# Support Varieties and Stable Categories for Algebraic Groups

Eric M. Friedlander∗

Dedicated to the memory of Brian Parshall  Department of Mathematics,
University of Southern California, Los Angeles, CA<EMAIL_ADDRESS>
###### Abstract.

We consider rational representations of a connected linear algebraic group
$\mathbb{G}$ over a field $k$ of positive characteristic $p>0$. We introduce a
natural extension $M\mapsto\Pi(\mathbb{G})_{M}$ to $\mathbb{G}$-modules of the
$\pi$-point support theory for modules $M$ for a finite group scheme $G$ and
show that this theory is essentially equivalent to the more “intrinsic” and
“explicit” theory $M\mapsto\mathbb{P}\mathfrak{C}(\mathbb{G})_{M}$ of supports
for an algebraic group of exponential type, a theory which uses 1-parameter
subgroups $\mathbb{G}_{a}\to\mathbb{G}$.

We extend our support theory to bounded complexes of $\mathbb{G}$-modules,
$C^{\bullet}\ \mapsto\ \Pi(\mathbb{G})_{C^{\bullet}}$. We introduce the tensor
triangulated category $StMod(\mathbb{G})$, the Verdier quotient of the bounded
derived category $D^{b}(Mod(\mathbb{G}))$ by the thick subcategory of mock
injective modules. Our support theory satisfies all the “standard properties”
for a theory of supports for $StMod(\mathbb{G})$.

As an application, we employ $C^{\bullet}\
\mapsto\Pi(\mathbb{G})_{C^{\bullet}}$ to establish the classification of
$(r)$-complete, thick tensor ideals of $stmod(\mathbb{G})$ in terms of
$stmod(\mathbb{G})$-realizable subsets of $\Pi(\mathbb{G})$ and the
classification of $(r)$-complete, localizing subcategories of
$StMod(\mathbb{G})$ in terms of $StMod(\mathbb{G})$-realizable subsets of
$\Pi(\mathbb{G})$.

###### Key words and phrases:

support variety, algebraic groups

###### 2000 Mathematics Subject Classification:

20G05, 20C20, 20G10

∗ partially supported by the Simons Foundation.

###### Contents

  1. 0 Introduction
  2. 1 Comparison of support theories for finite group schemes
  3. 2 Support theories $M\mapsto\Pi(\mathbb{G})_{M},\ M\mapsto\mathfrak{C}(\mathbb{G})_{M}$
  4. 3 Examples of $M\ \mapsto\ \mathbb{P}\mathfrak{C}(\mathbb{G})_{M}$
  5. 4 Stable module categories
  6. 5 Support theory for complexes of $\mathbb{G}$-modules
  7. 6 Classifying subcategories of $stmod(\mathbb{G})$ and $StMod(\mathbb{G})$
  8. 7 Questions and challenges

##  0\. Introduction

The goal of this work, refining and extending our earlier paper [13], is to
present a context and a point of view for the study of representations of
familiar linear algebraic groups $\mathbb{G}$ on vector spaces $V$ over a
field $k$. It constitutes a refinement and extension of our earlier work. We
work in the modular setting, fixing a field $k$ of prime characteristic $p>0$;
our linear algebraic groups are connected affine group schemes of finite type
over $k$; the vector spaces are vector spaces over $k$, not necessarily finite
dimensional. A typical example for $\mathbb{G}$ is the simple algebraic group
$SL_{n}$. The representations we consider are “rational”, formally defined as
comodules for the coordinate algebra $k[\mathbb{G}]$ with coproduct given by
the group structure of $\mathbb{G}$.

We demonstrate how the theory of support varieties, first developed for finite
groups and eventually extended to all finite group schemes, can be extended to
a good theory of supports for $\mathbb{G}$-modules. Our first formulation is
an evident extension of the theory of $\pi$-points, $M\mapsto\Pi(G)_{M}$, for
$\mathbb{G}$-modules $M$ with $G$ a finite group scheme as introduced by J.
Pevtsova and the author in [18]. This theory, $M\mapsto\Pi(\mathbb{G})_{M}$,
applies to any linear algebraic group $\mathbb{G}$ over an arbitrary field of
$k$ of positive characteristic and any (rational) $\mathbb{G}$-module $M$.
However, it is difficult to make explicit because a point of $\Pi(G)$ is an
equivalence class of maps to the group algebra of some Frobenius kernel
$\mathbb{G}_{(r)}\subset\mathbb{G}$. Our second formulation
$M\mapsto\mathbb{P}\mathfrak{C}(\mathbb{G})_{M}$, valid only if the algebraic
group $\mathbb{G}$ is of exponential type, uses varieties of 1-parameter
subgroups $\mathbb{G}_{a}\to\mathbb{G}$ as introduced in our earlier paper
[13].

The underlying justification for a theory of supports is that it is is
sensitive to extensions of representations. Most support theories are based on
cohomology, and support theory offers a geometric picture of cohomology
following foundational work of D. Quillen [29]. For example, the spectrum of
the cohomology of the $r$-th Frobenius kernel $(SL_{n})_{(r)}$ of $SL_{n}$ is
homeomorphic to the variety of $r$-tuples of pairwise commuting, $p$-nilpotent
$n\times n$ matrices of trace 0 [35]. Our theories are much influenced by J.
Carlson’s consideration of rank varieties for elementary abelian $p$-groups
[5].

As motivation for considering support theories for linear algebraic groups, we
remind the reader of a few consequences of support theory for finite groups
and finite group schemes. An important role of support theories is that they
offer a means of classifying categorical structures associated to
representation theories which are “wild” (e.g., [2], [3]). Support theories
have led to the identification and study of various interesting special
classes of representations such as “modules of constant Jordan type” [7] and
“mock injective” modules [14], provided new invariants for $G$-modules [21],
and enabled the construction of algebraic vector bundles [20], [4]. Another
fruitful theme has been the application of support theories to investigate
structural properties of the nilpotent cone and related algebro-geometric
objects [26].

There are obstructions to formulating a suitable support theory for
$\mathbb{G}$-modules. Not only is rational cohomology of $\mathbb{G}$
inadequate for this purpose, but also the abelian category $Mod(\mathbb{G})$
of $\mathbb{G}$-modules rarely has non-trivial projective objects (see [10])
and its injective objects are almost always infinite dimensional. The latter
obstruction persuades us to consider the bounded derived category
$D^{b}(Mod(\mathbb{G}))$ of the abelian category $Mod(\mathbb{G})$ and to
formulate associated stable categories. We verify that our constructions
$C^{\bullet}\mapsto\Pi(\mathbb{G})_{C^{\bullet}}$ and
$C^{\bullet}\mapsto\mathbb{P}\mathfrak{C}(\mathbb{G})_{C^{\bullet}}$ are well
defined on objects of the stable category $StMod(\mathbb{G})$ and satisfy the
standard properties expected of a good theory of supports. Although
$C^{\bullet}\mapsto\Pi(\mathbb{G})_{C^{\bullet}}$ satisfies numerous good
properties, we are far from recognizing which subspaces of $\Pi(\mathbb{G})$
are realizable as supports for a bounded complex of $\mathbb{G}$-modules. We
lack the analogue for $\mathbb{G}$-modules of the central result of J. Carlson
(easily extended from finite groups to arbitrary finite group scheme $G$) that
any closed subvariety of $\Pi(G)$ can be realized as the support of some
finite dimensional $G$-module [6].

We highlight some of the contents of this paper. In Section 1, we consider
four variations of support theories for modules for a finite group scheme:

$\quad M\mapsto\mathbb{P}|G|_{M};\quad M\mapsto\mathbb{P}V_{r}(G)_{M};\quad
M\mapsto\Pi(G)_{M};\quad
M\mapsto\mathbb{P}\mathfrak{C}_{r}(\mathbb{G}_{(r)})_{M}.$

After briefly recalling the formulations of each, we verify in Theorems 1.3,
1.5, 1.6 and 1.12 that these theories are equivalent for those finite group
schemes $G$ and $G$-modules $M$ for which they are defined. The $\pi$-point
theory $M\mapsto\Pi(G)_{M}$ applies to any finite group scheme; in Definition
2.2, we show that this theory naturally extends to a theory for linear
algebraic groups $\mathbb{G}$ and their $\mathbb{G}$-modules,
$M\mapsto\Pi(\mathbb{G})_{M}$.

In Section 2, we verify for a Frobenius kernel $\mathbb{G}_{(r)}$ of a linear
algebraic group $\mathbb{G}$ of exponential type that the exponential support
theory $M\mapsto\mathbb{P}\mathfrak{C}_{r}(\mathbb{G}_{(r)})_{M}$ also extends
to a theory for $\mathbb{G}$-modules
$M\mapsto\mathbb{P}\mathfrak{C}(\mathbb{G})_{M}$. In Theorem 2.6, we show that
this exponential theory $M\mapsto\mathbb{P}\mathfrak{C}(\mathbb{G})_{M}$ is
equivalent to the $\pi$-point theory $M\mapsto\Pi(\mathbb{G})_{M}$ for linear
algebraic groups of exponential type and arbitrary $\mathbb{G}$-modules.
Proposition 2.8 establishes further good properties of the theory
$M\mapsto\mathbb{P}\mathfrak{C}(\mathbb{G})_{M}$. In Examples 3.1, 3.2. 3.3,
and 3.4 of Section 3, we investigate examples of $Mod(\mathbb{G})$-realizable
subsets $\mathbb{P}\mathfrak{C}(\mathbb{G})_{M}\ \subset\
\mathbb{P}\mathfrak{C}(\mathbb{G})$.

For any finite group scheme $G$ over $k$, Theorem 4.2 extends to arbitrary
$G$-modules an important theorem of J. Rickard [30, Thm2.1] by giving an
explicit equivalence of the stable module category $StMod(G)$ with the
localization of the homotopy category $\mathcal{K}^{b}(Mod(G))$ of bounded
complexes of $G$ by the thick subcategory of bounded complexes of injective
$G$-modules. In Definition 4.7, we introduce the tensor triangulated category
$StMod(\mathbb{G})$ which serves as a natural domain for our support theory
for bounded complexes of arbitrary $\mathbb{G}$-modules. Motivated by
Rickard’s equivalence, we formulate our stable module category
$StMod(\mathbb{G})$ as a localization of the bounded derived category
$D^{b}(Mod(\mathbb{G}))$ by the thick subcategory of mock injective modules.
Defined similarly, the stable module category $stmod(\mathbb{G})$ of
Definition 4.7 is almost all examples equal to $D^{b}(mod(\mathbb{G}))$, the
bounded derived category of $mod(\mathbb{G})$.

We extend the support theories $M\mapsto\Pi(\mathbb{G})_{M}$ and
$M\mapsto\mathbb{P}\mathfrak{C}(\mathbb{G})_{M}$ to bounded complexes of
$\mathbb{G}_{(r)}$-modules and then to bounded complexes of
$\mathbb{G}$-modules in Section 5. As seen in Proposition 5.2, our
construction $C^{\bullet}\ \mapsto\Pi(\mathbb{G}_{(r)})_{C^{\bullet}}$
“agrees” with the usual $\pi$-support theory
$M\mapsto\Pi(\mathbb{G}_{(r)})_{M}$ for $\mathbb{G}_{(r)}$-modules. Theorem
5.4 shows that $C^{\bullet}\ \mapsto\ \Pi(\mathbb{G})_{C^{\bullet}}$ satisfies
all of the standard properties of a theory of supports for
$StMod(\mathbb{G})$. Among these properties, we mention the criterion for
$\Pi(\mathbb{G})_{C^{\bullet}}$ to be empty (if and only if $C^{\bullet}$ is
an object of $Mock^{b}(\mathbb{G})$), the more general fact that
$\Pi(\mathbb{G})_{C^{\bullet}}$ depends only upon the isomorphism class of
$C^{\bullet}$ in $StMod(\mathbb{G})$, and the tensor product property
$\Pi(\mathbb{G})_{C^{\bullet}\otimes D^{\bullet}}\ =\
\Pi(\mathbb{G})_{C^{\bullet}}\cap\Pi(\mathbb{G})_{D^{\bullet}}$.

In Section 6, we classify certain full subcategories of $stmod(\mathbb{G})$
and $StMod(\mathbb{G})$ in terms of certain subsets of $\Pi(\mathbb{G})$.
Namely, Theorem 6.5 establishes that $(r)$-complete, thick tensor ideals
$\mathfrak{C}\ \subset\ stmod(\mathbb{G})$ are classified by locally
$stmod(\mathbb{G})$-realizable subsets of $\mathbb{P}(\mathbb{G})$, and
Theorem 6.10 demonstrates that $(r)$-complete, localizing subcategories
$\tilde{\mathfrak{C}}\ \subset\ StMod(\mathbb{G})$ are classified by
$StMod(\mathbb{G})$-realizable subsets of $\Pi(\mathbb{G})$. The technique of
proof of both these theorems is to pass from $\mathbb{G}$ to the family of
$\\{\mathbb{G}_{(r)},r>0\\}$ of Frobenius kernels of $\mathbb{G}$. For
$stmod(\mathbb{G})$, we use the classification of thick tensor ideals of
$stmod(\mathbb{G}_{(r)})$ given by J. Pevtsova and the author [18, Thm6.3]
which in turn utilizes J. Rickard’s idempotents [31]. For $StMod(\mathbb{G})$,
we use the classification of localizing subcategories of
$StMod(\mathbb{G}_{(r)})$ proved by D. Benson, H. Krause, S. Iyengar, and J.
Pevtsova [3, Thm10.1].

In Section 7, we mention various questions and challenges concerning our
support theory which would enhance our understanding of $\mathbb{G}$-modules.

Throughout this paper, $k$ will denote a field of characteristic $p>0$ and all
linear algebraic groups $\mathbb{G}$ will be assumed to be connected; in other
words $\mathbb{G}$ will be assumed to be a reduced, connected affine group
scheme of finite type over $k$.

I am especially indebted to Julia Pevtsova whose suggestion of consideration
of colimits of supports of $\mathbb{G}_{(r)}$-modules I finally appreciated. I
also thank Cris Negron for helpful discussions on matters categorical and Paul
Sobaje for patiently answering my questions. Finally, I dedicate this paper to
the memory of my friend and collaborator Brian Parshall, who contributed much
to this mathematics.

##  1\. Comparison of support theories for finite group schemes

We review a few salient features of four formulations of support theories for
modules $M$ for a finite group scheme $G$

$M\mapsto\mathbb{P}|G|_{M};\quad M\mapsto\mathbb{P}V_{r}(G)_{M};\quad
M\mapsto\Pi(G)_{M};\quad
M\mapsto\mathbb{P}\mathfrak{C}_{r}(\mathbb{G}_{(r)})_{M}$

and establish their close relationships. In Theorem 1.3, we recall that the
cohomological support theory $M\mapsto\mathbb{P}|G|_{M}$ agrees with the
infinitesimal 1-parameter support theory $M\mapsto\mathbb{P}V_{r}(G)_{M}$ for
an infinitesimal group scheme of height $\leq r$ provided that $M$ is finite
dimensional. In Theorem 1.5, we recall that the cohomological support theory
$M\mapsto\mathbb{P}|G|_{M}$ is equivalent to the $\pi$-point support theory
$M\mapsto\Pi(G)_{M}$ for any finite group scheme provided $M$ is finite
dimensional. In Theorem 1.6, we make explicit the equivalence between
$M\mapsto\Pi(\mathbb{G}_{(r)})_{M}$ and
$M\mapsto\mathbb{P}V_{r}(\mathbb{G}_{(r)})_{M}$.

This section concludes with the formulation the exponential support theory
$M\mapsto\mathbb{P}\mathfrak{C}_{r}(\mathbb{G}_{(r)})_{M}$ for a Frobenius
kernel of a linear algebraic group $\mathbb{G}$ of exponential type. Our
definition of a linear algebraic group of exponential type in Definition 1.8
allows the exponential
$\mathcal{E}:\mathcal{N}_{p}(\mathfrak{g})\times\mathbb{G}_{a}\ \to\
\mathbb{G}$ to be a continuous algebraic map (a rational map with unique
specialization at every geometric point). In Theorem 1.12, we demonstrate that
the exponential support theory
$M\mapsto\mathbb{P}\mathfrak{C}_{r}(\mathbb{G}_{(r)})_{M}$ agrees with the
infinitesimal 1-parameter support theory
$M\mapsto\mathbb{P}V_{r}(\mathbb{G}_{(r)})_{M}$ for Frobenius kernels
$\mathbb{G}_{(r)}$ of linear algebraic groups of exponential type. This
exponential support theory is an an extension of joint work with A. Suslin and
C. Bendel for infinitesimal group schemes [34], [35] which itself extends
earlier joint work with B. Parshall [16]. In some sense,
$M\mapsto\mathbb{P}\mathfrak{C}(\mathbb{G})_{M}$ is the “rank variety”
formulation of $M\mapsto\Pi(\mathbb{G})_{M}$.

######  Recollection 1.1.

Let $\mathbb{G}$ be a finite group scheme. We denote by $H^{\bullet}(G,k)$ the
commutative, graded $k$-algebra equal to $H^{*}(G,k)$ if $p=2$ and otherwise
equal to the subalgebra of $H^{*}(G,k)$ generated by classes of even degree.
We set $\mathbb{P}|G|$ equal to $\operatorname{Proj}\nolimits
H^{\bullet}(G,k)$. For $M$ finite dimensional, we denote by
$\mathbb{P}|G|_{M}\ \subset\ \mathbb{P}|G|$ the closed, reduced subscheme
determined by (the radical of) the homogenous ideal $ker\\{H^{\bullet}(G,k)\to
Ext_{G}^{*}(M,M)\\}$.

The additive group $\mathbb{G}_{a}$ plays a central role in what follows. The
coordinate algebra of $\mathbb{G}_{a}$ is the polynomial algebra $k[T]$ on one
variable, whereas the group algebra $k\mathbb{G}_{a}$ is the truncated
polynomial algebra $k[u_{0},\ldots,u_{n},\ldots]/(u_{i}^{p})$ on countably
infinite generators each of whose $p$-th powers is 0. Here, $u_{r}:k[T]\ \to\
k$ is the functional sending a polynomial $p(T)$ to its coefficient of
$T^{p^{r}}$. If we denote by $v_{i}$ the dual basis element to $T^{i}\in
k[\mathbb{G}_{a}]$ so that $v_{i}(p(T))$ reads off the coefficient of $T^{i}$
in $p(T)$, then $u_{r}=v_{p^{r}}$. We denote by $\epsilon_{r}$ the map of
$k$-algebras (but not Hopf algebras, unless $r=1$)

(1.1.1) $\epsilon_{r}:k\mathbb{G}_{a(1)}\simeq k[t]/t^{p}\to
k\mathbb{G}_{a},\quad t\mapsto u_{r-1}.$

We also use $\epsilon_{r}$ to denote the factorization of (1.1.1) through
$k\mathbb{G}_{a(r)}\hookrightarrow k\mathbb{G}_{a}$,
$\epsilon_{r}:k\mathbb{G}_{a(1)}\ \to\ k\mathbb{G}_{a(r)}$.

The analysis of $M\mapsto\mathbb{P}|G|_{M}$ was achieved for the very special
case in which $G\simeq\mathbb{Z}/p^{\times r}$ by replacing cohomological
support varieties for modules for these elementary abelian $p$-group by more
accessible rank varieties. This was first generalized to restricted Lie
algebras (see [16]) and then greatly expanded to all infinitesimal group
schemes in joint work with A. Suslin and C. Bendel in [34], [35] as we next
recall.

######  Recollection 1.2.

Let $G$ be an infinitesimal group scheme of height $\leq r$. The functor on
commutative $k$-algebras sending $A$ to the set of maps of group schemes
$\phi:\mathbb{G}_{a(r),A}\to G_{A}$ is representable by a graded affine
$k$-scheme $V_{r}(G)$ whose projectivization we denote by
$\mathbb{P}V_{r}(G)$. For each finite dimensional $G$-modules $M$, one
considers the closed subscheme $\mathbb{P}V_{r}(G)_{M}\ \subset\
\mathbb{P}V_{r}(G)$ whose $K$-points for any field extension $K/k$ are
represented by (infinitesimal) 1-parameter subgroups
$\phi:\mathbb{G}_{a(r),K}\to G_{K}$ such that $\phi_{*}(u_{r-1})$ acts freely
on $M_{K}$ (in the sense that $(\phi\circ\epsilon_{r})*(M_{K})$ is a free
$K\mathbb{G}_{a(1)}\simeq K\mathbb{Z}/p$-module).

