects; see Proposition 2.14. 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. ∎

### 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

(8.3)
${\operatorname{Rep}(G,R)}$${\operatorname{Rep}(G_{k},k)}$$\scriptstyle{\pi}$$\scriptstyle{\pi_{r}}$

where $\pi(X)\colonequals X\otimes_{R}k$. 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 [3,
Theorem 1.3].

######  Lemma 8.4.

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 that $\pi$ is a tensor functor,
and in particular that

$\pi(\mathbf{i}R)=(\mathbf{i}R)\otimes_{R}k\cong\mathbf{i}_{G_{k}}k\,.$

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 verify easily 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
$\pi_{r}(\mathbf{i}_{G_{k}}k)$. Indeed

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

by [14, Proposition 2.11]), so 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

${\operatorname{Rep}(G,R)}$${\mathbf{D}(\operatorname{Mod}G)}$${\operatorname{Rep}(G_{k},k)}$${\mathbf{D}(\operatorname{Mod}G_{k})\,.}$$\scriptstyle{\pi}$$\scriptstyle{\mathbf{q}}$$\scriptstyle{-\otimes^{\mathrm{L}}_{R}k}$$\scriptstyle{\mathbf{i}}$$\scriptstyle{\pi_{r}}$$\scriptstyle{\mathbf{q}_{G_{k}}}$$\scriptstyle{\mathrm{restriction}}$$\scriptstyle{\mathbf{i}_{G_{k}}}$

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

$(X\otimes_{R}Y)_{k(\mathfrak{p})}\cong
X_{k(\mathfrak{p})}\otimes_{k(\mathfrak{p})}Y_{k(\mathfrak{p})}\quad\text{in}\quad\operatorname{Rep}(G_{k(\mathfrak{p})},k(\mathfrak{p}))\,.$

(2) $\Rightarrow$ (1) Assume first that $R$ is local and consider the adjoint
pair (8.3). 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

$\displaystyle\pi_{r}\pi(X)$
$\displaystyle\in\operatorname{Loc}(\pi_{r}(\operatorname{rep}(G_{k(\mathfrak{p})},k(\mathfrak{p}))\otimes\pi(Y)))$
$\displaystyle=\operatorname{Loc}(\pi_{r}(\operatorname{rep}(G_{k(\mathfrak{p})},k(\mathfrak{p})))\otimes
Y)$ $\displaystyle\subseteq\operatorname{Loc}^{\otimes}(Y)\,.$

Then Lemma 8.4 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\. Stratification

Let $S$ be a graded-commutative noetherian ring and $\mathcal{T}$ a rigidly-
compactly generated tensor triangulated category. Assume that $S$ acts on
$\mathcal{T}$ via a map of graded rings
$S\to\operatorname{Hom}^{*}_{\mathcal{T}}(\mathds{1},\mathds{1})$. We say that
the tensor triangulated category $\mathcal{T}$ is _stratified_ by the action
of $S$ if for any pair of objects $X,Y$ in $\mathcal{T}$ there are
equivalences

$\displaystyle\operatorname{supp}_{S}X\subseteq\operatorname{supp}_{S}Y$
$\displaystyle\iff\operatorname{Loc}^{\otimes}(X)\subseteq\operatorname{Loc}^{\otimes}(Y)$
$\displaystyle\iff X\in\operatorname{Loc}^{\otimes}(Y)\,.$

Lemma 4.12 in [17] reconciles this notion with the one in [14, Section 7.2].

Let $R$ be a noetherian commutative ring and $G$ a finite flat group scheme
over $R$. We consider the cohomology algebra
$S\colonequals\operatorname{Ext}^{*}_{G}(R,R)$ of $G$ and the cohomology
algebras of the fibres of the $R$-algebra $RG$. For each $\mathfrak{p}$ in
$\operatorname{Spec}R$ we set

$S(\mathfrak{p})\colonequals\operatorname{Ext}_{{G}_{k(\mathfrak{p})}}^{*}(k(\mathfrak{p}),k(\mathfrak{p}))\,.$

The functor $-\otimes^{\mathrm{L}}_{R}k(\mathfrak{p})$ induces a map of graded
$R$-algebras $S\to S(\mathfrak{p})$ and this induces a natural map of graded
$k(\mathfrak{p})$-algebras

$\kappa_{\mathfrak{p}}\colon S\otimes_{R}k(\mathfrak{p})\longrightarrow
S(\mathfrak{p})\,.$

The result below contains the first part of Theorem A. Recall that
$\operatorname{Rep}(G,R)$ is an abbreviation for
$\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj}G,R)$.