######  Theorem 1.3.

Let $G$ be an infinitesimal group scheme of height $\leq r$. There is a
natural (in $G$) map $\psi:H^{\bullet}(G,k)\to V_{r}(G)$ inducing a universal
homeomorphism

(1.3.1) $\Psi:\mathbb{P}V_{r}(G)\ \to\ \mathbb{P}|G|,\quad\text{restricting
to}\quad\Psi:\mathbb{P}V_{r}(G)_{M}\ \to\ \mathbb{P}|G|_{M}$

(where $\mathbb{P}V_{r}(G)$ denotes $\operatorname{Proj}\nolimits
k[V_{r}(G)]$) for each finite dimensional $G$-modules $M$.

The most general construction of a support theory for finite group schemes is
the $\pi$-point theory $M\mapsto\Pi(G)_{M}$ introduced by J. Pevtsova and the
author [18] which we next recall.

######  Recollection 1.4.

Let $G$ be a finite group scheme $G$ over $k$. Elements of $\Pi(G)$ are
equivalence classes of flat maps of $K$-algebras for some field extension
$K/k$, $\alpha_{K}:K[t]/t^{p}\ \to\ KG$, which factor through some unipotent
abelian subgroup scheme $U_{K}\subset G_{K}$. Two such flat maps
$\alpha_{K}:K[t]/t^{p}\ \to\ KG,\ \beta_{L}:L[t]/t^{p}\ \to\ LG$ are
equivalent if there exists a common field extension $\Omega$ of $K,\ L$ such
that $\alpha_{\Omega}^{*}(M_{\Omega})$ is free if and only if
$\beta_{\Omega}^{*}(M_{\Omega})$ is free for any finite dimensional $G$-module
$M$.

We consider subsets of $\Pi(G)$ of the form

$\Pi(G)_{M}\ \equiv\ \\{[\alpha_{K}]:\alpha_{K}^{*}(M_{K})\ is\ not\ free\ as\
a\ K[t]/t^{p}{\text{-}}module\\}$

for an arbitrary $\mathbb{G}$-module $M$, well defined by [13, Thm6.6]. The
closed subsets of $\Pi(G)$ are the subsets of this form with $M$ finite
dimensional.

The following theorem tell us us that $M\mapsto\mathbb{P}|G|_{M}$ is
equivalent to $M\mapsto\Pi(G)_{M}$ provided that $M$ is finite dimensional.

######  Theorem 1.5.

[18, Thm 7.4] There exists a natural scheme structure on $\mathbb{P}|G|$ and a
scheme-theoretic isomorphism

(1.5.1) $\Phi:\mathbb{P}|G|\ \stackrel{{\scriptstyle\sim}}{{\to}}\
\Pi(G),\quad\text{restricting to}\quad\mathbb{P}|G|_{M}\
\stackrel{{\scriptstyle\sim}}{{\to}}\ \Pi(G)_{M}$

for any finite dimensional $G$-module $M$.

The isomorphisms $\Psi$ of (1.3.1) and $\Phi$ of (1.5.1) are quite abstract.
The natural map $\psi:H^{\bullet}(G,k)\ \to\ k[V(G)]$ of [34, Thm1.14]
determined $\Psi$ for an infinitesimal group scheme of height $\leq r$ uses
the universal height $r$ 1-parameter subgroup for $G$. The isomorphism $\Phi$
of [18, Thm7.5] entails consideration of a sheafified version of the
endomorphisms of the stable module category of finite dimensional $G$-modules.

The following proposition makes $\Psi,\ \Phi$ more concrete.

######  Theorem 1.6.

Let $G$ be an infinitesimal group scheme over $k$ of height $\leq r$. Then the
composition

$\mathbb{P}V_{r}(G)\quad\stackrel{{\scriptstyle\Psi}}{{\to}}\quad\mathbb{P}|G|\quad\stackrel{{\scriptstyle\Phi}}{{\to}}\Pi(G)$

sends the $K$-point represented by the 1-parameter subgroup
$\eta_{K}:\mathbb{G}_{a(r),K}\to G_{K}$ to the point represented by the
$\pi$-point $\eta_{K*}\circ\epsilon_{r}:K[t]/t^{p}\to K\mathbb{G}_{a(r),K}\to
KG_{K}$.

###### Proof.

By [18, Thm3.6], the inverse of $\Phi$ (as a homeomorphism) sends a
$\Pi$-point $\alpha_{K}:K[t]/t^{p}$ to

(1.6.1) $ker\\{\alpha_{K}^{*}:H^{\bullet}(G_{K},K)\to
H^{\bullet}(\mathbb{G}_{a,K},K\\}\cap H^{\bullet}(G,k).$

To identify $\Psi$, we first consider the special case $G=GL_{N(r)}$. By [17,
Prop2.9], each equivalence class of $\pi$-points of $GL_{N(r)}$ contains a
representative of the form

$\alpha_{K}\ =\ \mathcal{E}_{\underline{B}*}\circ\epsilon_{r}:K[T]/t^{p}\to
K\mathbb{G}_{a(r),K}\to k\mathbb{G}_{K},$

unique up to base change by $L/K$ and up to scalar multiples. Then $\Psi$
sends a $K$-point $k[V_{r}(G)]\to K$ given by evaluation at the 1-parameter
subgroup $\mathcal{E}_{\underline{B}}$ defined over $K$ to

(1.6.2) $ker\\{ev_{\underline{B}}\circ\psi:H^{\bullet}(G,k)\to k[V_{r}(G)]\to
K\\}.$

Thus, for $G=GL_{n(r)}$, it suffices to show the equality of the radicals of
the kernels of (1.6.2) and of

(1.6.3) $ev_{\underline{B}}\circ\psi:H^{\bullet}(G,k)\to k[V_{r}(G)]\to
K,\quad(\mathcal{E}_{\underline{B}*}\circ\epsilon_{r})^{*}:H^{\bullet}(G_{K},K)\to
H^{\bullet}(\mathbb{G}_{a,K},K\\}\cap H^{\bullet}(G,k).$

This follows from [34, Thm5.2].

For a general infinitesimal group scheme $G$, we embed $G$ in some $GL_{N}$
and use [35, Thm5.2] which asserts that the map $\psi:H^{\bullet}(G,k)\to
k[V_{r}(G)]$ has nilpotent kernel and [35, Prop4.3] which implies that
$H^{\bullet}(GL_{N},k)\to H^{\bullet}(G,k)$ has image containing all
$p^{r}$-th powers so that $\mathbb{P}|G|\ \to\ \mathbb{P}|GL_{N(r)}|$ is
injective. We consider the diagram

(1.6.4)
$\textstyle{\mathbb{P}V_{r}(GL_{N(r)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\textstyle{\mathbb{P}|GL_{N(r)}|\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{\Pi(GL_{N(r)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}V_{r}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\textstyle{\mathbb{P}|G|\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{\Pi(G)}$

which commutes by the naturality of $\psi$ (which determines $\Psi$) and
$\Phi$. Thus, a diagram chase verifies that the radicals of the two kernels of
the maps of algebras (1.6.1), (1.6.2) are equal. ∎

The abstract formulation of $M\mapsto\Pi(G)_{M}$ does not easily lead to
computations. Following [34], we consider the following affine $k$-varieties,
worthy of study in their own right.

######  Definition 1.7.

Let $\mathbb{G}$ be a linear algebraic group with Lie algebra $\mathfrak{g}$.
We denote by $\mathcal{N}_{p}(\mathfrak{g})\subset\mathfrak{g}$ the
$p$-nilpotent variety of $\mathfrak{g}$, the (reduced) closed subvariety of
the affine space $\operatorname{Spec}\nolimits
S(\mathfrak{g}^{*})\simeq\mathbb{A}^{d}$ whose $K$-points are the elements
$X\in\mathfrak{g}_{K}$ such that $X^{[p]}=0$ for any field extension $K/k$.
For any $r>0$, we define

$\mathfrak{C}_{r}(\mathbb{G})\ \equiv\
\mathfrak{C}_{r}(\mathcal{N}_{p}(\mathfrak{g}))\ \subset\
\mathcal{N}_{p}(\mathfrak{g})^{\times r}$

to be the (reduced) closed subvariety of
$\mathcal{N}_{p}(\mathfrak{g})^{\times r}$ whose $K$-points are $r$-tuples
$\underline{B}=(B_{0},\ldots,B_{r-1})\in\mathcal{N}_{p}(\mathfrak{g}_{K})^{\times
r}$ satisfying the condition the $[B_{i},B_{j}]=0$ for all $i,j$ with $0\leq
i<j<r$.

The following definition of a linear algebraic group of exponential type is a
slight modification of that of [13] in that we allow $\mathcal{E}$ to be a
continuous algebraic map and we explicitly require $\mathcal{E}$ to be
$\mathbb{G}$-equivariant. Following [12], we define a continuous algebraic map
$f:X\ \dashrightarrow\ Y$ to be a “roof” $X\stackrel{{\scriptstyle
p}}{{\leftarrow}}\tilde{X}\stackrel{{\scriptstyle g}}{{\to}}Y$ of $k$-schemes
such that $p:\tilde{X}\to X$ is a universal homeomorphism (i.e., a finite,
surjective map such that $k(\tilde{x})$ is purely inseparable of
$k(p(\tilde{x}))$ for all points $\tilde{x}\in\tilde{X}$). A typical example
is a rational map from $X$ to $Y$ (i.e., a morphism with domain a dense open
set) whose graph in $X\times Y$ projects to $X$ via a universal homeomorphism.
A bicontinuous algebraic map of irreducible varieties $X,\ Y$,
$f:X\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}Y$, is given by a finite,
purely separable extension $k(X)$ over $k(Y)$ inducing a bijection between the
$K$-points of $X$ and the $K$-points of $Y$ for any algebraically closed
extension $K$ of $k$. Examples include the Frobenius map
$F:\mathbb{G}\to\mathbb{G}^{(1)}$ and the normalization $\tilde{N}(sl_{2})\to
N(sl_{2})$.

As remarked in [13, Rem1.7], if the linear algebraic group $\mathbb{G}$ admits
the structure $(\mathbb{G},\mathcal{E})$ of an algebraic group of exponential
type, two such structures are isomorphic via an automorphism of
$\mathcal{N}_{p}(\mathfrak{g})$.

######  Definition 1.8.

Let $\mathbb{G}$ be a linear algebraic group over $k$ with Lie algebra
$\mathfrak{g}$ equipped with a $\mathbb{G}$-equivariant, continuous algebraic
map $\mathcal{E}:\mathcal{N}_{p}(\mathfrak{g})\times\mathbb{G}_{a}\
\dashrightarrow\ \mathbb{G}$ sending a geometric point $(B,\alpha)$ of
$\mathcal{N}_{p}(\mathfrak{g})\times\mathbb{G}_{a}$ to
$\mathcal{E}_{B}(\alpha)$. Then $(\mathbb{G},\mathcal{E})$ is said to be an
algebraic group of exponential type provided that

  1. (1)

For each Frobenius kernel $\mathbb{G}_{(r)}\ \subset\mathbb{G}$ and geometric
point $\operatorname{Spec}\nolimits K\to V(\mathbb{G}_{(r)})$, the
corresponding 1-parameter subgroup $\mathbb{G}_{a(r),K}\ \to\
\mathbb{G}_{(r),K}$ admits a unique representation of the form

$\mathcal{E}_{\underline{B}}\ \equiv\
\large\prod_{s=0}^{r-1}\mathcal{E}_{B_{s}}\circ F^{s}:\mathbb{G}_{a(r),K}\
\to\ \mathbb{G}_{(r),K}.$

for some $K$-point
$\underline{B}=(B_{0},\ldots,B_{r-1})\in\mathfrak{C}_{r}(\mathbb{G}).$ Here,
$F:\mathbb{G}_{a}\to\mathbb{G}_{a}$ is the Frobenius map, and
$\mathcal{E}_{B}:\mathbb{G}_{a,K}\to\mathbb{G}_{K}$ is the 1-parameter
subgroup associated to the map of $K$-points determined by $\mathcal{E}$
restricted to $\\{B\\}\times\mathbb{G}_{a}$.

  2. (2)

For each algebraically closed field extension $K/k$, a 1-parameter subgroup
$\mathbb{G}_{a,K}\to\mathbb{G}_{K}$ has a unique representation of the form

$\mathcal{E}_{\underline{B}}\ \equiv\ \large\prod_{s\geq
0}\mathcal{E}_{B_{s}}\circ F^{s}:\mathbb{G}_{a,K}\ \to\ \mathbb{G}_{K}.$

for some $K$-point $\underline{B}=(B_{0},\ldots,B_{n},\ldots)$ of
$\mathfrak{C}(\mathbb{G})\ \equiv\
\varinjlim_{r}\mathfrak{C}_{r}(\mathbb{G})$.

We are much indebted to P. Sobaje for guiding us to the following results in
the literature.

######  Theorem 1.9.

The following are examples of linear algebraic groups $\mathbb{G}$ of
exponential type:

  1. (1)

Simple algebraic groups $\mathbb{G}$ of classical type, their standard
parabolic subgroups, and the unipotent radicals of these standard parabolic
subgroups. See [34].

  2. (2)

Connected, reductive groups $\mathbb{G}$ over an algebraically closed field
$k$ whose derived subgroup has no factor of exceptional type with Coxeter
number $h$ with $p\leq 2h-2$ . See [25, Thm9.5] for a somewhat sharper result,
with an explicit list of problematic exceptional simple factors.

  3. (3)

$\mathbb{U}$ is the unipotent radical of a parabolic subgroup
$\mathbb{P}\subset\mathbb{G}$ of a connected, reductive group over an
algebraically closed field $k$ with the property that the nilpotent class of
$\mathbb{U}$ is $<p$. See [32, Prop5.3].

###### Proof.

The examples of (1) are given in [34, Lem1.8], where
$\mathcal{E}:\mathcal{N}_{p}(\mathfrak{g})\times\mathbb{G}_{a}\to\mathbb{G}$
is constructed as a scheme-theoretic morphism and no condition is placed on
the field $k$.

In the context of (2), G. McNinch establishes
$\mathcal{E}:\mathcal{N}_{p}(\mathfrak{g})\times\mathbb{G}_{a}\to\mathbb{G}$
as an “isomorphism of varieties” (but not of schemes). Moreover, $\mathcal{E}$
is given by the restriction of the exponential
$exp:\mathcal{N}_{p}(gl_{N})\times\mathbb{G}_{a}\to GL_{N}$ associated to some
faithful representation $(\rho,V)$ of $\mathbb{G}$ of some dimension $N$.
Although McNinch only considers infinitesimal 1-parameter subgroups, his
results also apply to establish condition (2) of Definition 1.8 when
supplemented by the following observation. Given a 1-parameter subgroup
$\phi:\mathbb{G}_{a}\to\mathbb{G}$, condition (1) of Definition 1.8 tells us
that $(\rho\circ\phi)_{|\mathbb{G}_{(r)}}$ is given uniquely by a product of
exponentials; using the observation of [34] that 1-parameter subgroups
$\mathbb{G}_{a}\to GL_{N}$ have a unique, finite product representation, we
conclude that the product representations for
$(\rho\circ\phi)_{|\mathbb{G}_{(r)}}$ agree for sufficiently large $r$ and
thus that $\rho\circ\phi$ agrees with this “stable” product representation of
$(\rho\circ\phi)_{|\mathbb{G}_{(r)}}$.

In the context of (3), G. Seitz proves in [32, Prop5.3] that there is a unique
$\mathbb{P}$-equivariant isomorphism $\theta:\mathfrak{u}\
\stackrel{{\scriptstyle\sim}}{{\to}}\ \mathbb{U}$ of algebraic groups, where
$\mathfrak{u}$ is viewed as a vector group over $k$. Thus, any 1-parameter
subgroup $\mathbb{G}_{a}\to\mathbb{U}$ is given by composing an additive map
$\mathbb{G}_{a}\to\mathfrak{u}$ with $\theta$. ∎

In the following definition, we consider $p$-nilpotent linear operators on
$K[\mathbb{G}]$ of the form $\mathcal{E}_{B*}(u_{r})$. We recall that
$u_{r}:k[\mathbb{G}_{a}]\to k$ reads of the coefficient of $T^{p^{r-1}}$ of a
polynomial in in $k[T]=k[\mathbb{G}_{a}]$ and $\mathcal{E}_{B*}(u_{r})$ is the
composition $u_{r}\circ\mathcal{E}_{B}^{*}:K[\mathbb{G}]\to
K[\mathbb{G}_{a}]\to K$.

######  Definition 1.10.

Let $(\mathbb{G},\mathcal{E})$ be a linear algebraic group of exponential
type. For any $\mathbb{G}$-module $M$, we define the Jordan type of $M$ at a
$K$-point $\mathcal{E}_{\underline{B}}$ of
$\mathfrak{C}_{r}(\mathcal{N}_{p}(\mathfrak{g}))$ to be block sum
decomposition

$JT(M)_{\mathcal{E}_{\underline{B}}}\ \equiv\ [1]^{\oplus
m_{1}}\oplus\cdots\oplus[p]^{m_{p}},\ 0\leq m_{i}\leq\infty$

for the action on $M_{K}$ of the $p$-nilpotent element $\sum_{s\geq
0}(\mathcal{E}_{B_{s}})_{*}(u_{s})\in K\mathbb{G};$ we further define the
“stable Jordan type of $M$ at $\mathcal{E}_{\underline{B}}$ to be

$sJT(M)_{\mathcal{E}_{\underline{B}}}\ \equiv\ [1]^{\oplus
m_{1}}\oplus\cdots\oplus[p-1]^{m_{p-1}},\ 0\leq m_{i}\leq\infty.$

We define $\mathfrak{C}_{r}(\mathcal{N}_{p}(\mathfrak{g}))_{M}\ \subset\
\mathfrak{C}_{r}(\mathcal{N}_{p}(\mathfrak{g}))$ to be the subspace whose set
of $K$-points is the subset of those $K$-points $\underline{B}$ of
$\mathfrak{C}_{r}(\mathcal{N}_{p}(\mathfrak{g}))$ with the property that the
stable Jordan type $sJT(M)_{\mathcal{E}_{\underline{B}}}$ is not 0.

Equivalently (see[13, Defn4.4]), the $K$-point $\underline{B}$ is a $K$-point
of $\mathfrak{C}_{r}(\mathcal{N}_{p}(\mathfrak{g}))_{M}$ $\iff$
$(\mathcal{E}_{\Lambda_{r}(\underline{B})})\circ\epsilon_{r})^{*}(M_{K})$ is
not free as a $K[t]/t^{p}$-module.

The natural grading on the affine scheme $V_{r}(\mathbb{G}_{(r)})$ for
$(\mathbb{G},\mathcal{E})$ a linear algebraic group of exponential type
corresponds to the monoid action

$V_{r}(\mathbb{G}_{(r)})\times\mathbb{A}^{1}\ \to\
V_{r}(\mathbb{G}_{(r)}),\quad(\mathcal{E}_{\underline{B}},a)\
\mapsto\mathcal{E}_{a\bullet\underline{B}},$

where $\mathcal{E}_{a\bullet\underline{B}}\ =\
\mathcal{E}_{\underline{B}}(a\cdot t)$. Since $(\mathcal{E}_{B}\circ
F^{s})(a\cdot t)=\mathcal{E}_{a^{s}\cdot B}(t)$, we conclude that
$V(\mathbb{G}_{(r)})\times\mathbb{A}^{1}\ \to\ V(\mathbb{G}_{(r)})$ is given
by

(1.10.1) $(\mathcal{E}_{\underline{B}},\alpha)\ \mapsto\
\mathcal{E}_{(\alpha\bullet\underline{B})},\quad\alpha\bullet(B_{0},\ldots
B_{r-1})\equiv(a\cdot B_{0},\ldots,a^{p^{r-1}}\cdot B_{r-1}).$

We denote by $\mathbb{P}V_{r}(\mathbb{G}_{(r)})$ the projective variety
associated to $V_{r}(\mathbb{G}_{(r)})$.

We introduce a grading on $\mathfrak{C}_{r}(\mathbb{G})$ which enables
bicontinuous algebraic maps
$\rho_{r}\circ\Lambda_{r}:\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})\to\
\mathbb{P}V_{r}(\mathbb{G}_{(r)})$. Namely, we define the monoid action
$\mathfrak{C}_{r}(\mathbb{G})\times\mathbb{A}^{1}\ \to\
\mathfrak{C}_{r}(\mathbb{G})$ as the restriction of the monoid action
$\mathfrak{g}^{\times r}\times\mathbb{A}^{1}\to\mathfrak{g}^{\times r}$ given
by

(1.10.2) $(\underline{B},a)\ \mapsto\ a*\underline{B},\quad a*(B_{0},\ldots
B_{r-1})\equiv(a^{p^{r-1}}\cdot B_{0},\ldots,a\cdot B_{r-1}).$

We denote by $\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})$ the associated
projective variety. For any $\mathbb{G}$-module $M$ and any algebraically
closed field extension $K/k$, we define the $K$-points of
$\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})_{M}$ to be the image $K$-points of
$\mathfrak{C}_{r}(\mathbb{G})_{M}\backslash\\{0\\}$.