######  Theorem 9.1.

Let $R$ be a noetherian commutative ring, $G$ a finite flat group scheme over
$R$. The tensor triangulated category $\operatorname{Rep}(G,R)$ is stratified
by the action of $S\colonequals\operatorname{Ext}^{*}_{G}(R,R)$, with support
equal to $\operatorname{Spec}S$. In particular one gets a computation of the
Balmer spectrum of the derived category of lattices:

$\operatorname{Spc}(\operatorname{rep}(G,R))\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\operatorname{Spec}H^{*}(G,R)\,.$

As noted before, Lau [33] computes the Balmer spectrum of
$\operatorname{rep}(G,R)$ when $G$ is a finite group, and where the $G$-action
on $R$ is even allowed to be nontrivial.

The proof of the theorem above requires some preparation.

### Support and fibres

As we will explain in the proof of Theorem 9.1 below, Theorem 8.1 combined
with the main theorem of [18] 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 $\eta\colon R\to S$, which induces a
map

$\eta^{a}\colon\operatorname{Spec}S\longrightarrow\operatorname{Spec}R\,.$

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 $\kappa_{\mathfrak{p}}\colon
S\otimes_{R}k(\mathfrak{p})\to S(\mathfrak{p})$ which induces the map

$\kappa^{a}_{\mathfrak{p}}\colon\operatorname{Spec}S(\mathfrak{p})\longrightarrow\operatorname{Spec}(S\otimes_{R}k(\mathfrak{p}))\subseteq\operatorname{Spec}S\,.$

Theorem 6.3 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

$\kappa^{a}_{\mathfrak{p}}(\operatorname{supp}_{S(\mathfrak{p})}{X}_{k(\mathfrak{p})})=\operatorname{supp}_{S}X\cap(\eta^{a})^{-1}(\mathfrak{p})\,.$

###### Proof.

By Lemma 8.2, we can assume $(R,\mathfrak{m},k)$ is a local ring and
$\mathfrak{p}=\mathfrak{m}$, the maximal ideal of $R$. Set $S_{k}\colonequals
S\otimes_{R}k=S/\mathfrak{m}S$. Via the map $\kappa_{\mathfrak{m}}\colon
S_{k}\to S(\mathfrak{m})$ the $S(\mathfrak{m})$-action on
$\mathbf{K}(\operatorname{Proj}G_{k})$ induces an $S_{k}$-action. Applying
[16, Corollary 7.8(1)] to the identity functor on
$\mathbf{K}(\operatorname{Proj}G_{k})$ yields the first equality below

$\displaystyle\kappa^{a}_{\mathfrak{m}}(\operatorname{supp}_{S(\mathfrak{m})}X_{k})$
$\displaystyle=\operatorname{supp}_{S_{k}}X_{k}$
$\displaystyle=\operatorname{supp}_{S}X_{k}$
$\displaystyle=\operatorname{supp}_{S}(\pi_{r}\pi X)$
$\displaystyle=\operatorname{supp}_{S}(\varGamma_{\mathfrak{m}}X)$
$\displaystyle=\operatorname{supp}_{S}(X)\cap V(\mathfrak{m}S)$
$\displaystyle=\operatorname{supp}_{S}(X)\cap(\eta^{a})^{-1}(\mathfrak{m})\,.$

The same observation applied to the map $S\to S_{k}$ yields the second
equality. Now consider the adjunction (8.3). Then Lemma 7.1 gives the third
equality, and the next one follows from Lemma 8.4. Now observe that
$\varGamma_{\mathfrak{m}}=\varGamma_{V(\mathfrak{m})}=\varGamma_{V(\mathfrak{m}S)}$
by [16, Proposition 7.5]. Then [13, 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
$\operatorname{supp}_{S}X\subseteq\operatorname{supp}_{S}Y$, one has
$X\in\operatorname{Loc}^{\otimes}(Y)$. By Theorem 6.3, 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 it cohomology algebra, $S(\mathfrak{p})$; this is the main result of
[18]. 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
$\operatorname{supp}_{S}\mathds{1}=\operatorname{Spec}S$. This follows from
Lemma 9.2, for ${\mathds{1}}_{k(\mathfrak{p})}$ 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, [14, 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\. Costratification and other applications

The next result establishes costratification for $\operatorname{Rep}(G,R)$ in
the sense of [16]. Together with Theorem 9.1, this completes the proof of
Theorem A.

######  Theorem 10.1.

Let $R$ be a noetherian commutative ring and $G$ a finite flat group scheme
over $R$. The tensor triangulated category $\operatorname{Rep}(G,R)$ is
costratified by the action of $\operatorname{Ext}^{*}_{G}(R,R)$.