######  Definition 1.11.

Let $(\mathbb{G},\mathcal{E})$ be a linear algebraic group of exponential
type. We denote by

(1.11.1) $\rho_{r}:\mathfrak{C}_{r}(\mathbb{G})\quad\dashrightarrow\quad
V_{r}(\mathbb{G}_{(r)}).$

the continuous algebraic map associating to a $K$ point $\underline{B}$ of
$\mathfrak{C}_{r}(\mathbb{G})$ the $K$-point
$\mathcal{E}_{\underline{B}}:\mathbb{G}_{a(r),K}\to\mathbb{G}_{(r),K}$ of
$V_{r}(\mathbb{G}_{(r)})$; so defined, $\rho_{r}$ is a bicontinuous algebraic
map and thus a universal homeomorphism. Here, we have abused notation by using
$\mathcal{E}_{\underline{B}}$ to denote both the 1-parameter subgroup
$\mathbb{G}_{a,K}\to\mathbb{G}_{K}$ and its restriction
$(\mathcal{E}_{\underline{B}})_{(r)}:\mathbb{G}_{a(r),K}\to\mathbb{G}_{(r),K}$.

We denote by $\Lambda_{r}:\mathfrak{C}_{r}(\mathbb{G})\to\
\mathfrak{C}_{r}(\mathbb{G})$ the isomorphism sending $(B_{0},\ldots,B_{r-1})$
to $(B_{r-1},\ldots,B_{0})$.

######  Theorem 1.12.

Let $(\mathbb{G},\mathcal{E})$ be an algebraic group of exponential type and
let $M$ be a $\mathbb{G}$-module. The bicontinuous algebraic map given as the
composition

(1.12.1) $\rho_{r}\circ\Lambda_{r}:\mathfrak{C}_{r}(\mathbb{G})\
\stackrel{{\scriptstyle\sim}}{{\to}}\ \mathfrak{C}_{r}(\mathbb{G})\
\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}\ V_{r}(\mathbb{G}_{(r)})$

commutes with the gradings of $\mathfrak{C}_{r}(\mathbb{G})$ given by (1.10.2)
and of $V_{r}(G)$ given by (1.10.1).

Furthermore, $\rho_{r}\circ\Lambda_{r}$ satisfies the following properties:

  1. (1)

For all $\mathbb{G}$-modules $M$, $\rho_{r}\circ\Lambda_{r}$ determines a
bicontinuous algebraic map

(1.12.2) $\rho_{r}\circ\Lambda_{r}:\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})\
\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}\
\mathbb{P}V_{r}(\mathbb{G}_{(r)}),\quad\text{restricting to}\ \
\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})_{M}\
\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}\
\mathbb{P}V_{r}(\mathbb{G}_{(r)})_{M_{|\mathbb{G}_{(r)}}}$

where $M_{|\mathbb{G}_{(r)}}$ denotes the restriction of $M$ to
$\mathbb{G}_{(r)}$.

  2. (2)

If $M$ is finite dimensional, then $\rho_{r}\circ\Lambda_{r}$ restricts to a
universal homeomorphism from $\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})_{M}\
\subset\ \mathbb{P}\mathfrak{C}_{r}(\mathbb{G})$ to
$\mathbb{P}V_{r}(\mathbb{G}_{(r)})_{M_{|\mathbb{G}_{(r)}}}\ \subset\
\mathbb{P}V_{r}(\mathbb{G}_{(r)})$.

  3. (3)

The adjoint action of $\mathbb{G}$ on $\mathfrak{g}$ induces a
$\mathbb{G}(K)$-action on the $K$-points of
$\mathbb{P}\mathfrak{C}(\mathbb{G})$ for any field extension $K/k$. This
action stabilizes the $K$-points of
$\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})_{M}$ for any $\mathbb{G}$-module $M$.

  4. (4)

The composition
$\Psi\circ(\rho_{r}\circ\Lambda_{r}):\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})\
\simeq\ \mathbb{P}|\mathbb{G}_{(r)}|$ is $\mathbb{G}(K)$ equivariant when
evaluated at $K$-points for any field extension $K/k$.

###### Proof.

Comparing the actions of $\mathbb{A}^{1}$ on $\mathfrak{C}_{r}(\mathbb{G})$ as
in (1.10.2) and of $\mathbb{A}^{1}$ on $V_{r}\mathbb{G})$ as in (1.10.1), we
see that $\rho_{r}\circ\Lambda_{r}$ respects degrees of homogeneous elements.

By [13, Prop4.3], the $\pi$-point $K[u]/u^{p}\to K\mathbb{G}_{(r)}$ given by
$u\mapsto\sum_{s=0}^{r-1}(\mathcal{E}_{B_{s}})_{*}(u_{s})$ is equivalent to
the $\pi$-point sending
$u\mapsto(\mathcal{E}_{\Lambda_{r}(\underline{B})})_{*}(u_{r-1})$ provided
that $B_{s}=0$ for $s\geq r$. Thus, the Jordan type of $M$ at the geometric
point $\mathcal{E}_{\underline{B}}$ of
$\mathfrak{C}_{r}(\mathcal{N}_{p}(\mathfrak{g}))$ as defined in Definition
1.10 has a block of size $<p$ consists if and only if the 1-parameter subgroup
$\mathcal{E}_{\Lambda_{r}(\underline{B})}:\mathbb{G}_{a(r),K}\to\mathbb{G}_{(r),K}$
lies in $V_{r}(\mathbb{G}_{(r)})_{M}$. This implies that (1.12.2) induces a
bijection on $K$ points for any $K/k$ and thus is a bicontinuous map.

To prove that $\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})_{M}\subset\
\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})$ is closed if $M$ is finite
dimensional, we observe that the condition on a $K$-point $\underline{B}$ of
$\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})$ to lie in
$\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})_{M}$ is the condition that the rank of
$(\mathcal{E}_{\Lambda_{r}(\underline{B})})_{*}(u_{r-1})\in K\mathbb{G}$
acting on $M_{K}$ is strictly less that $\frac{p-1}{p}dim(M)$.

Observe that $(M_{K})^{\tau}$ is isomorphic to $M_{K}$ as a
$\mathbb{G}$-module for any $\mathbb{G}$-module $M$ and any
$\tau\in\mathbb{G}(K)$. Moreover, for any $\tau\in\mathbb{G}(K)$, the
uniqueness argument of [13, Remark1.7] and the $\mathbb{G}$-equivariance of
Springer isomorphisms imply that $(\mathcal{E}_{\underline{B}})^{\tau}\ =\
\mathcal{E}_{\underline{B}^{\tau}}$. Thus,
$((\mathcal{E}_{\underline{B}})^{\tau})^{*}(M_{K})$ is isomorphic to
$(\mathcal{E}_{\underline{B}})^{*}(M_{K})$, thereby proving (3).

Finally, $\Psi\circ(\rho_{r}\circ\Lambda_{r})$ sends a $K$-point
$\underline{B}$ of $\mathfrak{C}_{r}(\mathbb{G})$ to the intersection with
$H^{\bullet}(G,k)$ with the kernel of
$((\mathcal{E}_{\Lambda_{r}(\underline{B})})_{*}\circ\epsilon_{r})^{*}:H^{\bullet}(\mathbb{G}_{(r)},K)\to
H^{\bullet}(K[t]/t^{p},K)$. For any $\tau\in\mathbb{G}(K)$,
$\tau\circ\mathcal{E}_{\underline{B}}\ =\ \mathcal{E}_{\underline{B}^{\tau}}$
by $\mathbb{G}$-equivariance for $\mathcal{E}$. Thus, $\tau$ applied to this
kernel equals the intersection with $H^{\bullet}(G,k)$ with the kernel of
$(\mathcal{E}_{\Lambda_{r}(\underline{B}^{\tau})})_{*}\circ\epsilon_{r})^{*}$

∎

##  2\. Support theories $M\mapsto\Pi(\mathbb{G})_{M},\
M\mapsto\mathfrak{C}(\mathbb{G})_{M}$

In this section, we consider support theories for $\mathbb{G}$-modules for a
linear algebraic group. Our first theory, $M\mapsto\Pi(\mathbb{G})_{M}$ is a
natural extension of the $\pi$-point support theory for finite group schemes
recalled in Recollection 1.4. Although simple to define and good for
establishing general properties, $M\mapsto\Pi(\mathbb{G})_{M}$ does not lend
itself to computation. With this in mind, we also consider the natural
extension $M\mapsto\mathbb{P}\mathfrak{C}(\mathbb{G})_{M}$ of the exponential
theory for Frobenius kernels of linear algebraic groups of exponential type
formulated in Definition 1.7.

For notational convenience, we shall frequently denote the restriction
$M_{|\mathbb{G}_{(r)}}$ of a $\mathbb{G}$-module $M$ to some Frobenus
kernel$\mathbb{G}_{(r)}\ \subset\ \mathbb{G}$ by $M$.

We utilize $M\mapsto\mathbb{P}\mathfrak{C}(\mathbb{G})_{M}$ to verify various
properties of $M\mapsto\Pi(\mathbb{G})_{M}$ for $\mathbb{G}$ a linear
algebraic group of exponential type. We also use
$M\mapsto\mathbb{P}\mathfrak{C}(\mathbb{G})_{M}$ in order to consider
$\mathbb{G}$-modules of bounded exponential degree in Proposition 2.12.

We begin with an observation about closed embeddings of infinitesimal group
schemes. This observation is contrary to the behavior of cohomological support
varieties for finite groups. For example, if $\tau$ is a finite group and
$\kappa\subset\tau$ is a $p$-Sylow subgroup, then $|\kappa|\to|\tau|$ is
rarely injective.

######  Proposition 2.1.

Let $i:H\hookrightarrow G$ be a closed embedding of infinitesimal group
schemes of height $\leq r$. Then for any $G$-module $M$, $i$ induces an
embedding $i_{*}:\Pi(H)\ \subset\ \Pi(G)$ which restricts to
$\Pi(H)_{i^{*}M}\subset\ \Pi(G)_{M}$.

###### Proof.

Clearly, composition with $i$ induces an embedding $V_{r}(H)\to V_{r}(G)$ and
thus by Theorem 1.6 an embedding $i:\Pi(H)\to\Pi(G)$ also given by composition
with $i$. In other words, sending a $\pi$-point $\alpha_{K}:K[t]/t^{p}\to KH$
to $i_{*}\circ\alpha_{K}:K[t]/t^{p}\to KH\to KG$ is well defined and injective
on equivalence classes of $\pi$-points. (Recall that $i_{*}:kH\to kG$ is flat
whenever $i:H\to G$ is a closed embedding of finite group schemes, so that
$i\circ\alpha_{K}$ is always a $\pi$-point whenever $\alpha_{K}$ is a
$\pi$-point for any closed embedding of finite group schemes; see [24,
8.16.2].)

Let $M$ be a $G$-module and $\alpha_{K}$ be a $\pi$-point of $H$. Then
$i\circ\alpha_{K}$ represents a point in $\Pi(G)_{M}$ $\iff$
$(i\circ\alpha_{K})^{*}(M_{K})$ is not free as a $K[t]/t^{p}$-module $\iff$
$\alpha_{K}^{*}(i^{*}M_{K})$ is not not free as a $K[t]/t^{p}$-module $\iff$
$\alpha_{K}$ represents a point of $\Pi(H)_{i^{*}M}$. Thus,
$i_{*}:\Pi(H)\to\Pi(G)$ restricts to $\Pi(H)_{i^{*}M}\to\Pi(G)_{M}$ for any
$G$-module $M$. ∎

Proposition 2.1 justifies the constructions of the following definition of
$M\mapsto\Pi(\mathbb{G})_{M}$ as a colimit with respect to $r$ of
$M\mapsto\Pi(\mathbb{G}_{(r)})_{M}$. One can view
$M\mapsto\Pi(\mathbb{G})_{M}$ as a support theory for modules for the
hyperalgebra $k\mathbb{G}\ \equiv\ \varinjlim_{r}\mathbb{G}_{(r)}$.

######  Definition 2.2.

Let $\mathbb{G}$ be a linear algebraic group over $k$. We define
$\Pi(\mathbb{G})$ to be the topological space $\Pi(\mathbb{G})\quad\
\equiv\quad\varinjlim_{r}\Pi(\mathbb{G}_{(r)})$, the colimit with the colimit
topology whose connecting maps
$\Pi(\mathbb{G}_{(r)})\to\Pi(\mathbb{G}_{(r+1)})$ are given by sending a
$\pi$-point $\alpha_{K}:K[t]/t^{P}\to K\mathbb{G}_{(r)}$ to its composition
with the flat map $K\mathbb{G}_{(r)}\to K\mathbb{G}_{(r+1)}$ induced by the
closed embedding $\mathbb{G}_{(r)}\hookrightarrow\mathbb{G}_{(r+1)}$.

For a $\mathbb{G}$-module $M$, the $\Pi$-support space $\Pi(\mathbb{G})_{M}$
is defined to be

$\Pi(\mathbb{G})_{M}\ \equiv\
\varinjlim_{r}\Pi(\mathbb{G}_{(r)})_{M_{|\mathbb{G}_{(r)}}}\quad\
\subset\quad\varinjlim_{r}\Pi(\mathbb{G}_{(r)})\ \equiv\Pi(\mathbb{G}).$

The fact that $\Pi(\mathbb{G}_{(r)})\hookrightarrow\Pi(\mathbb{G}_{(r+1)})$
restricts to
$\Pi(\mathbb{G}_{(r)})_{M_{|\mathbb{G}_{(r)}|}}\hookrightarrow\Pi(\mathbb{G}_{(r+1)})_{M_{|\mathbb{G}_{(r+1)}|}}$
for $M$ a finite dimensional $\mathbb{G}_{(r+1)}$-module is immediate from the
definition of the equivalence relation on $\pi$-points. For arbitrary
$\mathbb{G}_{(r+1)}$-modules $M$, one appeals to [18, Thm4.6] which asserts
that equivalence of $\pi$-points implies strong equivalence.

######  Remark 2.3.

We compare our current construction of $M\to\Pi(\mathbb{G})_{M}$ with the
construction of $M\to V(\mathbb{G})_{M}$ considered in [13, Defn4.4]. First of
all, [13] requires that the linear algebraic group $\mathbb{G}$ be of
exponential type. Second, in [13], $k$ is required to be algebraically closed
and the formulation of support varieties is as a topological space of closed
points, whereas $\Pi(\mathbb{G})_{M}$ includes non-closed points. Finally,
$\Pi(\mathbb{G})_{M}$ as defined in Definition 2.2 is a colimit with respect
to $r$ of $\Pi(\mathbb{G}_{(r)})_{M_{|\mathbb{G}_{(r)}|}}$, whereas limits
were taken in [13] rather than colimits.

We recall the definition of a mock injective module for a linear algebraic
group $\mathbb{G}$ introduced in [13, Defn4.3].. The subcategory of
$Mod(\mathbb{G})$ consisting of mock injective modules plays a key role in our
formulation of stable module categories in Section 4. Interesting examples of
such modules, even for $\mathbb{G}_{a}$, are constructed in [13] and [23].

######  Definition 2.4.

Let $\mathbb{G}$ be a linear algebraic group. A $\mathbb{G}$-module $J$ is
said to be mock injective if the restriction of $J$ to each Frobenius kernel
$\mathbb{G}_{(r)}$ of $G$ is an injective $\mathbb{G}_{(r)}$-module.

The properties for our support theory $M\mapsto\Pi(\mathbb{G})_{M}$ stated in
the following theorem are the basic properties required of a theory of
support. Perhaps it is worth mentioning that support theories do not determine
functors from categories of $G$-modules to subsets of some space. For example,
one could consider a $\mathbb{G}$-module $M$ with non-trivial support and an
embedding $M\hookrightarrow I$ of $M$ into an injective $\mathbb{G}$-module;
in this case, there is no conceivable map from the support of $M$ to the empty
set (which is the support of $I$).

Since $M\to\Pi(\mathbb{G})_{M}$ is the colimit of
$M\to\Pi(\mathbb{G}_{(r)})_{M_{|Gr}}$, the properties of
$M\mapsto\Pi(\mathbb{G})_{M}$ stated in Theorem 2.5 are immediate consequences
of the corresponding properties for each Frobenius kernel $\mathbb{G}_{(r)}$
of $\mathbb{G}$ as established in [18] together with the definition of Mock
injective modules.

######  Theorem 2.5.

Let $(\mathbb{G},\mathcal{E})$ be a linear algebraic group and consider
$\mathbb{G}$-modules $M,M_{i},N$. Then sending a $\mathbb{G}$-module $M$ to
the subspace $\Pi(\mathbb{G})_{M}\subset\Pi(\mathbb{G})$ satisfies the
following properties:

  1. (1)

Isomorphism: If $M$ and $N$ are isomorphic $\mathbb{G}$-modules, then (as
subsets of $\Pi(G)$)

$\Pi(G)_{M}\quad=\quad\Pi(G)_{N}.$

.

  2. (2)

Arbitrary direct sums: For any family $\\{M_{i},i\in I\\}$ of
$\mathbb{G}$-modules,

$\Pi(\mathbb{G})_{\oplus_{I}M_{i}}\quad=\quad\bigcup_{i\in
I}\Pi(\mathbb{G})_{M_{i}}.$

  3. (3)

Tensor products: For any pair of $\mathbb{G}$-modules $M$ and $N$,

$\Pi(\mathbb{G})_{M\otimes N}\ =\ \Pi(\mathbb{G})_{M}\cap V(\mathbb{G})_{N}.$

  4. (4)

Two out of three: For any short exact sequence of $\mathbb{G}$-modules $0\to
M_{1}\to M_{2}\to M_{3}\to 0$ and any permutation $\sigma$ of $\\{1,2,3\\}$,

$\Pi(\mathbb{G})_{M_{\sigma(2)}}\ \subset\
\Pi(\mathbb{G})_{M_{\sigma(1)}}\cup\Pi(\mathbb{G})_{M_{\sigma(3)}}.$

  5. (5)

Trivial module: $\Pi(\mathbb{G})_{k}\quad=\quad\Pi(\mathbb{G})$.

  6. (6)

Closed: If $M$ is finite dimensional, then $\Pi(\mathbb{G})_{M}\ \subset\
\Pi(\mathbb{G})$ is a closed subspace.

  7. (7)

Detection: $\Pi(\mathbb{G})_{M}=\emptyset$ $\quad\iff\quad$ $M$ is mock
injective.

To establish further properties and to compute examples, we shall consider the
colimit of the exponential support theory
$M\mapsto\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})_{M}$ of Definition 1.10. We
observe that the natural embedding $\mathfrak{C}_{r}(\mathbb{G})\
\hookrightarrow\ \mathfrak{C}_{r+1}(\mathbb{G})$ (determined by sending a
$K$-point $\underline{B}=(B_{0},\ldots,B_{r-1})$ to
$(\underline{B},0)\equiv(B_{0},\ldots,B_{r-1},0)$ multiplies the degrees of
homogeneous elements (specified by the $\mathbb{A}^{1}$-action given in
(1.10.2)) by $p$ and thus induces $\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})\
\hookrightarrow\ \mathbb{P}\mathfrak{C}_{r+1}(\mathbb{G})$. Moreover, for
$(\mathbb{G},\mathcal{E})$ an algebraic group of exponential type, one checks
by inspection the equality for any $K$-point $\underline{B}$ of
$\mathfrak{C}_{r}(\mathbb{G})$

(2.5.1)
$\mathcal{E}_{\Lambda_{r}(\underline{B})}\circ\epsilon_{r}\quad=\quad\mathcal{E}_{\Lambda_{r+1}((\underline{B},0))}\circ\epsilon_{r+1}:K[t]/t^{p}\to\
\mathbb{G}.$

######  Theorem 2.6.

Let $(\mathbb{G},\mathcal{E})$ be an algebraic group of exponential type and
consider the bicontinuous algebraic map
$\rho_{r}\circ\Lambda_{r}:\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})\
\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}\
\mathbb{P}V_{r}(\mathbb{G}_{(r)})$ of Theorem 1.12 for each $r>0$. Then the
following diagram commutes

(2.6.1)
$\textstyle{\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{r}\circ\Lambda_{r}}$$\textstyle{\mathbb{P}V(\mathbb{G}_{(r)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi\circ\Psi}$$\textstyle{\Pi(\mathbb{G}_{(r)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}\mathfrak{C}_{r+1}(\mathbb{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{r+1}\circ\Lambda_{r+1}}$$\textstyle{\mathbb{P}V(\mathbb{G}_{(r+1)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi\circ\Psi}$$\textstyle{\Pi(\mathbb{G}_{(r+1}),}$

where the horizontal maps are bicontinuous algebraic maps and the vertical
maps are the natural embeddings.