###### Proof.

We again set $S\colonequals\operatorname{Ext}^{*}_{G}(R,R)$. We verify the
conditions of the bootstrap theorem [7, Theorem 17.2], that is to say that the
collection of functors

$\pi_{\mathfrak{p}}\colon\operatorname{Rep}(G,R)\longrightarrow\operatorname{Rep}(G_{k(\mathfrak{p})},k(\mathfrak{p})),$

indexed on $\mathfrak{p}\in\operatorname{Spec}R$, has the following
properties:

  1. (1)

the images of the maps
$(\varphi_{\mathfrak{p}}\coloneqq\operatorname{Spc}(\pi_{\mathfrak{p}}))_{\mathfrak{p}\in\operatorname{Spec}R}$
cover $\operatorname{Spc}(\operatorname{rep}(G,R))$;

  2. (2)

$\operatorname{Rep}(G,R)$ is stratified by the action of $S$; and

  3. (3)

the categories $\operatorname{Rep}(G_{k(\mathfrak{p})},k(\mathfrak{p}))$ are
costratified for all $\mathfrak{p}\in\operatorname{Spec}R$.

Then $\operatorname{Rep}(G,R)$ is costratified over
$\operatorname{Spc}(\operatorname{rep}(G,R))$, and hence is also costratified
by the action of $S$ in light of the homeomorphism
$\operatorname{Spc}(\operatorname{rep}(G,R))\cong\operatorname{Spec}S$.

Condition (2) holds by Theorem 9.1, while the costratification of
$\operatorname{Rep}(G_{k(\mathfrak{p})},k(\mathfrak{p}))$ has been established
in [16]. Appealing to [8, Theorem 1.3], Condition (1) follows from Proposition
3.6. Here is a direct argument for (1). Indeed, consider the commutative
diagram

${{\operatorname{Spc}(\operatorname{rep}(G_{k(\mathfrak{p})},k(\mathfrak{p})))}}$${{\operatorname{Spc}(\operatorname{rep}(G,R))}}$${{\operatorname{Spec}\operatorname{Ext}_{{G}_{k(\mathfrak{p})}}^{*}(k(\mathfrak{p}),k(\mathfrak{p}))}}$${{\operatorname{Spec}S\otimes_{R}k(\mathfrak{p})}}$${\operatorname{Spec}S}$${{\operatorname{Spec}k(\mathfrak{p})}}$${{\operatorname{Spec}R,}}$$\scriptstyle{\varphi_{\mathfrak{p}}}$$\scriptstyle{\rho}$$\scriptstyle{\sim}$$\scriptstyle{\operatorname{Spec}\kappa_{\mathfrak{p}}}$$\scriptstyle{\sim}$$\scriptstyle{\rho}$$\scriptstyle{\sim}$

in which the maps $\rho$ are Balmer’s comparison maps from [2], the bottom
right vertical map is induced by the degree $0$ inclusion $R\to S$, and the
horizontal maps are the ones induced by base change. By construction, the
right lower square is a pullback in topological spaces. Theorem 9.1 shows that
the maps $\rho$ are homeomorphisms; moreover,
$\operatorname{Spec}\kappa_{\mathfrak{p}}$ is a homeomorphism, by Theorem 6.3.
For $\mathfrak{p}$ varying through the points of $\operatorname{Spec}R$, this
implies that the images of $\varphi_{\mathfrak{p}}$ jointly cover
$\operatorname{Spc}(\operatorname{rep}(G,R))$, as desired. ∎

######  10.2.

It is possible to give a proof of this theorem by transcribing the argument
from [7] to the theory of costratification as developed in [16].

### Applications

We record some consequences of the Theorems 9.1 and 10.1. The one below is by
[16, 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

$\left\\{\begin{gathered}\text{tensor closed localising}\\\
\text{subcategories of
$\operatorname{Rep}(G,R)$}\end{gathered}\;\right\\}\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\left\\{\begin{gathered}\text{Hom closed colocalising}\\\ \text{subcategories
of $\operatorname{Rep}(G,R)$}\end{gathered}\right\\}\,.$

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

One gets also (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

$\operatorname{Spc}(\operatorname{stmod}(G,R))\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\operatorname{Proj}H^{*}(G,R).$

###### Proof.

One can prove this along the lines of Theorem 10.1. It can also be deduced
from Corollary 2.15, which implies that the canonical functor
$\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj}G,R)\to\operatorname{StMod}^{\mathrm{lf}}(G,R)$
is a finite localisation inducing the inclusion
$\operatorname{Proj}S\to\operatorname{Spec}S$. ∎