Moreover, for every $\mathbb{G}$-module $M$ and every $r>0$, the composition
$\Phi\circ\Psi\circ(\rho_{r}\circ\lambda_{r}):\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})\
\to\ \Pi(\mathbb{G})$ restricts to a bijection

(2.6.2)
$\Phi\circ\Psi\circ(\rho_{r}\circ\lambda_{r}):\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})_{M}\
\stackrel{{\scriptstyle\sim}}{{\to}}\
\Pi(\mathbb{G}_{(r)})_{M_{|\mathbb{G}_{(r)}}}.$

Taking colimits with respect to $r$ determines a homeomorphism derived from
bicontinuous algebraic map

(2.6.3) $\Phi:\mathbb{P}\mathfrak{C}(\mathbb{G})\
\stackrel{{\scriptstyle\sim}}{{\to}}\ \Pi(\mathbb{G}),\quad\text{restricting
to}\quad\mathbb{P}\mathfrak{C}(\mathbb{G})_{M}\
\stackrel{{\scriptstyle\sim}}{{\to}}\ \Pi(\mathbb{G})_{M}$

for any $\mathbb{G}$-module $M$.

For each field extension $K/k$, the adjoint action of $\mathbb{G}(K)$ on the
$K$ points of each $\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})$ determines an
adjoint action of $\mathbb{G}(K)$ on the $K$ points
$\mathbb{P}\mathfrak{C}(\mathbb{G})$

###### Proof.

To verify the commutativity of left hand square of (2.6.1), it suffice to
check at geometric points. This check is done by comparing
$a*\langle\underline{B},0\rangle$ and $\langle a*\underline{B},0\rangle$ using
(1.10.1) and (1.10.2), where $\langle-,0\rangle$ sends an $r$-tuple to the
$r+1$-tuple obtained by adding 0 to the right. The equality (2.5.1 implies the
commutativity of the right hand square of (2.6.1).

To verify that
$\Phi\circ\Psi\circ(\rho_{r}\circ\lambda_{r}):\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})\
\to\ \Pi(\mathbb{G})$ restricts as in (2.6.2), we use the explicit definitions
of $\mathbb{P}\mathfrak{C}(\mathbb{G})_{M}$ in Definition 1.10 and of
$\Pi(G)_{M}$ in Recollection 1.2. This restriction is verified by juxtaposing
the determination of
$\Psi\circ(\rho_{r}\circ\lambda_{r}):\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})\to\mathbb{P}|\mathbb{G}_{(r)}|$
in Theorem 1.12(4), the definition of $\rho_{r}$ in Definition 1.11, and the
determination of
$\Phi\circ\Psi:\mathbb{P}V_{r}(\mathbb{G}_{(r)})\to\Pi(\mathbb{G}_{(r)})$ in
Theorem 1.6. Observe that this argument remains valid for any $M$ even though
we consider $\mathbb{P}|\mathbb{G}_{(r)}|_{M}$ only for finite dimensional
$\mathbb{G}_{(r)}$-modules. ∎

To formulate certain functoriality properties (with respect to $\mathbb{G}$)
of $M\mapsto\mathbb{P}\mathfrak{C}(\mathbb{G})_{M}$, we require the following
definition.

######  Definition 2.7.

Let $(\mathbb{G},\mathcal{E})$ and
$(\mathbb{G}^{\prime},\mathcal{E}^{\prime})$ be linear algebraic groups of
exponential type. A smooth closed embedding
$\mathbb{G}\hookrightarrow\mathbb{G}^{\prime}$ is said to be an embedding of
exponential type if $\mathcal{E}^{\prime}$ restricted along $f\times id$ to
$\mathcal{N}_{p}(\mathbb{G})\times\mathbb{G}_{a}$ equals
$\mathcal{E}:\mathcal{N}_{p}(\mathbb{G})\times\mathbb{G}_{a}\to\mathbb{G}$.

######  Proposition 2.8.

Let $f:(\mathbb{G},\mathcal{E})\ \to\
(\mathbb{G}^{\prime},\mathcal{E}^{\prime})$ be an embedding of linear
algebraic groups of exponential type.

  1. (1)

The embedding $df:\mathfrak{g}\hookrightarrow\mathfrak{g}^{\prime}$ determines
morphisms
$\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})\to\mathbb{P}\mathfrak{C}_{r}(\mathbb{G}^{\prime})$
sending $\underline{B}$ to $df_{*}(\underline{B}))$, with colimit sending
$K$-points of $\mathbb{P}\mathfrak{C}(\mathbb{G})$ to $K$-points of
$\mathbb{P}\mathfrak{C}(\mathbb{G}^{\prime})$.

  2. (2)

Composition with $f$ determines $f_{*}:\Pi(\mathbb{G})_{f^{*}(N)}\ \to\
\Pi(\mathbb{G}^{\prime})_{N}$ for any $\mathbb{G}^{\prime}$-module $N$.

  3. (3)

Moreover, $f_{*}(\Pi(\mathbb{G})_{f^{*}(N)})\ =\
f_{*}(\Pi(\mathbb{G}))\cap\Pi(\mathbb{G}^{\prime})_{N}$.

  4. (4)

If $J^{\prime}$ is a mock injective $\mathbb{G}^{\prime}$-module, then
$f^{*}(J^{\prime})$ is a mock injective $\mathbb{G}$-module.

###### Proof.

The first statement is self-evident. Since
$f_{*}\circ\mathcal{E}_{\underline{B}*}\circ\epsilon_{r}:K\mathbb{G}_{a(1)}\to
K\mathbb{G}_{a}\to KG\to KG^{\prime}$ equals
$\mathcal{E}^{\prime}_{df_{*}(\underline{B})}\circ\epsilon_{r}:K\mathbb{G}_{a(1)}\to
K\mathbb{G}_{a}\ \to\ K\mathbb{G}^{\prime}$, the action of
$(\mathcal{E}_{\Lambda_{r}(\underline{B}})_{*}(u_{r-1})$ on $f^{*}N$ equals
the action on $N$ by the image of
$(\mathcal{E}_{\Lambda_{r}(\underline{B}})_{*}(u_{r-1})$ under
$f_{*}:K\mathbb{G}\to K\mathbb{G}^{\prime}$. This immediately implies
assertions (2) and (3). Assertion (4) following immediately from assertion (3)
and the detection property of Theorem 2.5(7). ∎

######  Definition 2.9.

We recall that base change along the $p^{r}$-th power map $k\to k$ associates
to a scheme $X$ over $k$ a map $F^{r}:X\ \to\ X^{(r)}$. For a group scheme $G$
and a $G$-module $M$ this leads to the definition of the Frobenius twist
$M^{(r)}$ of $M$ given as the restriction along $F^{r}$ of the
$G^{(r)}$-module $M^{(r)}$.

If the group scheme $G$ is defined over $\mathbb{F}_{p^{r}}$, then we can
identify $G^{(r)}$ with $G$ and we can associate to a $G$-module $M$ the
external Frobenius twist $M^{[r]}$ defined by the restriction along $F^{r}$ of
the $G$-module $M$. If the action of $G$ on $M$ is defined over $F_{p^{r}}$,
then $M^{(r)}\ \simeq\ M^{[r]}$ (see [24, II.3.16]).

The following proposition is complementary to Proposition 2.8. One major
difference is $F^{r}:\mathbb{G}\to\mathbb{G}$ is far from smooth; in fact, its
associated differential map on Lie algebras is the 0-map. Another difference
is that Proposition 2.10 gives an explicit relationship between elements of
the support varieties of $M^{[r]}$ and of $M$.

######  Proposition 2.10.

Let $(\mathbb{G},\mathcal{E})\ \hookrightarrow\ (GL_{N},exp)$ be an embedding
of exponential type defined over some finite field $\mathbb{F}_{p^{d}}$. Then
for any $\mathbb{G}$-module $M$, any $p$-nilpotent element $B$ of
$\mathfrak{g}_{K}$ and any $r\geq d$, the action of
$(\mathcal{E}_{B^{(r)}})_{*}(u_{s})$ on $M_{K}$ equals the action
$(\mathcal{E}_{B})_{*}(u_{r+s})$ on $M_{K}^{[r]}$ and the action of
$(\mathcal{E}_{B})_{*}(u_{s})$ on $M_{K}^{[r]}$ is trivial for $s<r$.

Consequently, for any $r\geq d$, the $K$-points of
$\mathbb{P}\mathfrak{C}(\mathbb{G})_{M^{[r]}}$ are the $K$-points of
$\operatorname{Proj}\nolimits(\delta_{r}^{-1}(\mathfrak{C}(\mathbb{G})_{M}))$,
where $\delta_{r}:\mathfrak{C}(\mathbb{G})\to\mathfrak{C}(\mathbb{G})$ sends a
$K$-point $(A_{0},\ldots,A_{r-1},B_{0},\ldots,B_{n},\ldots)$ to
$(B_{0},\ldots,B_{n},\ldots)$.

###### Proof.

The action $(\mathcal{E}_{B})_{*}(u_{r+s})$ on $M_{K}^{[r]}$ is given by the
composition

$M_{K}\ \to\ M_{K}\otimes K[\mathbb{G}]\ \stackrel{{\scriptstyle 1\otimes
F^{r}}}{{\to}}\ M_{K}\otimes K[\mathbb{G}]\
\stackrel{{\scriptstyle(\mathcal{E}_{B})^{*}}}{{\to}}\ M_{K}\otimes K[t]\
\stackrel{{\scriptstyle 1\otimes u_{s}}}{{\to}}M_{K}.$

Our condition on $(\mathbb{G},\mathcal{E})$ as the restriction (defined over
$\mathbb{F}_{p^{r}}$) of the exponential structure on $GL_{N}$ implies that
the composition
$F^{r}\circ\mathcal{E}_{B}:\mathbb{G}_{a}\to\mathbb{G}\to\mathbb{G}$ equals
$\mathcal{E}_{B^{(r)}}\circ
F^{r}:\mathbb{G}_{a}\to\mathbb{G}_{a}\to\mathbb{G}$. (See [13, Prop1.11].)
Thus, the equality of the actions of $(\mathcal{E}_{B^{(r)}})_{*}(u_{s})$ on
$M_{K}$ and $(\mathcal{E}_{B})_{*}(u_{r+s})$ on $M^{[r]}_{K}$ and the
triviality of the actions of $(\mathcal{E}_{B})_{*}(u_{s})$ on $M^{[r]}_{K}$
for $s<r$ both follow from the identification of $u_{s+r}$ with
$F^{r}_{*}(u_{s})$.

This immediately implies the identification of $K$-points of
$\mathfrak{C}(\mathbb{G})_{M^{[r]}}$ with
$\delta_{r}^{-1}(\mathfrak{C}(\mathbb{G})_{M})$ in terms of the $K$-points,
and thus the asserted identification of
$\mathbb{P}\mathfrak{C}(\mathbb{G})_{M^{[r]}}$. ∎

The following is an immediate consequence of Proposition 2.10 and Theorem
2.5(3),(7).

######  Corollary 2.11.

Let $(\mathbb{G},\mathcal{E})\ \hookrightarrow\ (GL_{N},exp)$ be an embedding
of exponential type defined over some finite field $\mathbb{F}_{p^{d}}$, let
$r\geq d$, and let $J$ be a mock injective $\mathbb{G}$-module.

  1. (1)

$\mathbb{P}\mathfrak{C}(\mathbb{G})_{J^{[r]}}\ \hookrightarrow\
\mathbb{P}\mathfrak{C}(\mathbb{G})$ equals
$\mathbb{P}\mathfrak{C}(\mathbb{G}_{(r)})\ \hookrightarrow\
\mathbb{P}\mathfrak{C}(\mathbb{G})$.

  2. (2)

If $M$ is a $\mathbb{G}$-module, then
$\mathbb{P}\mathfrak{C}(\mathbb{G})_{M\otimes J^{[r]}}\ \hookrightarrow\
\mathbb{P}\mathfrak{C}(\mathbb{G})$ equals
$(\mathbb{P}\mathfrak{C}(\mathbb{G})_{M}\cap\mathbb{P}\mathfrak{C}(\mathbb{G}_{(r)}))\
\hookrightarrow\ \mathbb{P}\mathfrak{C}(\mathbb{G}).$

Let $\mathbb{G}$ be a linear algebraic group of exponential type. A
$\mathbb{G}$-module $M$ has exponential degree $<p^{r}$ if every $u_{s}\in
K\mathbb{G}_{a}$ with $s\geq r$ acts trivially on $\mathcal{E}_{B}^{*}(M)$ for
every $B$ a geometric point of $\mathcal{N}_{p}(\mathfrak{g})$ (see [13,
Defn4.5]).

As we see in Proposition 2.12(1), $\mathbb{G}$-modules of bounded exponential
degree have a strong condition on their
$\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})$-support. Observe that other
$\mathbb{G}$-modules also satisfy this condition. For example, if $M$ has
exponential degree $<p^{r}$ and if $M\hookrightarrow J$ is an embedding of $M$
in a mock injective $\mathbb{G}$-module $J$, then $J/M$ also has exponential
degree $<p^{r}$ by the “two out of three” property of Theorem 2.5.4.

######  Proposition 2.12.

Let $(\mathbb{G},\mathcal{E})$ be an algebraic group of exponential type and
$M$ be a $\mathbb{G}$-module.

  1. (1)

If $M$ has exponential degree $<p^{r}$, then
$\mathbb{P}\mathfrak{C}(\mathbb{G})_{M}\ =\
\operatorname{Proj}\nolimits(\pi_{r}^{-1}(\pi_{r}(\mathfrak{C}_{r}(\mathbb{G})_{M})))$,
where $\pi_{r}:\mathfrak{C}(\mathbb{G})\to\mathfrak{C}_{r}(\mathbb{G})$ sends
the $K$-point $(B_{0},\ldots,B_{n},\ldots)$ to $(B_{0},\ldots,B_{r-1})$.

  2. (2)

Any finite dimensional $\mathbb{G}$-module has bounded exponential degree.

  3. (3)

The subcategory of $\mathbb{G}$-modules of exponential degree $<p^{r}$ is an
abelian subcategory of the category $Mod(\mathbb{G})$ of $\mathbb{G}$-modules
which is closed under extensions and tensor products by arbitrary
$\mathbb{G}$-modules.

###### Proof.

Assume that $M$ has exponential degree $<p^{r}$. Then assertion (1) follows
immediately from the observation for any $t\geq r$ that the action of
$\sum_{s=0}^{t-1}\mathcal{E}_{B_{s}*}(u_{s})$ on $M_{K}$ equals that of
$\sum_{s=0}^{r-1}\mathcal{E}_{B_{s}*}(u_{s})$ on $M_{K}$.

Assertion 2 is proved in [13, Prop2.6].

A straight-forward verification shows that if $M$ has exponential degree
$<p^{r}$ and if $N$ has exponential degree $<p^{s}$, then any extension of $M$
by $N$ has exponential degree $<p^{max(r,s)}$ and the tensor product $M\otimes
N$ has exponential degreee $<p^{r+s}$. ∎

Observe that if $\mathbb{G}$ is as in Proposition 2.10 and if the
$\mathbb{G}$-module does not have exponential degree $<p^{s}$, then $M^{[r]}$
does not have exponential degree $<p^{r+s}$.

##  3\. Examples of $M\ \mapsto\ \mathbb{P}\mathfrak{C}(\mathbb{G})_{M}$

In the examples below, we frequently encounter projectivized versions of the
projection $\pi_{r}:\mathfrak{C}(\mathbb{G})\ \to\
\mathfrak{C}(\mathbb{G}_{(r)})$ sending a $K$-point
$(B_{0},\ldots,B_{n},\ldots)$ to $(B_{0},\ldots,B_{r-1})$. Starting with a
subspace $X\subset\mathbb{P}\mathfrak{C}_{r}(\mathbb{G}))$, we abuse notation
by writing $W\ =\ \pi_{r}^{-1}(X)\ \subset\
\mathbb{P}\mathfrak{C}(\mathbb{G})$ determined by the condition that a
$K$-point $\underline{B}$ of $\mathfrak{C}(\mathbb{G})$ represents a $K$-point
of $W$ if and only if $(B_{0},\ldots,B_{r-1})$ represents a $K$-point of $X$.
For examples, see Proposotion 2.12(1).

######  Example 3.1.

We consider the special case $\mathbb{G}=\mathbb{G}_{a}$. The variety
$\mathfrak{C}(\mathbb{G}_{a})$ is identified with $\mathbb{A}^{\infty}$. The
1-parameter subgroup
$\mathcal{E}_{\underline{b}}:\mathbb{G}_{a,K}\to\mathbb{G}_{a,K}$ given by the
$K$-point $\underline{b}=(b_{0},b_{1},\ldots b_{n}.\ldots)$ of
$\mathbb{A}^{\infty}$ is defined by
$(\mathcal{E}_{\underline{b}})^{*}:K[\mathbb{G}_{a}]\ \to\ K[\mathbb{G}_{a}]$
sending $T\in K[T]=K[\mathbb{G}_{a}]$ to $b_{0}T+b_{1}T^{p}\ldots
b_{n}T^{p^{n}}+\ldots$; the $\mathbb{A}^{1}$-action is given by sending a
$K$-point $(a,\underline{b})$ to $(a\cdot b_{0},a^{p}\cdot
b_{1},\ldots,a^{p^{n}}\cdot b_{n}\ldots)$. For any $\mathbb{G}_{a(r)}$-module
$M$, the restriction along the quotient map of algebras (but not a map of Hopf
algebras) $k\mathbb{G}_{a}\twoheadrightarrow k\mathbb{G}_{a(r)}$ provides $M$
with the structure of a $\mathbb{G}_{a}$-module such that
$\mathfrak{C}(\mathbb{G}_{a})_{M}\ =\
\pi_{r}^{-1}(\mathfrak{C}_{r}(\mathbb{G}_{a})_{M})\ \subset\
\mathfrak{C}(\mathbb{G}_{a})$ (where
$\pi_{r}:\mathfrak{C}(\mathbb{G}_{a})\to\mathfrak{C}_{r}(\mathbb{G}_{a})$ is
the projection onto the first $r$ factors). If $W\ \subset\
\mathbb{P}^{r-1}=\mathbb{P}\mathfrak{C}_{r}(\mathbb{G}_{a})$ is any closed
subset, some tensor product of Carlson $L_{\zeta}$-modules for
$\mathbb{G}_{a(r)}$, $M=\otimes_{i}L_{\zeta_{i}}$, has the property that
$\Pi(\mathbb{G}_{(r)})_{M}=W$ and thus
$\mathbb{P}\mathfrak{C}(\mathbb{G}_{a})_{M}\ =\pi_{r}^{-1}(W)$. Consequently,
a closed subset $W\ \subset\ \mathbb{P}\mathfrak{C}(\mathbb{G}_{a})$ is of the
form $\mathbb{P}\mathfrak{C}(\mathbb{G}_{a})_{M}$ for some finite dimensional
$\mathbb{G}$-module $M$ if and only if $W\ =\
\operatorname{Proj}\nolimits\pi_{r}^{-1}(\pi_{r}(W))$ for some $r>0$.

More generally, using Rickard idempotent modules, we may realize any subset
$X\ \subset\ \mathbb{P}\mathfrak{C}_{r}(\mathbb{G}_{a})$ as
$\Pi(\mathbb{G}_{a(r)})_{M}$ for some (not necessarily finite dimensional)
$\mathbb{G}_{a(r)}$-module $M$. Thus, given any subset $X\ \subset\
\mathbb{P}\mathfrak{C}_{r}(\mathbb{G}_{a})$, we may find some
$\mathbb{G}_{a}$-module $M$ with the property that
$\mathbb{P}\mathfrak{C}(\mathbb{G}_{a})_{M}\ =\ \pi_{r}^{-1}(X)$.

We explicitly construct infinite dimensional $\mathbb{G}_{a}$-modules $M$ with
the property that $X=\mathbb{P}\mathfrak{C}(\mathbb{G}_{a})_{M}$ does not
satisfy $X=\pi_{r}^{-1}(W)$ for any
$W\subset\mathbb{P}\mathfrak{C}_{r}(\mathbb{G}_{a})$. Let $L$ denote the
regular representation of $\mathbb{G}_{a}$ on $k[\mathbb{G}_{a}]\simeq k[T]$.
Thus, $k\mathbb{G}_{a}=k[u_{0},\ldots,u_{n},\ldots]/(u_{i}^{p})$ acts on
$k[T]$ with each $u_{i}$ a derivation and $u_{i}(T^{p^{j}})=\delta_{i,j}$. For
any subset $S\subset\mathbb{N}$, we define the $\mathbb{G}_{a}$-module $L_{S}$
by identifying $L_{S}$ with $k[T]$ as a $k$-vector space and defining $u_{i}$
on $L_{S}$ to be 0 if $i\notin S$ and $u_{i}$ as the derivation
$u_{i}(T^{p^{j}})=\delta_{i,j}$ if $i\in S$. For any $\underline{B}\not=0$,
$\mathcal{E}_{\underline{b}}$ represents a $K$-point of
$\mathbb{P}\mathfrak{C}(\mathbb{G}_{a})_{L_{S}}$ if and only if $b_{i}=0,\
\forall i\in S$. This is verified by observing that if $b_{i}=0$ for all $i\in
S$, then the Jordan type of $L_{S}$ at $\mathcal{E}_{\underline{b}}$ consists
only of blocks of size 1, whereas if $b_{i}\not=0$ for some $i\in S$, then
$\mathcal{E}_{\underline{b}}$ determines a non-zero $\pi$-point of
$\mathbb{P}\mathfrak{C}(\mathbb{G}_{a(i+1})_{L_{S}}$.

The preceding analysis applies with little change if one replaces
$\mathbb{G}=\mathbb{G}_{a}$ by $\mathbb{G}=\mathbb{G}_{a}^{\times s}$ for any
positive integer $s$.

######  Example 3.2.

Let $\mathbb{G}$ be a split, reductive algebraic group which admits an
embedding $(\mathbb{G},\mathcal{E})\ \hookrightarrow\ (GL_{N},exp)$ of
exponential type. For any $s>0$, let $St_{s}$ be the $\mathbb{G}$-module given
by

$St_{s}\quad\equiv\quad L((p^{s}-1)\rho)\quad=\quad
H^{0}(\mathbb{G}/\mathbb{B},(p^{s}-1)\rho)).$

The restriction of $St_{s}$ to $\mathbb{G}_{(s)}$ is both irreducible and
injective. Consequently, the only $K$-point of
$\mathfrak{C}_{s}(\mathbb{G})_{St_{s}}$ is 0. Since $St_{s}$ is a block of the
$\mathbb{G}$-module $k\mathbb{G}_{(s)}$, $u_{r}$ acts trivially on $St_{s}$
for $r\geq s$ . Thus, $\mathbb{P}\mathfrak{C}(\mathbb{G})_{St_{s}}\ =\
\pi_{s}^{-1}(\\{\emptyset\\})$; in other words,
$\mathbb{P}\mathfrak{C}(\mathbb{G})_{St_{s}}$ is the center of the projection
$\pi_{s}:\mathbb{P}\mathfrak{C}(\mathbb{G})\dashrightarrow\mathbb{P}\mathfrak{C}_{s}(\mathbb{G})$.

Following [35, Prop7.8], we can use this computation to determine
$\mathbb{P}\mathfrak{C}(\mathbb{G})_{H^{0}(\lambda)}$ for any dominant weight
$\lambda$ of the form $n\rho$.

######  Example 3.3.

Let $\mathbb{U}_{3}$ denote the Heisenberg group, the unipotent radical of a
split Borel subgroup of $GL_{3}$. The computations in [15] give the
identification $\mathfrak{C}_{r}(\mathbb{U}_{3})\ =\
Y_{r}\times\mathbb{A}^{r}\ \subset\ \mathbb{A}^{2r}\times\mathbb{A}^{1},$
where $Y_{r}$ is the closed subvariety of $\mathbb{A}^{2r}$ with generators
$x_{1,2}(\ell),\ x_{2,3}(\ell^{\prime}),0\leq\ell,\ell^{\prime}<r$ subject to
the relations $x_{1,2}(\ell)\cdot x_{2,3}(\ell^{\prime})\ =\
x_{2,3}(\ell)\cdot x_{1,2}(\ell^{\prime}),\quad 0\leq\ell<\ell^{\prime}.$ In
fact, $Y_{r}$ is shown to be a rational variety of dimension $r+1$, smooth
outside of the origin [15, Prop5.2].

Thus, $\mathbb{P}\mathfrak{C}_{r}(\mathbb{U}_{3})\ =\
(\mathbb{P}Y_{r})\\#\mathbb{P}^{r-1}\ \subset\ \mathbb{P}^{3r-1}$.
Furthermore, the colimit $\mathbb{P}\mathfrak{C}(\mathbb{U}_{3})$ is
identified with $(\mathbb{P}Y_{\infty})\\#\mathbb{P}^{\infty}$.

Observe that the restriction to $\mathfrak{C}_{r}(\mathbb{U}_{3})$ of the
adjoint action of $\mathbb{U}_{3}$ to
$\mathcal{N}_{p}(\mathfrak{u}_{3})^{\times r}$ is non-trivial. Since the
adjoint action must stabilize subsets of the form
$\mathbb{P}\mathfrak{C}(\mathbb{U}_{3})_{M}$ for a $\mathbb{U}_{3}$-module $M$
by Theorem 1.12(3), this constrains which subspaces are of the form
$\mathbb{P}\mathfrak{C}(\mathbb{U}_{3})_{M}$.

One class of examples of $\mathbb{U}_{3}$-modules is given by inflation
$pr^{*}(N)$ of $\mathbb{G}_{a}^{\times 2}$-modules $N$ along the projection
homomorphism $pr:\mathbb{U}_{3}\to\mathbb{G}_{a}^{\times 2}$. This projection
induces the projection map

$p_{r}:\mathbb{P}\mathfrak{C}_{r}(\mathbb{U}_{3})\ \simeq\
(\mathbb{P}Y_{r})\\#\mathbb{P}^{r-1}\ \dashrightarrow\ \mathbb{P}Y_{r}\
\hookrightarrow\mathbb{P}^{2r-1}.$

Consequently, for any (Zariski) closed subspace $W\subset\mathbb{P}Y_{r}$, we
may realize $p_{r}^{-1}(W)\subset\mathbb{P}\mathfrak{C}_{r}(\mathbb{U}_{3})$
as $\mathbb{P}\mathfrak{C}(\mathbb{U}_{3})_{p_{r}^{*}(N)}$ for some finite
dimensional $\mathbb{U}_{3}/Z$-module $N$ by Example 3.1.

Interesting infinite dimensional examples are given by induction from the
central subgroup $\mathbb{G}_{a}\ \simeq\ Z\ \subset\ \mathbb{U}_{3}$. For
example, let $M=\operatorname{ind}\nolimits_{Z}^{\mathbb{U}_{3}}k\simeq
k[\mathbb{U}_{3}/Z]$. The action of $\mathbb{U}_{3}$ on $M$ is given by the
projection to $\mathbb{U}_{3}/Z$ followed by the left regular representation
of $\mathbb{U}_{2}/Z$. Thus, if
$\mathcal{E}_{\underline{B}}\in\mathfrak{C}_{r}(\mathbb{U}_{3})$ factors
through $Z$, then the action of $(\mathcal{E}_{\underline{B}})_{*}(u_{r-1})$
on $M$ is trivial and therefore
$\underline{B}\in\mathfrak{C}_{r}(\mathbb{U}_{3})$ lies in
$\mathfrak{C}_{r}(\mathbb{U}_{3})_{M}$. Otherwise, the composition
$pr\circ\mathcal{E}_{\underline{B}}:\mathbb{G}_{a}\ \to\ \mathbb{U}_{3}\ \to\
\mathbb{U}_{3}/Z$ represents a non-zero point of
$\mathfrak{C}_{r}(\mathbb{U}_{3}/Z)$; since
$\mathfrak{C}(\mathbb{U}_{3}/Z)_{k[\mathbb{U}_{3}/Z]}=0$, the action at the
1-parameter subgroup $\mathcal{E}_{\underline{B}}\to\mathbb{U}_{3}$ on $M$ can
be identifiied with the action at
$p_{r}\circ\mathcal{E}_{\underline{B}}\to\mathbb{U}_{3}/Z$ for the regular
representation of $\mathbb{U}_{3}/Z$ which is free. We conclude that
$Z\hookrightarrow\mathbb{U}_{3}$ induces a homemorphism
$\mathbb{P}\mathfrak{C}(Z)\ \stackrel{{\scriptstyle\sim}}{{\to}}\
\mathbb{P}\mathfrak{C}(\mathbb{U}_{3})_{M}$.

######  Example 3.4.

Let $\mathbb{G}$ be a simple algebraic group over an algebraically closed
field $k$ with Coexter number $h<p$. Assuming the Lusztig character formula is
valid for all restricted dominant weights $\lambda$, C. Drupieski, D. Nakano,
and B. Parshall showed in [11, Thm3.3] for any restricted dominant weight
$\lambda$ that

$\mathfrak{C}_{1}(\mathbb{G})_{L(\lambda)}\ =\
\mathbb{G}\cdot\mathfrak{u}_{J(\lambda)}$

where $\mathfrak{u}_{J(\lambda)}$ is the Lie algebra of the unipotent radical
of a parabolic subgroup $\mathbb{P}_{J(\lambda)}$ explicitly determined by
$\lambda$ and $L(\lambda)$ is the irreducible $\mathbb{G}$-module of high
weight $\lambda$.

For $\mathbb{G}$ a classical simple algebraic group of rank $\ell$ satisfying
$p>\frac{\ell^{2}}{2}+1$, P. Sobaje in [33] explicitly computed
$\mathfrak{C}_{r}(\mathbb{G})_{L(\lambda)}$ for any dominant weight
$\lambda=\sum_{i\geq 0}p^{i}\lambda_{i}$ (with each $\lambda_{i}$ a restricted
dominant weight). The form of Sobaje’s determination is

$\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})_{L_{\lambda}}\ =\
\\{\underline{B}\in\mathbb{P}\mathfrak{C}_{r}(\mathbb{G}):B_{i}\in\mathbb{G}\cdot\mathfrak{u}_{J(\lambda_{i})}\\}.$

Assume further that $\lambda=\sum p^{i}\lambda_{i}$ satisfies the condition
$2\sum_{j=1}^{\ell}\langle\lambda_{i},\omega_{j}^{\vee}\rangle<p(p-1)$ for
each $i$. Then, as argued in [13, Prop5.1], we can identify the $K$-points of
$\mathfrak{C}(\mathbb{G})_{L_{\lambda}}$ as those $K$-points
$\mathcal{E}_{\underline{B}}$ of $\mathfrak{C}(\mathbb{G})$ such that $B_{i}$
is a $K$-point of $\mathbb{G}\cdot\mathfrak{u}_{J(\lambda_{i})}.$

Similar results are obtained by these authors with the irreducible module
$L(\lambda)$ replaced by $H^{0}(\lambda)\ =\
ind_{\mathbb{B}}^{\mathbb{G}}(k_{\lambda})$, where $\mathbb{B}\
\subset\mathbb{G}$ is a Borel subgroup of $\mathbb{G}$.

##  4\. Stable module categories

For a finite group scheme $G$, a natural domain for support theory is the
stable module category $StMod(G)$, with triangulated category structure
introduced by Happel [22]. This triangulated structure utilizes the fact that
the group algebra $kG$ is self-injective, implying that projective objects are
the same as injective objects in the abelian category $Mod(G)$. Our primary
objective in this section is to introduce in Definition 4.7 the triangulated
categories $StMod(\mathbb{G})$ and $stmod(\mathbb{G})$ associated to the
triangulated structures of the homotopy categories of (cochain) complexes
$\mathcal{K}^{b}(Mod(\mathbb{G}))$ and $\mathcal{K}^{b}(mod(\mathbb{G}))$.

We begin by recalling various constructions associated to the abelian category
$Mod(G)$ of $G$-modules and the full abelian subcategory $mod(G)$ of finite
dimensional $G$-modules for an affine group scheme $G$ over $k$. We denote by
$CH(Mod(G))$ the abelian category of (cochain) complexes of $G$-modules and by
$\mathcal{K}(Mod(G))$ the associated homotopy category with respect to cochain
homotopy equivalence. The additive category $\mathcal{K}(Mod(G))$ has the
natural structure of a triangulated category with basic exact triangles
$C^{\bullet}\to D^{\bullet}\to E^{\bullet}\to C^{\bullet}[1]$ associated to
short exact sequences $0\to C^{\bullet}\to D^{\bullet}\to E^{\bullet}\to 0$ in
$CH(Mod(G))$. We are primarily interested in the analogous structures
$CH^{b}(Mod(G)),\ \mathcal{K}^{b}(Mod(G))$ whose objects are bounded complexes
of $\mathbb{G}$-modules and their full subcategories $CH^{b}(mod(G)),\
\mathcal{K}^{b}(mod(G))$ of bounded complexes of finite dimensional
$G$-modules,

We say that a bounded complex $C^{\bullet}$ of $G$-modules in $CH^{b}(Mod(G))$
has length $\leq d$ if there exists some integer $m$ such that $C^{i}=0$ for
$i<m$ or $i>m+d$. In particular, complexes $M[n]$ which have the $G$-module
$M$ in some degree $n$ and 0 in all other degrees have length 0.

We recall that the stable module category $StMod(G)$ of a finite group scheme
$G$ is the additive category whose objects are $G$-modules and whose abelian
group of maps $Hom_{StMod(G)}(M,N)$ is the quotient of $Hom_{Mod(G)}(M,N)$ by
the subgroup of maps $M\to N$ which factor through an injective $G$-module.
One equips $StMod(G)$ with a triangulated structure whose +1 shift is given by
sending $M$ to the cokernel $\Omega^{-1}(M)$ of a minimal embedding
$M\hookrightarrow I_{M}$ with $I_{M}$ an injective $G$-module, and whose -1
shift is given by sending $M$ to the kernel $\Omega^{1}(M)$ of a minimal
surjection $P_{M}\twoheadrightarrow M$ with $P_{M}$ a projective $G$-module.
We define $stmod(G)\subset StMod(G)$ to be the tensor triangulated subcategory
whose objects are finite dimensional $G$-modules.

If $\mathfrak{C}\ \hookrightarrow\ \mathfrak{D}$ is an embedding of
triangulated categories (which implies by definition that the image of
$\mathfrak{C}$ is a full subcategory of $\mathfrak{D}$), then the Verdier
quotient $\mathfrak{D}\ \to\ \mathfrak{D}/\mathfrak{C}$ is the universal
triangulated map from $\mathfrak{D}$ to a triangulated category $\mathcal{E}$
with kernel containing $\mathfrak{C}$. If $\mathfrak{C}$ is a thick
subcategory (i.e., containing any summand in $\mathfrak{D}$ of an object of
$\mathfrak{C}$), then the kernel of $\mathfrak{D}\to\mathfrak{D}/\mathfrak{C}$
equals $\mathfrak{C}$. The Verdier quotient is constructed as the category
whose objects are the same as objects of $\mathfrak{D}$ and whose maps $X\to
Y$ in $\mathfrak{D}/\mathfrak{C}$ are equivalence classes of “roofs”: triples
$X\leftarrow E\to Y$ in $\mathfrak{D}$ such that the cone of $E\to X$ is an
object of $\mathfrak{C}$. We refer the reader to [27] for a detailed
discussion of Verdier quotients of triangulated categories.

Granted the construction of the Verdier quotient, we readily observe that
$\mathfrak{D}\to\mathfrak{D}/\mathfrak{C}$ is the localization
$\mathfrak{D}\to\mathfrak{D}[mor_{\mathfrak{C}}^{-1}]$, where
$mor_{\mathfrak{C}}$ is the class of maps $X\to Y$ in $\mathfrak{D}$ whose
cone lies in $\mathfrak{C}$. Namely, $mor_{\mathfrak{C}}$ is a “multiplicative
system” as considered for example in [36, 10.3.4]. Since
$\mathfrak{D}\to\mathfrak{D}/\mathfrak{C}$ sends each $X\to Y$ in
$mor_{\mathfrak{C}}$ to an invertible map,
$\mathfrak{D}\to\mathfrak{D}[mor_{\mathfrak{C}}^{-1}]$ is well defined and
factors $\mathfrak{D}\ \to\ \mathfrak{D}/\mathfrak{C}$; on the other hand,
granted the existence of $\mathfrak{D}[mor_{\mathfrak{C}}^{-1}]$, the fact
that $\mathfrak{C}$ lies in the kernel of
$\mathfrak{D}\to\mathfrak{D}[mor_{\mathfrak{C}}^{-1}]$ provides the inverse
functor $\mathfrak{D}/\mathfrak{C}\to\mathfrak{D}[mor_{\mathfrak{C}}^{-1}]$
factoring $\mathfrak{D}\to\mathfrak{D}[mor_{\mathfrak{C}}^{-1}]$.

We remind the reader that the bounded derived category of an abelian category
$\mathcal{A}$, $D^{b}(\mathcal{A})$, is the Verdier quotient of the homotopy
category $\mathcal{K}^{b}(\mathcal{A})$ of bounded chain complexes of
$\mathcal{A}$ by the thick triangulated subcategory $Quasi^{b}(\mathcal{A})$
of bounded complexes quasi-isomorphic to 0 (i.e., which are exact). If
$\mathcal{A}$ has enough injective objects (for example, if $\mathcal{A}$ is
the category of $G$-modules for an affine group scheme), then
$D^{b}(\mathcal{A})$ can be defined more simply by first explicitly defining
the derived category $D^{+}(\mathcal{A})$ of complexes bounded below using
injective resolutions and then restricting to objects which are bounded
complexes. If $f:I^{\bullet}\to J^{\bullet}$ is a quasi-isomorphism (i.e.,
induces an isomorphism in cohomology) of bounded complexes of injective
$G$-modules, then $f$ is a chain homotopy equivalence. Consequently, the
homotopy category $\mathcal{K}^{\bullet}(Inj)$ of bounded complexes for the
full additive subcategory $Inj(Mod(G))\subset Mod(G)$ embeds as a full
subcategory of $D(Mod(G))$.

The following well-known theorem (see, for example, [30, Thm2.1]) provides
motivation for our constructions. We recall that a perfect complex of
$G$-modules for a finite group scheme $G$ is a bounded complex of finite
dimensional $G$-modules quasi-isomorphic to a bounded complex of finite
dimensional injective $G$-modules. Thus, the subcategory $Perf(G)\ \subset\
D^{b}(mod(G))$ of perfect complexes is the essential image in $D^{b}(mod(G))$
of the homotopy category of bounded complexes of finite dimensional injective
$G$-modules. $\mathcal{K}^{b}(inj(G))\ \subset\ D^{b}(mod(G))$.

######  Theorem 4.1.

If $G$ is any finite group scheme, then the natural map $mod(G)\ \to\
\mathcal{K}^{b}(mod(G))$ induces an equivalence of tensor triangulated
categories

$\Psi:stmod(G)\quad\stackrel{{\scriptstyle\sim}}{{\to}}\quad
D^{b}(mod(G))/Perf(G).$

This theorem is somewhat surprising, even in the case of $\mathbb{G}_{a(1)}$
(i.e., $k[t]/t^{p}$-modules). Consider the complex $C^{\bullet}$ with
$C^{0}=k[t]/t^{p}$, $C^{1}=k[t]/t^{p}$ and $d:C^{0}\to C^{1}$ given by
multiplication by $t$; thus $C^{\bullet}\not=0$ in $D^{b}(mod(G))$ but is 0 in
$stmod(G)$. In the proof of Theorem 4.2, the construction of
$\tilde{\Phi}:D^{b}(mod(G))\to stmod(G)$ applied to the complex $C^{\bullet}$
sends $C^{\bullet}$ to the cone of $\tilde{\Phi}(C^{0})=0\in stmod(G)$ mapping
to $\tilde{\Phi}(C^{1})=0\in stmod(G)$ which is 0.

We shall utilize the truncation functors

$\tau_{\leq n},\tau_{\geq m}:CH(Mod(G))\ \to CH(Mod(G)),\quad
C^{\bullet}\mapsto\tau_{\leq n}(C^{\bullet})\hookrightarrow C^{\bullet},\quad
C^{\bullet}\twoheadrightarrow\tau_{\geq m}(C^{\bullet}).$

Here, $\tau_{\leq n}(C^{\bullet})$ is the subcomplex of $C^{\bullet}$ such
that $\tau_{\leq n}(C^{\bullet})^{i}=0$ for $i>n$, $\tau_{\leq
n}(C^{\bullet})^{n}=ker\\{d^{n}\\}$, and $\tau_{\leq
n}(C^{\bullet})^{i}=C^{i}$ for $i<n$; and $\tau_{\geq m}(C^{\bullet})$ is the
quotient complex of $C^{\bullet}$ such that $\tau_{\geq m}(C^{\bullet})^{n}=0$
for $n<m$, $\tau_{\geq m}(C^{\bullet})^{m}=C^{m}/im\\{d^{m-1}\\}$, and
$\tau_{\geq m}(C^{\bullet})^{n}=C^{n}$ for $n>m$. So defined, if
$C^{\bullet}\to D^{\bullet}$ is a quasi-isomorphism, then $\tau_{\leq
n}(C^{\bullet})\to\tau_{\leq n}(D^{\bullet})$ and $\tau_{\geq
m}(C^{\bullet})\to\tau_{\geq m}(D^{\bullet})$ are also quasi-isomorphisms.