Given an object $M$ in $\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj}G,R)$ we
write $\operatorname{supp}_{G}M$ and $\mathrm{cosupp}_{G}M$ for its support
and cosupport, respectively, computed using the action of $H^{*}(G,R)$ on
$\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj}G,R)$; see Section 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

(10.5)
$\begin{split}\operatorname{supp}_{G}(M\otimes_{R}N)&=\operatorname{supp}_{G}M\cap\operatorname{supp}_{G}N\\\
\mathrm{cosupp}_{G}\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(M,N)&=\operatorname{supp}_{G}M\cap\mathrm{cosupp}_{G}N\,.\end{split}$

The formula for the support of the tensor product can be deduced from
Corollary 10.4 as in the proof of [15, Theorem 11.1]. The cosupport of the
function objects follows from [16, Theorem 9.5].

Given a subgroup scheme $H\leq G$, 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

(10.6)
$\begin{split}\operatorname{supp}_{H}(M\downarrow_{H})&=(\mathrm{res}^{*}_{G,H})^{-1}(\operatorname{supp}_{G}M)\\\
\mathrm{cosupp}_{H}(M\downarrow_{H})&=(\mathrm{res}^{*}_{G,H})^{-1}(\mathrm{cosupp}_{G}M)\,.\end{split}$

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 [14, 16], which deal with the
case when $R$ is a field.

##  11\. An example

In this section we give an example, promised in Section 2 of a group $G$ and a
noetherian commutative ring $R$ with the property that the category
$\operatorname{Mod}^{\mathrm{lf}}(G,R)$ is properly contained in
$\operatorname{Mod}(G,R)$.

Let $k$ be a field of characteristic $2$, set $R\colonequals k[t]/(t^{2})$ and
$G\colonequals\mathbb{Z}/2$, the elementary abelian $2$-group of rank one.
Setting $x\colonequals g-1$, where $g$ is the generator of $G$, one gets an
isomorphism

$RG\cong\frac{k[t,x]}{(t^{2},x^{2})}\,,$

which identifies with the group algebra over $k$ of
$E\colonequals(\mathbb{Z}/2)^{2}$, the Klein $4$-group. The cohomology algebra
is

$H\colonequals H^{*}(E,k)\cong k[t^{*},x^{*}]\,,$

where the basis $t^{*},x^{*}$ of $H^{1}$ is dual to the basis $t,x$ of
$(t,x)/(t,x)^{2}$. One has

$\operatorname{Mod}^{\mathrm{lf}}(G,R)\subseteq\operatorname{Mod}(G,R)\subseteq\operatorname{Mod}(E,k)\,.$

This induces full and faithful inclusions

(11.1)
$\operatorname{StMod}^{\mathrm{lf}}(G,R)\subseteq\operatorname{StMod}(G,R)\subseteq\operatorname{StMod}(E,k)\,.$

Here is a key observation about these categories.

######  Lemma 11.2.

Both $\operatorname{StMod}(G,R)$ and $\operatorname{StMod}^{\mathrm{lf}}(G,R)$
are localising subcategories of $\operatorname{StMod}(E,k)$, with

$\displaystyle\operatorname{supp}_{H}\operatorname{StMod}(G,R)$
$\displaystyle=\operatorname{Proj}H\setminus\\{(t^{*})\\}$
$\displaystyle\operatorname{supp}_{H}\operatorname{StMod}^{\mathrm{lf}}(G,R)$
$\displaystyle=\operatorname{Proj}H\setminus\\{(0),(t^{*})\\}\,.$

###### Proof.

A $kE$-module $M$ is in $\operatorname{Mod}(G,R)$ if and only if it is
projective when restricted to $R$; equivalently, its support does not contain
$(t^{*})$. This justifies the claim about the support of
$\operatorname{StMod}(G,R)$.

The support of any finite dimensional $kE$-module $M$ is closed, so $M$ is in
$\operatorname{mod}(G,R)$ if and only if the subset $\\{(0),(t^{*})\\}$ is not
in its support. Thus this subset is not in the support of any module that is
in the localising subcategory of $\operatorname{Mod}(G,R)$ generated by
$\operatorname{mod}(G,R)$. This justifies the second equality. ∎

It follows from the result above that the _generic module_ in
$\operatorname{StMod}(E,k)$, that is to say, the $kE$-module
$\varGamma_{(0)}k$ is in $\operatorname{StMod}(G,R)$ but not in
$\operatorname{StMod}^{\mathrm{lf}}(G,R)$.