In the following theorem, we consider the thick tensor triangulated
subcategory $\mathcal{I}nj^{b}(Mod(G))\ \subset\ D^{b}(Mod(G))$ of complexes
quasi-isomorphic to bounded complexes of injective $G$-modules. This is the
“essential image” of $\mathcal{K}^{b}(Inj(G))\ \subset D^{b}(Mod(G))$.

######  Theorem 4.2.

(see [30, Thm2.1]) For any finite group scheme $G$, the natural map $Mod(G)\
\to\ \mathcal{K}^{b}(Mod(G))$ (sending the $G$-module $M$ to the chain complex
$M[0]$ concentrated in degree 0) induces an equivalence of tensor triangulated
categories

$\Psi:StMod(G)\quad\stackrel{{\scriptstyle\sim}}{{\to}}\quad
D^{b}(Mod(G))/\mathcal{I}nj^{b}(Mod(G)).$

Furthermore, we construct $\tilde{\Phi}:D^{b}(Mod(G))\ \to\ StMod(G)$ inducing
the inverse

(4.2.1)
$\Phi=\Psi^{-1}:D^{b}(Mod(G))/\mathcal{I}nj^{b}(Mod(G))\quad\stackrel{{\scriptstyle\sim}}{{\to}}\quad
StMod(G)$

sending $M[n]$ to $\Omega^{-n}(M)$.

###### Proof.

Necessarily, $\Psi(\Omega^{n}(M))\ =\ M[-n]\in D^{b}(Mod(G))/Inj^{b}(Mod(G))$.
As explained in Rickard’s proof of [30, Thm2.1], an exact triangle $X\to Y\to
Z\to X[1]$ in $StMod(G)$ arises from a pushout diagram in
$Mod(\mathbb{G}_{(r)})$ from the short exact sequence $0\to X\to I\to X[1]\to
0$ (with $I$ injective) along $X\to Y$ to the short exact sequence $0\to Y\to
Z\to X[1]\to 0$. Consequently, the result of applying $\Psi$ to a such an
exact triangle in $StMod(G)$, $\Psi(X)\to\Psi(Y)\to\Psi(Z)\to\Psi(X)[1]$, is
isomorphic in $D^{b}(Mod(G))/Inj^{b}(Mod(G))$ to the image of the exact
triangle in $\mathcal{K}^{b}(Mod(G))$ arising from the short exact sequence
$0\to X\to Y\oplus I\to(Y\oplus I)/X\to 0$ in $Mod(G)$. Thus, $\Psi$ is exact;
in other words, it sends exact triangles to exact triangles.

To show that $\Psi$ is a full functor, we first show that $Mod(G)\to
D^{b}(Mod(G))$ is full, then observe that $D^{b}(Mod(G))\to
D^{b}(Mod(G))/\mathcal{I}nj^{b}(Mod(G))$ is full, so that $\Psi$ must also be
full. Let $M,\ N$ be $G$-modules and consider a map from $M[0]$ to $N[0]$ in
$D^{b}(Mod(G))$ represented by a “roof” $M[0]\ \leftarrow C^{\bullet}\ \to\
N[0]$ in $\mathcal{K}^{b}(Mod(G))$. This roof factors through the quotient map
$C^{\bullet}\twoheadrightarrow\tau_{\geq 0}(C^{\bullet})$ (which is a quasi-
isomorphism). On the other hand, $M\ \simeq\
ker\\{d^{0}\\}/im\\{d^{-1}\\}\hookrightarrow\tau_{\geq 0}(C^{\bullet})^{0}$.
Thus, there is a natural map of roofs from
$(M[0]\stackrel{{\scriptstyle=}}{{\leftarrow}}M[0]\to N[0)]$ to $(M[0]\
\leftarrow\tau_{\geq 0}(C^{\bullet})\ \to\ N[0])$, thereby establishing the
fullness of $Mod(G)\to D^{b}(Mod(G))$.

We next show that $Hom_{D^{b}(Mod(G))}(M,N)\ \to\
Hom_{D^{b}(Mod(G))/\mathcal{I}nj^{b}(Mod(G))}(M,N)$ is surjective. Consider a
roof of maps $M[0]\ \leftarrow C^{\bullet}\ \to\ N[0]$ in $D^{b}(Mod(G))$,
with associated exact triangle $J^{\bullet}\to C^{\bullet}\to M[0]$ in
$D^{b}(Mod(\mathbb{G}))$ for some bounded complex $J^{\bullet}$ of injective
$G$-modules. Replacing $C^{\bullet}$ by the mapping cylinder of
$J^{\bullet}\to C^{\bullet}$ if necessary, we may assume that
$J^{\bullet}\hookrightarrow C^{\bullet}$ so that there is an isomorphism
$C^{\bullet}\to((C^{\bullet}/J^{\bullet})\oplus J^{\bullet})$ and
$C^{\bullet}\to M[0]$ factors as
$C^{\bullet}\to((C^{\bullet}/J^{\bullet})\oplus J^{\bullet})\to M[0]$ with
$C^{\bullet}/J^{\bullet}\to M[0]$ an isomorphism in $D^{b}(Mod(G))$. Thus, the
given roof is equal as a map in $D^{b}(Mod(G))/\mathcal{I}nj^{b}(Mod(G))$ to a
roof of the type considered in the previous paragraph and thus is the image
under $\Psi$ of a map $M\to N$.

As shown in the proof of [30, Thm2.1], exactness and fullness of $\Psi$ imply
that $\Psi$ is also faithful. Since any summand of an injective
$\mathbb{G}$-module is itself injective, we conclude that
$\mathcal{I}nj^{b}(Mod(G))$ is a thick subcategory of $D^{b}(Mod(G))$. To
check that every object
$C^{\bullet}\in\mathcal{K}^{b}(Mod(G))/\mathcal{I}nj^{b}(Mod(G))$ lies in the
image of $\Psi$ (up to isomorphism), we argue by induction on the length $d$
of $C^{\bullet}$, recognizing that any complex of length $0$ is of the form
$M[n]$ and thus is in the image of $\Psi$. We construct
$\tilde{\Phi}:D^{b}(Mod(G))\ \to\ StMod(G)$ (which induces $\Phi$ of (4.2.1))
with the property that $\Psi\circ\tilde{\Phi}$ sends $X^{\bullet}\in
D^{b}(Mod(G))$ to its image in $D^{b}(Mod(G))/\mathcal{I}nj^{b}(Mod(G))$ under
the quotient map. We proceed by induction, assuming that $\tilde{\Phi}$ has
been defined on the full subcategory of $D^{b}(Mod(G))$ whose objects are
quasi-isomorphic to complexes of length $\leq d$.

Let $X^{\bullet}\in D^{b}(Mod(G))$ be a complex of length $d+1$ with
$X^{n+1}\not=0$ and $X^{i}=0$ for $i>n+1$. We consider the exact triangle in
$\mathcal{K}^{b}(Mod(G))$

(4.2.2) $H^{n+1}(X^{\bullet})[n]\to\tau_{\leq n}(X^{\bullet})\to
X^{\bullet}\to H^{n+1}(X^{\bullet})[n+1],$

and recognize that $\tau_{\leq n}(X^{\bullet})$ is a complex of length $\leq
d$. We extend $\tilde{\Phi}(H^{n+1}(X^{\bullet})[n])\to\tilde{\Phi}(\tau_{\leq
n}(X^{\bullet}))$ to an exact triangle in $StMod(G)$

(4.2.3) $\tilde{\Phi}(H^{n+1}(X^{\bullet})[n])\ \to\ \tilde{\Phi}(\tau_{\leq
n}(X^{\bullet}))\ \to M\ \to\ \tilde{\Phi}(H^{n+1}(X^{\bullet})[n+1]).$

Then $\Psi$ defines an isomorphism of exact triangles in $D^{b}(Mod(G))$ from
$\Psi$(4.2.3) to (4.2.2). We define $\tilde{\Phi}(X^{\bullet})\equiv M$. In
particular, $X^{\bullet}\ =\ \Psi(M)$ is in the image of $\Psi$.

Assume we have defined $\tilde{\Phi}(X^{\bullet})$ for all complexes of length
$d+1$ (with arbitrary $n$ as top non-vanishing degree). Let $Y^{\bullet}$ be
another bounded complex of length $d+1$. For any map $f:X^{\bullet}\to
Y^{\bullet}\in D^{b}(Mod(G))$, we define
$\tilde{\Phi}(f):\tilde{\Phi}(X^{\bullet})\to\tilde{\Phi}(Y^{\bullet})$ to be
the unique map satisfying the condition that $\Psi(\tilde{\Phi}(f))$ equals
the image of $f$ in $D^{b}(Mod(G))/Inj^{b}(Mod(G))$. One readily verifies that
$\tilde{\Phi}$ so defined is a functor which preserves exact triangles and
induces $\Phi$ inverse to $\Psi$. ∎

######  Remark 4.3.

Let $G$ be a finite group scheme. Although the category $StMod(G)$ has
arbitrary direct sums, the direct sum $\bigoplus_{j\in J}C^{\bullet}_{j}\ \in\
\mathcal{K}(Mod(G))$ of an infinite family of bounded complexes of $G$-modules
does not represent the direct sum in $StMod(G)$.

Arbitrary direct sums in $D^{b}(G)/\mathcal{I}nj(Mod(G))$ are determined by
using the triangulated structure to represent each bounded complex
$C^{\bullet}_{j}$ as $M_{j}[0]$ for some $\mathbb{G}_{(r)}$-module $M_{j}$ and
then taking the arbitrary sum $\bigoplus_{j}M_{j}$ of $G$-modules.

######  Remark 4.4.

The tensor triangulated category $D^{b}(Mod(G))/\mathcal{I}nj^{b}(Mod(G))$ of
Theorem 4.2 has the following two alternate formulations. (Analogous
formulations are valid for $D^{b}(mod(G))/Perf(G)$.)

The first is as the Verdier quotient
$\mathcal{K}^{b}(Mod(G))/\mathcal{I}nj^{b}(Mod(G))$, provided that
$\mathcal{I}nj^{b}(Mod(G))$ is viewed as a thick tensor subcategory of
$\mathcal{K}^{b}(Mod(G))$ rather than as a subcategory of $D^{b}(Mod(G))$.
This is essentially immediate from the universal property satisfied by a
Verdier quotient.

The second is the somewhat more “elementary” formulation as the category of
left (or right) fractions of $D^{b}(Mod(G))$ for the multiplicative system
$\mathcal{S}\ \subset Mor(D^{b}(Mod(G)))$ consisting of morphisms of
$f:C^{\bullet}\to D^{\bullet}$ whose cone $c(f)$ is quasi-isomorphic to a
bounded complex of injective $G$-modules. (See [36, Chap10].)

We obtain the following useful corollary of Theorem 4.2.

######  Corollary 4.5.

Let $G$ be a finite group scheme.

  1. (1)

A complex $C^{\bullet}\in\mathcal{K}^{b}(Mod(G))$ is quasi-isomorphic to a
bounded complex of injective $G$-modules if and only if
$\tilde{\Phi}(C^{\bullet})\ =\ 0$ in $St(Mod(G))$.

  2. (2)

For any map $f:C^{\bullet}\to D^{\bullet}$ in $CH^{b}(Mod(G))$, the map
$\Phi(f)$ is an isomorphism in $StMod(G)$ if and only if the cone of $f$ in
$\mathcal{K}^{b}(Mod(\mathbb{G}))$ is an object of $Inj^{b}(Mod(G))$.

  3. (3)

If $C^{\bullet}\to D^{\bullet}\to E^{\bullet}\to C^{\bullet}[1]$ is an exact
triangle in $D^{b}(Mod(G))$ and if $C^{\bullet}$ is quasi-isomorphic to a
bounded complex of injective $G$-modules, then $D^{\bullet}$ is quasi-
isomorphic to a bounded complex of injective $G$-modules $\iff$ $E^{\bullet}$
is quasi-isomorphic to a bounded complex of injective $G$-modules.

###### Proof.

Assertion (1) is immediate from Theorem 4.2 and the fact that the kernel of
the Verdier quotient $\mathfrak{D}\to\mathfrak{D}/\mathfrak{C}$ is
$\mathfrak{C}$ whenever the triangulated subcategory
$\mathfrak{C}\subset\mathfrak{D}$ is a thick subcategory. Assertion (2)
follows immediately from Theorem 4.2 and Remark 4.4.

Since $\tilde{\Phi}$ is exact, we conclude that
$\tilde{\Phi}(C^{\bullet})\to\tilde{\Phi}(D^{\bullet})\to\tilde{\Phi}(E^{\bullet})\to\tilde{\Phi}(C^{\bullet})[1]$
is an exact triangle in $St(Mod(G))$. Thus, if $\tilde{\Phi}(C^{\bullet})=0$,
then $\tilde{\Phi}(D^{\bullet})=0$ $\iff$ $\tilde{\Phi}(E^{\bullet})=0$. This,
together with (1) implies assertion (3). ∎

In our approach to stable module categories for $\mathbb{G}$-modules, mock
injective $\mathbb{G}$-modules as in Definition 2.4 play the role that
injective $\mathbb{G}_{(r)}$-modules play in defining
$StMod(\mathbb{G}_{(r)})$.

######  Definition 4.6.

Let $\mathbb{G}$ be a connected linear algebraic group. We define
$\mathcal{M}ock^{b}(\mathbb{G})$ to be the full subcategory of
$D^{b}(Mod(\mathbb{G}))$ consisting of those bounded complexes $C^{\bullet}$
of $\mathbb{G}$-modules each of whose restrictions
$C^{\bullet}_{|\mathbb{G}_{(r)}}$ is an object of
$\mathcal{I}nj^{b}(\mathbb{G}_{(r)})$.

We define $mock^{b}(\mathbb{G})$ to be the full subcategory of
$D^{b}(mod(\mathbb{G}))$ consisting of those of bounded complexes
$C^{\bullet}$ of finite dimensional $\mathbb{G}$-modules each of whose
restrictions $C^{\bullet}_{|\mathbb{G}_{(r)}}$ is an object of
$\mathcal{I}nj^{b}(\mathbb{G}_{(r)})$.

We easily verify that $\mathcal{M}ock^{b}(\mathbb{G})\ \hookrightarrow\
D^{b}(\mathbb{G})$ is a thick tensor ideal by observing that
$\mathcal{M}ock^{b}(\mathbb{G}_{(r)})\hookrightarrow D^{b}(\mathbb{G}_{(r)})$
is a thick tensor ideal for every $r>0$.

######  Definition 4.7.

Let $\mathbb{G}$ be a connected linear algebraic group. We define
$StMod(\mathbb{G})$ to be the tensor triangulated category defined as the
Verdier quotient

$StMod(\mathbb{G})\quad\equiv\quad
D^{b}(Mod(\mathbb{G}))/\mathcal{M}ock^{b}(\mathbb{G})$

and denote by $q_{Mock}:D^{b}(Mod(\mathbb{G}))\to StMod(\mathbb{G})$ the
quotient functor.

We define $stmod(\mathbb{G})$ to be the tensor triangulated category defined
as the Verdier quotient

$stmod(\mathbb{G})\quad\equiv\quad
D^{b}(mod(\mathbb{G}))/mock^{b}(\mathbb{G})$

and denote by $q_{mock}:D^{b}(mod(\mathbb{G}))\to stmod(\mathbb{G})$ the
quotient functor.

######  Remark 4.8.

Let $\mathbb{G}$ be a connected linear algebraic group with the property that
the minimal dimension of a non-trivial injective $\mathbb{G}_{(r)}$-module
becomes arbitrarily large as $r$ increases. Then every mock injective
$\mathbb{G}$-module is infinite dimensional, so that $stmod(\mathbb{G})\ =\
D^{b}(mod(\mathbb{G}))$. This is the case for each of the examples in Section
3 assuming that $\mathbb{G}$ does not have a factor which is a torus.

We shall frequently abuse notation in what follows by denoting by $M$ both a
$\mathbb{G}$-module $M$ and its restriction $M_{|\mathbb{G}_{(r)}}$ to a
Frobenius kernel $\mathbb{G}_{(r)}\hookrightarrow\mathbb{G}$.

######  Proposition 4.9.

Let $\mathbb{G}$ be an irreducible linear algebraic group and $C^{\bullet},\
D^{\bullet}$ be two bounded complexes of finite dimensional
$\mathbb{G}$-modules. Then for $r$ sufficiently large, the restriction map is
an isomorphism:

(4.9.1) $Hom_{stmod(\mathbb{G})}(C^{\bullet},D^{\bullet})\ \to\
Hom_{stmod(\mathbb{G}_{(r)})}(C^{\bullet}_{|\mathbb{G}_{(r)}},D^{\bullet}_{|\mathbb{G}_{(r)}}).$

###### Proof.

For any finite dimensional $\mathbb{G}$-modules $M,\ N$, the restriction map
$Hom_{Mod(\mathbb{G})}(M,N)\ \to\
Hom_{mod(\mathbb{G}_{(r)})}(M_{|\mathbb{G}_{(r)}},N_{|\mathbb{G}_{(r)}})$ is
an isomorphism for $r>>0$ by [24, I.9.8(6)]. This readily implies that

(4.9.2) $Hom_{\mathcal{K}^{b}(mod(\mathbb{G}))}(C^{\bullet},D^{\bullet})\
\stackrel{{\scriptstyle\sim}}{{\to}}\
Hom_{\mathcal{K}^{b}(mod(\mathbb{G}_{(r)}))}(C^{\bullet}_{|\mathbb{G}_{(r)}},D^{\bullet}_{|\mathbb{G}_{(r)}})\quad
r>>0$

for any pair $C^{\bullet},\ D^{\bullet}$ of bounded complexes of finite
dimensional $\mathbb{G}$-modules. The isomorphism (4.9.2) clearly preserves
quasi-isomorphisms, thus easily implies that the restriction map

(4.9.3) $Hom_{D^{b}(mod(\mathbb{G}))}(C^{\bullet},D^{\bullet})\ \to\
Hom_{D^{b}(mod(\mathbb{G}_{(r)}))}(C^{\bullet}_{|\mathbb{G}_{(r)}},D^{\bullet}_{|\mathbb{G}_{(r)}})$

is also an isomorphism.

Two maps $f,g\in Hom_{D^{b}(mod(\mathbb{G}))}(C^{\bullet},D^{\bullet})$ are
equal in $Hom_{stmod(\mathbb{G})}(C^{\bullet},D^{\bullet})$ if and only if the
exact triangle $D^{\bullet}\to cone(f-g)\to C[1]$ splits. Using (4.9.3), we
conclude such a splitting exists if only if this exact triangle restricted to
$\mathbb{G}_{(r)}$ admits a splitting for all $r>>0$. This, together with the
isomorphism (4.9.3) implies the injectivity of (4.9.1). The surjectivity of
(4.9.1). follows from the isomorphism (4.9.2) and Theorem 4.2. ∎

In the following proposition, we do not rule out the existence of a chain of
maps in $D^{b}(Mod(\mathbb{G}))$ beginning and ending with bounded complexes
of finite dimensional modules $C^{\bullet}\leftarrow X^{\bullet}_{1}\to
X^{\bullet}_{2}\leftarrow\cdots\to D^{\bullet}$ which represents a map in
$StMod(\mathbb{G})$ which is not a map in $stmod(\mathbb{G})$.

######  Proposition 4.10.

Let $\mathbb{G}$ be a connected, irreducible linear algebraic group such that
every non-trivial mock injective $\mathbb{G}$ is infinite dimensional. Then
$stmod(\mathbb{G})\ \hookrightarrow\ StMod(\mathbb{G})$ is a faithful
embedding of tensor triangulated categories.

###### Proof.

We utilize the following commutative square for each $r>0$

(4.10.1)
$\textstyle{stmod(\mathbb{G}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{StMod(\mathbb{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{stmod(\mathbb{G}_{(r)}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{StMod(\mathbb{G}_{(r)})).}$

Observe that $stmod(\mathbb{G}_{(r)})\ \to\ StMod(\mathbb{G}_{(r)})$ is fully
faithful for all $r$ (utilizing the fact that maps in these categories are
equivalence classes of maps of $\mathbb{G}_{(r)}$-modules). By Proposition
4.9, if $f\not=g\in Hom_{stmod(\mathbb{G}))}(C^{\bullet},D^{\bullet})$, then
$f,\ g$ have unequal images in
$Hom_{stmod(\mathbb{G}_{(r)})}(C^{\bullet}_{|\mathbb{G}_{(r)}},D^{\bullet}_{|\mathbb{G}_{(r)}})$
for $r>>0$ and thus (by the previous observation) $f,\ g$ have unequal images
in
$Hom_{StMod(\mathbb{G}_{(r)})}(C^{\bullet}_{|\mathbb{G}_{(r)}},D^{\bullet}_{|\mathbb{G}_{(r)}})$.
Consequently, the commutativity of (4.10.1) implies that $stmod(\mathbb{G})\
\to\ StMod(\mathbb{G})$ is faithful. ∎

We summarize various categories we have considered, together with functors
relating these categories, in the following commutative diagram:

(4.10.2)
$\textstyle{\mathcal{I}nj^{b}(Mod(\mathbb{G}_{(r)}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D^{b}(Mod(\mathbb{G}_{(r)}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{StMod(\mathbb{G}_{(r)})}$$\textstyle{\mathcal{M}ock^{b}(\mathbb{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D^{b}(Mod(\mathbb{G}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{StMod(\mathbb{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{mock^{b}(\mathbb{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D^{b}(mod(\mathbb{G}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{stmod(\mathbb{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Perf(\mathbb{G}_{(r)}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D^{b}(mod(\mathbb{G}_{(r)}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{stmod(\mathbb{G}_{(r)}).}$

The functors established next with target $StMod(\mathbb{G})$ might be useful
in future applications.

######  Proposition 4.11.

Let $f:(\mathbb{G},\mathcal{E})\to(\mathbb{G}^{\prime},\mathcal{E}^{\prime})$
be a map of algebraic groups of exponential type as in Definition 2.7. Then
restriction determines a well defined functor

$f^{*}:StMod(\mathbb{G}^{\prime})\quad\to\quad StMod(\mathbb{G})$

of tensor triangulated categories.

For any algebraic group $\mathbb{G}$ of exponential type and any $r>0$,
induction determines a well defined functor

$ind_{\mathbb{G}_{(r)}}^{\mathbb{G}}:StMod(\mathbb{G}_{(r)})\quad\to\quad
StMod(\mathbb{G})$

of triangulated categories.

If, in addition, $\mathbb{G}$ is defined over $\mathbb{F}_{q}$ for some $p$-th
power $q$, then induction determines a well defined functor

$ind_{\mathbb{G}(\mathbb{F}_{q})}^{\mathbb{G}}:StMod(\mathbb{G}(\mathbb{F}_{q}))\quad\to\quad
StMod(\mathbb{G})$

of triangulated categories.

###### Proof.

Since $f^{*}:\mathcal{K}^{b}(Mod(\mathbb{G}^{\prime}))\ \to\
\mathcal{K}^{b}(Mod(\mathbb{G}))$ is exact and preserves tensor products, it
is a tensor triangulated functor. By Proposition 2.8, $f^{*}$ sends mock
injective modules to mock injective modules, and thus determines
$f^{*}:StMod(\mathbb{G}^{\prime})\to StMod(\mathbb{G})$.

We recall from [8] that a closed subgroup $H\subset\mathbb{G}$ is called exact
if $ind_{H}^{\mathbb{G}}:Mod(H)\to Mod(\mathbb{G})$ is exact (i.e., preserves
short exact sequences). By [8, Thm4.3],
$\mathbb{G}_{(r)}\hookrightarrow\mathbb{G}$ is exact since
$\mathbb{G}/\mathbb{G}_{(r)}\ =\ \mathbb{G}^{(r)}$ is affine. Moreover, the
induction functor $ind_{\mathbb{G}_{(r)}}^{\mathbb{G}}$ sends (mock) injective
$\mathbb{G}_{(r)}$-modules to injective $\mathbb{G}$-modules. We conclude that
$ind_{\mathbb{G}_{(r)}}^{\mathbb{G}}$ induces the triangulated map of Verdier
quotients

$ind_{\mathbb{G}_{(r)}}^{\mathbb{G}}:StMod(\mathbb{G}_{(r)})\simeq\mathcal{K}^{b}(Mod(\mathbb{G}_{(r)}))/\mathcal{I}nj^{b}(Mod(\mathbb{G}_{(r)}))$
$\ \to\ \mathcal{K}^{b}(Mod(\mathbb{G}))/\mathcal{M}ock^{b}(\mathbb{G})\simeq
StMod(\mathbb{G}).$

This same proof justifies the tensor triangulated functor
$ind_{\mathbb{G}(\mathbb{F}_{q})}^{\mathbb{G}}:StMod(\mathbb{G}(\mathbb{F}_{q}))\to
StMod(\mathbb{G})$ upon replacing $\mathbb{G}_{(r)}\hookrightarrow\mathbb{G}$
by $\mathbb{G}(\mathbb{F}_{q})\hookrightarrow\mathbb{G}$. ∎

##  5\. Support theory for complexes of $\mathbb{G}$-modules

In this section, we extend our formulation of the support theory
$M\mapsto\Pi(\mathbb{G})_{M}$ to bounded complexes $C^{\bullet}$ of
$\mathbb{G}$-modules following our consideration of various triangulated
categories in the previous section. As we show below, $C^{\bullet}\ \mapsto\
\Pi(\mathbb{G})_{C^{\bullet}}$ depends only upon the isomorphism class of
$C^{\bullet}$ in $StMod(\mathbb{G})$ and satisfies evident analogues of the
good properties for $M\mapsto\Pi(\mathbb{G})_{M}$ established in Theorem 2.5.
This intertwining of support theory and stable module categories will be
applied in Section 6 to provide a classification of certain tensor
triangulated subcategories of our stable module categories.

We proceed by first formulating in Definition 5.1 our support theory
$C^{\bullet}\ \mapsto\ \Pi(G)_{C^{\bullet}}$ for bounded complexes of
$G$-modules with $G$ a finite group scheme. Proposition 5.2 shows that this
formulation corresponds to the usual support theory for $G$-modules under the
homeomorphism of Theorem 4.2. After defining our support theory $C^{\bullet}\
\mapsto\Pi(\mathbb{G})_{C^{\bullet}}$ for bounded complexes of
$\mathbb{G}$-modules in Definition 5.3, we present in Theorem 5.4 numerous
good properties of this theory whose proofs are derived from the analogous
properties of $M\mapsto\Pi(\mathbb{G}_{(r)})_{M}$ using Proposition 5.2. For
$(\mathbb{G},\mathcal{E})$ an algebraic group of exponential type, we describe
the subspace $\mathbb{P}\mathfrak{C}(\mathbb{G})_{C^{\bullet}}\ \subset\
\mathbb{P}\mathfrak{C}(\mathbb{G})$ corresponding to
$\Pi(\mathbb{G})_{C^{\bullet}}\ \subset\Pi(\mathbb{G})$ in terms of “actions
at exponentials.”

We recall the map of $k$-algebras $\epsilon_{r}:k\mathbb{G}_{a(1)}\to
k\mathbb{G}_{a(r)}$ of (1.1.1) and observe that the restriction map
$\epsilon_{r}^{*}:\mathcal{K}^{b}(Mod(\mathbb{G}_{a(r)})\to\mathcal{K}^{b}(Mod(\mathbb{G}_{a(1)})$
is exact (i.e., a map of triangulated categories). Observe that the 0-module
is a free module of rank 0 for any $G$ (for any affine group scheme $G$). In
particular, if $C^{\bullet}\in CH^{b}(G)$ is contractible (i.e., homotopy
equivalent to the 0-complex), then $C^{\bullet}$ is quasi-isomorphic to the
0-complex and thus to a complex of free $G$-modules.

######  Definition 5.1.

Let $G$ be a finite group scheme over $k$ and let $C^{\bullet}$ be a bounded
complex of $G$-modules. Let $\alpha_{K}:K[t]/t^{p}\to KG$ be a $\pi$-point
whose equivalence class $[\alpha_{K}]$ is a (Zariski) point of $\Pi(G)$. Then
$[\alpha_{K}]$ is said to be a point of the $\Pi(G)_{C^{\bullet}}\ \subset\
\Pi(G)$ if $\alpha_{K}^{*}(C^{\bullet}_{K})$ is not quasi-isomorphic to a
bounded complex of free $K[t]/t^{p}$-modules.

The awkwardness of proving that $C^{\bullet}\to\Pi(G)_{C^{\bullet}}$ is well
defined on objects of $StMod(G)$ is partially caused by the fact that
$M\mapsto\Pi(G)_{M}$ is not a well defined functor on $Mod(G)$. For example, a
map $M\to N$ of $G$–modules and a $\pi$-point $\alpha_{K}:K[t]/t^{p}\to KG$
might be such that $\alpha_{K}^{*}(M)$ is not free (thus determining a point
$\alpha_{K}$ of $\Pi(G)_{M}$) whereas $\alpha_{K}^{*}(N)$ is free.

######  Proposition 5.2.

Let $G$ be a finite group scheme.

  1. (1)

If a map $f:C^{\bullet}\ \to\ D^{\bullet}$ of bounded complexes of $G$-modules
satisfies the condition that $cone(f)$ is quasi-isomorphic to a bounded
complex of injective $G$-modules, then $\Pi(G)_{C^{\bullet}}\ =\
\Pi(G)_{D^{\bullet}}$ (as subspaces of $\Pi(G)$).

  2. (2)

If $C^{\bullet},\ D^{\bullet}$ are bounded complexes of $G$-modules which are
isomorphic in $D^{b}(Mod(G))/\mathcal{I}nj^{b}(Mod(G))$, then
$\Pi(G)_{C^{\bullet}}\ =\ \Pi(G)_{D^{\bullet}}$.

  3. (3)

For any $C^{\bullet}$ in $D^{b}(Mod(G))$, the functor
$\tilde{\Phi}:D^{b}(Mod(G))\ \to\ StMod(G)$ of Theorem 4.2 satisfies the
property that

(5.2.1) $\Pi(G)_{C^{\bullet}}\quad=\quad\Pi(G)_{\tilde{\Phi}(C^{\bullet})}.$

  4. (4)

For any $C^{\bullet}$ in $D^{b}(Mod(G))$, $\Pi(G)_{C^{\bullet}}=\emptyset$ if
and only if $C^{\bullet}$ is an object of $\mathcal{I}nj^{b}(Mod(G))$.

  5. (5)

If $C^{\bullet}\to D^{\bullet}\to E^{\bullet}\to C^{\bullet}[1]$ is an exact
triangle in $D^{b}(Mod(G))$, then $\Pi(\mathbb{G})_{D^{\bullet}}\ \subset\
\Pi(\mathbb{G})_{C^{\bullet}}\cap\Pi(\mathbb{G})_{E^{\bullet}}$.

  6. (6)

For any $C^{\bullet}$ in $D^{b}(mod(G))$, $\Pi(G)_{C^{\bullet}}\ \subset\
\Pi(\mathbb{G})$ is closed.

###### Proof.

Assertion (1) follows from Corollary 4.5(3) since each $\pi$-point
$\alpha_{K}:K[t]/t^{p}\to KG$ determines a triangulated map
$\alpha_{K}^{*}:D^{b}(Mod(G))\to D^{b}(K[t]/t^{p})$, preserving quasi-
isomorphisms, commuting with taking cones, and sending injective $G$-modules
to free $K[T]/t^{p}$-modules.

Consider a “roof” $s^{-1}\circ f$ representing a map from $C^{\bullet}$ to
$D^{\bullet}$ in $StMod(G)$; namely, a pair of maps in $D^{b}(Mod(G))$,
$C^{\bullet}\stackrel{{\scriptstyle s}}{{\leftarrow}}E\stackrel{{\scriptstyle
f}}{{\to}}D^{\bullet}$, with the cone of $s$ quasi-isomorphic to a bounded
complex of injective modules. Since maps in $StMod(G)$ are given by a calculus
of fractions, we conclude that such a roof is an isomorphism in $StMod(G)$ if
and only $f$ is itself an isomorphism in $StMod(G)$. Thus, assertion (2)
follows from assertion (1).

Since the functor $\Psi$ of Theorem 4.1 is essentially surjective, it suffices
to verify the identification (5.2.1) for $C^{\bullet}$ of the form $M[0]$ for
some $\mathbb{G}$-module $M$. This is immediate from the definitions.

Assertion (4) follows from assertion (3) and the “projectivity test” for
$G$-modules which asserts that $\Pi(G)_{M}=\emptyset$ if and only if $M$ is a
projective $G$-modules [28]. Alternatively, (4) follows directly from
Corollary 4.5(1) and the fact that $\Phi$ is an equivalence of categories.

Assertion (5) follows from the exactness of $\Phi$ and the “two out of three”
property for $M\mapsto\Pi(G)_{M}$ given in Theorem 2.5(4).

Finally, if $C^{\bullet}$ in $D^{b}(mod(G))$, then $\tilde{\Phi}(C^{\bullet})$
is an object of $stmod(G)$ represented by a finite dimensional $G$-module.
Since $\Pi(G)_{M}\ \subset\ \Pi(G)$ is closed for any finite dimensional
$G$-module by Theorem 2.5(6), assertion (6) follows from assertion (3). ∎

The following definition of $C^{\bullet}\ \mapsto\
\Pi(\mathbb{G})_{C^{\bullet}}$ is a natural extension of
$M\mapsto\Pi(\mathbb{G})_{M}$ for a $\mathbb{G}$-module $M$ as defined in
Definition 2.2 and $C^{\bullet}\ \mapsto\ \Pi(\mathbb{G}_{(r)})_{C^{\bullet}}$
for a bounded complex of $\mathbb{G}_{(r)}$-modules as formulated in
Definition 5.1.

######  Definition 5.3.

Let $\mathbb{G}$ be a linear algebraic group and let $C^{\bullet}$ be bounded
complex of $\mathbb{G}$-modules. We define the subspace
$\Pi(\mathbb{G})_{C^{\bullet}}\ \subset\ \Pi(\mathbb{G})$ by

$\Pi(\mathbb{G})_{C^{\bullet}}\quad\equiv\quad\varinjlim_{r}\Pi(\mathbb{G}_{(r)})_{C^{\bullet}_{|\mathbb{G}_{(r)}}}\quad\subset\quad\varinjlim_{r}\Pi(\mathbb{G}_{(r)})\quad\equiv\quad\Pi(\mathbb{G})$

where $\Pi(\mathbb{G}_{(r)})_{C^{\bullet}_{|\mathbb{G}_{(r)}}}\subset\
\Pi(\mathbb{G}_{(r)})$ is defined in Definition 5.1.

We state and prove our generalization of Theorem 2.5 to bounded complexes of
$\mathbb{G}$-modules, thereby establishing the basic properties of the support
theory $C^{\bullet}\to\Pi(\mathbb{G})_{C^{\bullet}}$.

######  Theorem 5.4.

Let $\mathbb{G}$ be a linear algebraic group over $k$. Then

$C^{\bullet}\in CH^{b}(Mod(\mathbb{G}))\quad\ \text{maps
to}\quad\Pi(\mathbb{G})_{C^{\bullet}}\ \subset\ \Pi(\mathbb{G})$

as defined in Definition 5.3 satisfies the following properties.

  1. (1)

$\Pi(\mathbb{G})_{C^{\bullet}}\ =\ \emptyset$ if and only $C^{\bullet}\ \in\
\mathcal{M}ock^{b}(\mathbb{G})$.

  2. (2)

$\Pi(\mathbb{G})_{C^{\bullet}}\quad=\quad\Pi(\mathbb{G})_{C^{\bullet}[n]}$ as
subsets of $V(\mathbb{G})$

  3. (3)

If $C^{\bullet},\ D^{\bullet}$ are bounded complexes of $\mathbb{G}$-modules
which are isomorphic in $StMod(\mathbb{G})$, then
$\Pi(\mathbb{G})_{C^{\bullet}}\ =\ \Pi(\mathbb{G})_{D^{\bullet}}$.

  4. (4)

If $C^{\bullet}\to D^{\bullet}\to E^{\bullet}\to C^{\bullet}[1]$ is an exact
triangle in $StMod(\mathbb{G})$, then $\Pi(\mathbb{G})_{D^{\bullet}}\ \subset\
\Pi(\mathbb{G})_{C^{\bullet}}\cap\Pi(\mathbb{G})_{E^{\bullet}}$.

  5. (5)

If $C^{\bullet}\ =\ \bigoplus_{i\in I}C^{\bullet}_{i}$, then
$\Pi(\mathbb{G})_{C^{\bullet}}\ =\ \bigcup_{i\in
I}\Pi(\mathbb{G})_{C^{\bullet}_{i}}$.

  6. (6)

$\Pi(\mathbb{G})_{C^{\bullet}\otimes D^{\bullet}}\ =\
\Pi(\mathbb{G})_{C^{\bullet}}\cap\Pi(\mathbb{G})_{D^{\bullet}}$.

  7. (7)

If $C^{\bullet}\in CH^{b}(mod(\mathbb{G}))$, then
$\Pi(\mathbb{G})_{C^{\bullet}}\ \subset\ \Pi(\mathbb{G})$ is closed.

###### Proof.

As defined in Definition 4.6, $C^{\bullet}$ is an object of
$Mock^{b}(\mathbb{G})$ if and only if its restriction to each
$\mathbb{G}_{(r)}$ lies in $Inj^{b}(Mod(\mathbb{G}_{(r)})$) which is
equivalent by Proposition 5.2(4) to the condition that each
$\Pi(\mathbb{G}_{(r)})_{C^{\bullet}}$ is empty. This proves the first
assertion

The second assertion follows from the fact that
$(\mathcal{E}_{\Lambda_{r}(\underline{B})}\circ\epsilon_{r})^{*}(-)$ commutes
with shifts $(-)[n]$ of chain complexes and the fact that $(-)[n]$ preserves
quasi-isomorphisms.

Proposition 5.2(2) implies assertion 3. Assertion (4) follows from the
exactness of restriction of $\mathbb{G}$-modules to $\mathbb{G}_{(r)}$-modules
and Proposition 5.2(5). Similarly, assertions (5) and (6) follow from the
exactness of restriction of $\mathbb{G}$-modules to
$\mathbb{G}_{(r)}$-modules, Proposition 5.2(3), and the corresponding
properties for $M\to\Pi(\mathbb{G})_{M}$ given in Theorem 2.5.

Since $\Pi(\mathbb{G})$ is equipped with the colimit topology,
$\Pi(\mathbb{G})_{C^{\bullet}}\ \subset\ \Pi(\mathbb{G})$ is closed if and
only if each $\Pi(\mathbb{G}_{(r)})_{C^{\bullet}}\ \subset\
\Pi(\mathbb{G}_{(r)})$ is closed. Thus, assertion (7) follows from Proposition
5.2(6). ∎

######  Remark 5.5.

It is natural to ask what if any new closed subsets of $\Pi(\mathbb{G})$
(other than those of the form $\Pi(\mathbb{G})_{M[0]}$ for $M$ finite
dimensional) are realized as $\Pi(\mathbb{G})_{C^{\bullet}}$ for $C^{\bullet}$
a bounded complex of finite dimensional $\mathbb{G}$-modules.

A serious impediment to understanding what subsets are realized as
$\Pi(\mathbb{G})_{C^{\bullet}}$ is that the functor $\Phi_{(r)}$ for a given
$r>0$ (as in Theorem 4.2) depends upon the triangulated structure of
$StMod(\mathbb{G}_{(r)})$ which is not natural with respect to passing from
$\mathbb{G}_{(r)}$ to $\mathbb{G}_{(r+1)}$. Thus, if we take a bounded complex
of finite dimensional $\mathbb{G}$-modules $C^{\bullet}$, we may associate for
each $r>0$ a finite dimensional $\mathbb{G}_{(r)}$-module
$\Phi_{(r)}(C^{\bullet})$, but we lack a method to construct a (finite
dimensional) $\mathbb{G}$-module using the family
$\\{\Phi_{(r)}(C^{\bullet}),\ r>0\\}$.

We introduce the following terminology for “realizable” subsets of
$\Pi(\mathbb{G})$.

######  Terminology 5.6.

Let $\mathbb{G}$ be a linear algebraic group and consider a subspace $X\
\subset\ \Pi(\mathbb{G})$.

  * •

$X$ is said to be $Mod(\mathbb{G})$-realizable (respectively,
$mod(\mathbb{G})$-realizable) if there exists some $\mathbb{G}$-module (resp.,
finite dimensional $\mathbb{G}$-module) $M$ such that $X\ =\
\Pi(\mathbb{G})_{M}$.

  * •

$X$ is said to be $StMod(\mathbb{G})$-realizable (respectively,
$stmod(\mathbb{G})$-realizable) if there exists some bounded complex of
$\mathbb{G}$-modules (resp., bounded complex of finite dimensional
$\mathbb{G}$-modules) $C^{\bullet}$ such that $X\ =\
\Pi(\mathbb{G})_{C^{\bullet}}$.

  * •

$X$ is said to be locally $StMod(\mathbb{G})$-realizable (respectively,
locally $stmod(\mathbb{G})$-realizable) if there exists some collection of
bounded complexes of $\mathbb{G}$-modules (resp., collection of bounded
complexes of finite dimensional $\mathbb{G}$-modules)
$\\{C^{\bullet}_{\alpha}\\}$ such that $X\ =\
\bigcup_{\alpha}\Pi(\mathbb{G})_{C^{\bullet}_{\alpha}}$.