The inclusions in (11.1) do not respect the tensor triangular structures, for
the product on the two categories on the left is $-\otimes_{R}-$ whereas for
that on the right is $-\otimes_{k}-$. This is an important point, and is
discussed further below.

### 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 $H^{1}(G,R)=\operatorname{Ext}^{1}_{RG}(R,R)$; denote it
$\eta$. 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:

######  Lemma 11.3.

The map above is an isomorphism. ∎

Thus $\operatorname{Proj}H^{*}(G,R)$ consists of a single point, in contrast
with $\operatorname{Proj}H^{*}(E,k)=\mathbb{P}^{1}_{k}$. This means that there
is a single nonzero thick _ideal_ in $\operatorname{stmod}(G,R)$, whereas
thick subcategories are aplenty, and parameterised by specialisation closed
subsets of $\mathbb{P}^{1}_{k}\setminus\\{(0)\\}$. Here is an example: Assume
$k$ is algebraically closed and set

$M_{[a,b]}\colonequals RG/(at+bx)\quad\text{for
$[a,b]\in\mathbb{P}^{1}_{k}$.}$

Observe that these modules are in $\operatorname{mod}(G,R)$, except the one
for $[a,b]=[1,0]$. Since there is only a single thick ideal in
$\operatorname{stmod}(G,R)$, these modules must tensor-generate each other.
Indeed as long as $[a,b]\neq[1,0]$, as $G$-modules one has

$M_{[a,b]}\otimes_{R}M_{[-a,b]}\cong M_{[0,b]}\cong R\,.$

We deduce also that $\operatorname{StMod}^{\mathrm{lf}}(G,R)$ has no
nontrivial localising ideals. This is another measure of the difference
between this category and $\operatorname{StMod}(G,R)$: The modules in the
latter are precisely those $kE$-modules not supported on the closed point
$V\colonequals\\{[1,0]\\}$, so

$\operatorname{StMod}(G,R)=L_{V}\operatorname{StMod}(E,k)\,.$

It follows that $\operatorname{StMod}(G,R)$ is also compactly generated, with
compact objects generated by $L_{V}k$. Moreover the minimal localising
subcategories of $\operatorname{StMod}(G,R)$ are parameterised by the points
of $\mathbb{P}^{1}_{k}\setminus V$. This set consists of closed points and the
generic point $\\{0\\}$. The localising subcategories corresponding to the
closed points are compactly generated and hence contained in
$\operatorname{StMod}^{\mathrm{lf}}(G,R)$. Thus there are only two proper
localising ideals in $\operatorname{StMod}(G,R)$, namely,
$\operatorname{StMod}^{\mathrm{lf}}(G,R)$ and the localising subcategory
generated by the $G$-module $\varGamma_{(0)}k$ supported at $(0)$.

A similar analysis can be made of the categories
$\mathbf{K}^{\mathrm{lf}}(\operatorname{Proj}G,R)$ and
$\mathbf{K}(\operatorname{Proj}G)$. Now there is one more minimal localising
ideal to take into account, namely the $K$-projective complexes in
$\mathbf{K}(\operatorname{Proj}G)$.

######  11.4.

The preceding construction can be generalised as follows to yield Frobenius
$R$-algebras $A$ for which $\operatorname{StMod}^{\mathrm{lf}}(A,R)$ is
smaller than $\operatorname{StMod}(A,R)$.

Let $G$ be a commutative finite group scheme defined over a field $k$ of
positive characteristic $p$, such that the cohomology ring $H^{*}(G,k)$ has
Krull dimension at least $2$. Choose a $\pi$-point

$\alpha\colon R\colonequals k[t]/(t^{p})\longrightarrow kG\equalscolon A$

such that the corresponding point $[\alpha]$ in
$\operatorname{Proj}H^{*}(G,k)$ is closed and has codimension $\geq 1$. Then
$A$ is a Frobenius $R$-algebra and $\operatorname{StMod}(A,R)$ consists of
those $G$-modules whose support does not contain $[\alpha]$. Choose a non-
closed point $\mathfrak{p}$ in $\operatorname{Proj}H^{*}(G,k)$ whose closure
contains $[\alpha]$; it exists because the codimension of $[\alpha]$ is
positive. The module $\varGamma_{\mathfrak{p}}k$ is then in
$\operatorname{StMod}(A,R)$ but not in
$\operatorname{StMod}^{\mathrm{lf}}(A,R)$.

The construction above carries over to any finite flat group scheme $G$ and
$\pi$-point $\alpha$ whose image is central in $kG$.

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