The identification (2.6.3) of $M\to\Pi(\mathbb{G})_{M}$ with
$M\to\mathbb{P}\mathfrak{C}(\mathbb{G})_{M}$ for $(\mathbb{G},\mathcal{E})$ an
algebraic group of exponential type provides a more “concrete” interpretation
of $M\to\Pi(\mathbb{G})_{M}$. We proceed to show that if
$(\mathbb{G},\mathcal{E})$ an algebraic group of exponential type, then this
identification extends to bounded complexes of $\mathbb{G}$-modules.

######  Definition 5.7.

Consider an algebraic group $(\mathbb{G},\mathcal{E})$ of exponential type.
For any bounded complex $C^{\bullet}$ of $\mathbb{G}$-modules, we define the
subspace
$\mathbb{P}\mathfrak{C}(\mathbb{G})_{C^{\bullet}}\quad\subset\quad\mathbb{P}\mathfrak{C}(G)$
by identifying its $K$-points for any field extension $K/k$ to consist of
those $K$-points of $\mathbb{P}\mathfrak{C}(G)$ represented by a $K$-point
$\underline{B}\in\mathfrak{C}(\mathbb{G})$ such that

$(\mathcal{E}_{\Lambda_{r}(\underline{B})}\circ\epsilon_{r})^{*}(C^{\bullet})$
is not quasi-isomorphic to a complex of free modules
for any $r$ sufficiently large that $B_{i}=0$ for $i\geq r$.

We define $\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})_{C^{\bullet}}$ as the
intersection
$\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})\cap\mathbb{P}\mathfrak{C}(\mathbb{G})_{C^{\bullet}}$,
so that

$\mathbb{P}\mathfrak{C}(\mathbb{G})_{C^{\bullet}}\quad=\quad\bigcup_{r}\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})_{C^{\bullet}}.$

######  Proposition 5.8.

Let $(\mathbb{G},\mathcal{E})$ be an algebraic group of exponential type, fix
$r>0$, and let $C^{\bullet}$ be a a bounded complex of $\mathbb{G}$-modules.
The homeomorphism
$\Phi\circ\Psi\circ(\rho_{r}\circ\lambda):\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})\
\to\ \Pi(\mathbb{G}_{(r)})$ of Theorem 2.6 restricts to a homeomorphism

(5.8.1)
$\Phi\circ\Psi\circ(\rho_{r}\circ\lambda):\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})_{C^{\bullet}}\
\stackrel{{\scriptstyle\sim}}{{\to}}\
\Pi(\mathbb{G}_{(r)})_{C^{\bullet}_{|\mathbb{G}_{(r)}}},$

thereby determining the homeomorphism
$\mathbb{P}\mathfrak{C}(\mathbb{G})_{C^{\bullet}}\
\stackrel{{\scriptstyle\sim}}{{\to}}\ \Pi(\mathbb{G})_{C^{\bullet}}.$

###### Proof.

To prove (5.8.1), we must compare Definition 5.1 with Definition 5.7. Exactly
as in the proof of Theorem 2.6, this comparison is made by juxtaposing the
determination of
$\Psi\circ(\rho_{r}\circ\lambda_{r}):\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})\to\mathbb{P}|\mathbb{G}_{(r)}|$
in Theorem 1.12(4), the definition of $\rho_{r}$ in Definition 1.11, and the
determination of
$\phi\circ\Psi:\mathbb{P}V_{r}(\mathbb{G}_{(r)})\to\Pi(\mathbb{G}_{(r)})$ in
Theorem 1.6. ∎

The following proposition is the extension to complexes of Proposition
2.12(1). The proof follows immediately upon recalling that the $\pi$-point
associated to the 1-parameter subgroup $\mathcal{E}_{\underline{B}}$ with
$B_{i}=0,\ i\geq r$ is
$\alpha_{K}\equiv\mathcal{E}_{\Lambda_{r}(\underline{B})}\circ\epsilon_{r}:K[t]/t^{p}\to
K\mathbb{G}$ so that the conditions of Definition 5.1 and Definition 5.7 match
in light of the identification $\Pi(\mathbb{G})_{C^{\bullet}}\ \equiv\
\varinjlim_{r}\Pi(\mathbb{G}_{(r)})_{C^{\bullet}}$ of Definition 5.3.

######  Proposition 5.9.

Let $(\mathbb{G},\mathcal{E})$ be an algebraic group of exponential type and
let $C^{\bullet}$ be a a bounded complex of $\mathbb{G}$-modules. If
$C^{\bullet}$ is quasi-isomorphic to a bounded complex each of whose terms has
bounded exponential degree (as in Proposition 2.12, then

$\mathbb{P}\mathfrak{C}(\mathbb{G})_{C^{\bullet}}\ =\
\pi_{r}^{-1}(\mathbb{P}\mathfrak{C}_{r}(\mathbb{G})_{C^{\bullet}}),\quad
r>>0.$

In anticipation of the classification of certain subcategories in Section 6,
we remark that the full subcategory of $stmod(\mathbb{G})$ (respectively.
$StMod(\mathbb{G})$) consisting of bounded complexes of $\mathbb{G}$-modules
whose terms have bounded exponential degree is a thick triangulated
subcategory (resp., localizing subcategory), but not a tensor ideal.

##  6\. Classifying subcategories of $stmod(\mathbb{G})$ and
$StMod(\mathbb{G})$

In this section, we use the classification of thick, tensor ideals of
$stmod(\mathbb{G}_{(r)})$ (given, for example, in [30]) to classify the
$(r)$-complete thick tensor ideals of $stmod(\mathbb{G})$ in Theorem 6.5. The
formulation of the property “ $(r)$-complete” for a thick tensor ideal
$\mathfrak{C}\subset stmod(\mathbb{G})$ is introduced in Definition 6.1. This
property is naturally suggested by the relationship of the support varieties
for $\mathbb{G}$-modules and those associated to restrictions of
$\mathbb{G}$-modules to Frobenius kernels $\\{\mathbb{G}_{(r)},\ r>0\\}$. We
also present an analogous classification for the $(r)$-complete localizing
tensor triangulated subcategories of $StMod(\mathbb{G})$.

We remind the reader that a full triangulated subcategory $\mathfrak{C}$ of a
triangulated category $\mathfrak{D}$ is a thick subcategory if and only if
every object of $\mathfrak{D}$ which is a direct summand of an object of
$\mathfrak{C}$ is itself an object of $\mathfrak{C}$ [30]. If the triangulated
category $\mathfrak{D}$ has a (for convenience, symmetric) tensor structure,
then a full triangulated subcategory $\mathfrak{C}$ of $\mathfrak{D}$ is said
to be a tensor ideal if tensoring any object of $\mathfrak{C}$ with an object
of $\mathfrak{D}$ is again an object of $\mathfrak{C}$. The thick tensor ideal
in $\mathfrak{C}$ generated by a collection of objects of $\mathfrak{C}$ is
the full triangulated subcategory of $\mathfrak{C}$ whose objects are obtained
by repeatedly applying the operations of taking finite sums of objects, taking
summands of objects, taking cones of maps, and taking tensor products with
arbitrary objects of $\mathfrak{C}$.

######  Definition 6.1.

Consider a linear algebraic group $\mathbb{G}$ and let $\mathfrak{C}\subset
stmod(\mathbb{G})$ be a triangulated subcategory. For each $r>0$, we denote by
$\mathfrak{C}_{|\mathbb{G}_{(r)}}\ \subset\ stmod(\mathbb{G}_{(r)})$ the
essential image of $\mathfrak{C}$ under the restriction functor
$stmod(\mathbb{G})\to stmod(\mathbb{G}_{(r)})$. In other words,
$\mathfrak{C}_{|\mathbb{G}_{(r)}}$ is the full subcategory of $\mathfrak{C}$
whose objects are those objects of $\mathfrak{C}$ isomorphic to objects
obtained by restriction of objects in $stmod(\mathbb{G})$.

If $\mathfrak{C}\ \subset\ stmod(\mathbb{G})$ is a thick tensor ideal, then we
denote by $\mathfrak{C}_{(r)}$ the thick tensor ideal of
$stmod(\mathbb{G}_{(r)})$ generated by $\mathfrak{C}_{|\mathbb{G}_{(r)}}$.

We say that a thick tensor ideal $\mathfrak{C}\subset stmod(\mathbb{G})$ is
$(r)$-complete if the following condition is satisfied for every
$C^{\bullet}\in stmod(\mathbb{G})$:

(6.1.1) $C^{\bullet}\in\mathfrak{C}\quad\iff\quad
C^{\bullet}_{|\mathbb{G}_{(r)}}\in\ \mathfrak{C}_{(r)},\ \forall r>0.$

We remark that even if $\mathfrak{C}$ is $r$-complete,
$\mathfrak{C}_{|\mathbb{G}_{(r)}}$ is unlikely to be a tensor ideal in
$stmod(\mathbb{G}_{(r)})$ since typically there are objects of
$stmod(\mathbb{G}_{(r)})$ which are not restrictions of objects of
$stmod(\mathbb{G})$.

######  Definition 6.2.

Consider a linear algebraic group $\mathbb{G}$. For any collection
$\mathcal{S}$ of bounded complexes of finite dimensional $\mathbb{G}$-modules,
we define the locally closed subset

$\Pi(\mathbb{G},\mathfrak{C})\ \equiv\
\bigcup_{C^{\bullet}\in\mathcal{S}}\Pi(\mathbb{G})_{C^{\bullet}}\ \subset\
\Pi(\mathbb{G})$

and define

$\Pi(\mathbb{G}_{(r)},\mathfrak{C}_{|\mathbb{G}_{(r)}})\ \equiv\
\bigcup_{C^{\bullet}\in\mathcal{S}}\Pi(\mathbb{G}_{(r)})_{C^{\bullet}_{|\mathbb{G}_{(r)}}}\
\subset\ \Pi(\mathbb{G}_{(r)}),$

where $C^{\bullet}_{|\mathbb{G}_{(r)}}$ is the restriction of $C^{\bullet}$ to
$\mathbb{G}_{(r)}$.

For any subset $X\subset\Pi(\mathbb{G})$, we define the full subcategory

$\mathfrak{C}_{X}\ \equiv\ \langle\\{C^{\bullet}\in stmod(\mathbb{G}),\ such\
that\ \mathbb{P}V(\mathbb{G})_{C^{\bullet}}\subset X\\}\rangle\ \subset\
stmod(\mathbb{G}).$

######  Proposition 6.3.

Let $\mathbb{G}$ be a linear algebraic group and let $\mathfrak{C}\ \subset\
stmod(\mathbb{G})$ be a thick tensor ideal. We define the $(r)$-completion of
$\mathfrak{C}$, $\mathfrak{C}^{\vee}$, to be the full subcategory of
$stmod(\mathbb{G})$ whose objects are those bounded complexes $C^{\bullet}$ of
finite dimensional $\mathbb{G}$-modules such that the restriction of
$C^{\bullet}$ to $\mathbb{G}_{(r)}$ lies in $\mathfrak{C}_{(r)}$ for every
$r>0$.

Then $\mathfrak{C}\ \mapsto\ \mathfrak{C}^{\vee}$ satisfies the following
properties.

  1. (1)

$\mathfrak{C}^{\vee}$ is a thick tensor ideal of $stmod(\mathbb{G})$.

  2. (2)

$\mathfrak{C}_{(r)}\ =\ (\mathfrak{C}^{\vee})_{(r)}$ for all $r>0$, so that
$(\mathfrak{C}^{\vee})^{\vee}\ =\ \mathfrak{C}^{\vee}$.

  3. (3)

$\mathfrak{C}^{\vee}$ is the the minimal $(r)$-complete thick tensor ideal of
$stmod(\mathbb{G})$ containing $\mathfrak{C}$.

  4. (4)

The natural embedding $\Pi(\mathbb{G}_{(r)},\mathfrak{C}_{|\mathbb{G}_{(r)}})\
\hookrightarrow\Pi(\mathbb{G}_{(r)},\mathfrak{C}_{(r)})$ is the identity for
each $r>0.$

###### Proof.

If $C^{\bullet}\to D^{\bullet}\to E^{\bullet}\to C^{\bullet}[1]$ is an exact
triangle in $stmod(\mathbb{G})$ with $C^{\bullet}\to D^{\bullet}$ in
$\mathfrak{C}^{\vee}$, then the restriction of this exact triangle to
$stmod(\mathbb{G}_{(r)})$ is an exact triangle in $\mathfrak{C}_{(r)}$ for all
$r>0$. This implies that $E^{\bullet}$ is an object of $\mathfrak{C}^{\vee}$,
Similarly, if $C^{\bullet}\in C^{\vee}$ and $X^{\bullet}$ is an arbitrary
bounded complex of finite dimensional $\mathbb{G}$-modules, then any summand
of $C^{\bullet}$ is an object of $C^{\vee}$ and $C^{\bullet}\otimes
X^{\bullet}$ is an object of $C^{\vee}$ for any $X^{\bullet}\in
stmod(\mathbb{G})$, since each $\mathfrak{C}_{(r)}\subset
stmod(\mathbb{G}_{(r)})$ is a thick tensor ideal. Thus, $\mathfrak{C}^{\vee}$
is a thick tensor ideal of $stmod(\mathbb{G})$.

The operations on objects of $\mathfrak{C}$ which produce objects of
$\mathfrak{C}^{\vee}$ restrict to internal operations on $\mathfrak{C}_{(r)}$,
so that $\mathfrak{C}_{(r)}\ =\ (\mathfrak{C}^{\vee})_{(r)}$. This implies
that $(\mathfrak{C}^{\vee})^{\vee}\ =\ \mathfrak{C}^{\vee}$ and that
$\mathfrak{C}^{\vee}$ is $(r)$-complete.

Observe that objects of $\mathfrak{C}_{(r)}$ are obtained by starting with
objects of $\mathfrak{C}_{|\mathbb{G}_{(r)}}\ \subset\
stmod(\mathbb{G}_{(r)})$ and successively applying the operations of taking
cones, taking direct summands, and tensoring with objects of
$stmod(\mathbb{G})$. By Theorem 5.4, these operations preserve the property
that if the support of the input object of an operation is contained in
$\Pi(\mathbb{G}_{(r)},\mathfrak{C}_{|\mathbb{G}_{(r)}})$, then the support of
the output object is also contained in
$\Pi(\mathbb{G}_{(r)},\mathfrak{C}_{|\mathbb{G}_{(r)}})$. Consequently,
$\Pi(\mathbb{G}_{(r)},\mathfrak{C}_{|\mathbb{G}_{(r)}})\ =\
\Pi(\mathbb{G}_{(r)},\mathfrak{C}_{(r)})$. ∎

The definition of an $(r)$-complete thick tensor ideal of $stmod(\mathbb{G})$
was made in anticipation of the following result.

######  Proposition 6.4.

Let $\mathbb{G}$ be a linear algebraic group and let $X\ \subset\
\Pi(\mathbb{G})$ be a subspace. Then $\mathfrak{C}_{X}$ is a thick tensor
ideal of $stmod(\mathbb{G})$ which is $(r)$-complete.

###### Proof.

Theorem 5.4(4) easily implies that $\mathfrak{C}_{X}$ is a thick triangulated
subcategory of $stmod(\mathbb{G})$; consequently, Theorem 5.4(6) implies that
$\mathfrak{C}_{X}$ is a thick tensor ideal in $stmod(\mathbb{G})$.

Assume that $C^{\bullet}$ is an object of $\mathfrak{C}_{X}^{\vee}$; in other
words, $C^{\bullet}$ is a bounded complex of finite dimensional
$\mathbb{G}$-modules whose restriction to each $\mathbb{G}_{(r)}$ lies in
$(\mathfrak{C}_{X})_{(r)}$. By Proposition 6.3(4) and the evident inclusion
$\Pi(\mathbb{G}_{(r)},(\mathfrak{C}_{X})_{|\mathbb{G}_{(r)}})\ \subset\ X$, we
conclude that $\Pi(\mathbb{G}_{(r)},(\mathfrak{C}_{X})_{(r)})\ \subset\ X$ for
all $r>0$, so that $\Pi(\mathbb{G})_{C^{\bullet}}\ \subset\ X$ and thus
$C^{\bullet}$ is an object of $\mathfrak{C}_{X}$. ∎

We proceed to show that locally $stmod(\mathbb{G})$-realizable subsets of
$\Pi(\mathbb{G})$ as in Terminology 5.6 classify $(r)$-complete, thick tensor
ideals of $stmod(\mathbb{G})$. Our proof is heavily dependent upon the
classification of thick tensor ideals of $stmod(\mathbb{G}_{(r)})$ first
established in [2] for finite groups and then for general finite group schemes
in [18].

######  Theorem 6.5.

Let $\mathbb{G}$ be a linear algebraic group. The correspondences

$X\ \mapsto\ \mathfrak{C}_{X}\quad\mathfrak{C}\ \mapsto\
\Pi(\mathbb{G},\mathfrak{C}).$

restrict to give mutually inverse bijections

$\\{locally\ stmod(\mathbb{G}){\text{-}}realizable\ subsets\
X\subset\Pi(\mathbb{G})\\}\ \longleftrightarrow$ $\\{(r){\text{-}}complete,\
thick\ tensor\ ideals\ \mathfrak{C}\subset stmod(\mathbb{G})\\}.$

###### Proof.

Observe that $\Pi(\mathbb{G},\mathfrak{C}_{X})\ \subset\ X$ for any subspace
$X\subset\Pi(\mathbb{G})$. Assume now that $X$ is a locally
$stmod(\mathbb{G})$ realizable subspace of $\Pi(\mathbb{G})$. Then for any
point $x\in X$, there is there exists some bounded complex $C^{\bullet}_{x}$
of finite dimensional $\mathbb{G}$-modules with
$\Pi(\mathbb{G})_{C^{\bullet}_{x}}\ \subset\ X$ such that
$x\in\Pi(\mathbb{G})_{C^{\bullet}_{x}}$. Hence, $X\subset\
\Pi(\mathbb{G},\mathfrak{C}_{X})$ and thus $\Pi(\mathbb{G},\mathfrak{C}_{X})\
=\ X$

To complete the proof, we prove that the evident inclusion $\mathfrak{C}\
\subset\ \mathfrak{C}_{\Pi(\mathbb{G},\mathfrak{C})}$ of full, triangulated
subcategories of $stmod(\mathbb{G})$ is an equivalence if
$\mathfrak{C}\subset\operatorname{stmod}\nolimits(\mathbb{G})$ is an
$(r)$-complete tensor ideal. Assume that $\mathfrak{C}$ is a thick tensor
ideal and consider some object $E^{\bullet}$ of
$\mathfrak{C}_{\Pi(\mathbb{G},\mathfrak{C})}$. Since
$\mathfrak{C}_{|\mathbb{G}_{(r)}}\ \subset\ \mathfrak{C}_{(r)}$, the
restriction of $E^{\bullet}$ lies in
$\mathfrak{C}_{\mathbb{P}V(\mathbb{G}_{(r)},\mathfrak{C}_{(r)})}$. We apply
the classification of thick tensor ideals of $stmod(\mathbb{G}_{(r)})$, given
in [18, Thm6.3] based upon the construction of Rickard idempotents in [31],
which tells us for the thick tensor ideal $\mathfrak{C}_{(r)}\ \subset\
stmod(\mathbb{G}_{(r)})$ that
$\mathfrak{C}_{\mathbb{P}V(\mathbb{G}_{(r)},\mathfrak{C}_{(r)})}=\mathfrak{C}_{(r)}$
as subcategories of $stmod(\mathbb{G}_{(r)})$. Thus, the restriction of
$E^{\bullet}$ to each $\mathbb{G}_{(r)}$ lies in $\mathfrak{C}_{(r)}$ so that
$E^{\bullet}$ is an object of $\mathfrak{C}^{\vee}$. Hence, if $\mathfrak{C}$
is also $(r)$-complete, then $E^{\bullet}\in\mathfrak{C}$. ∎

Rickard constructs idempotent endofunctors on $StMod(\mathbb{G}_{(r)})$,
$\mathcal{E}_{\mathfrak{C}}(-)$ and $\mathcal{F}_{\mathfrak{C}}(-)$,
associated to a thick subcategory $\mathfrak{C}\subset
stmod(\mathbb{G}_{(r)})$. Among other properties, these functors satisfy the
condition for any $X\in StMod(\mathbb{G}_{(r)})$ that there is a natural exact
triangle

$E_{\mathfrak{C}_{(r)}}(X)\to X\to F_{\mathfrak{C}_{(r)}}(X)\to
E_{\mathfrak{C}_{(r)}}(X)[1]$

isomorphic to

$E_{\mathfrak{C}_{(r)}}(k)\otimes X\to X\to F_{\mathfrak{C}_{(r)}}(k)\otimes
X\to(E_{\mathfrak{C}_{(r)}}(k)\otimes X)[1].$

If $\mathfrak{C}$ is a tensor ideal inside $stmod(\mathbb{G}_{(r)})$, then
Rickard ’s arguments using adjunction relating
$Hom(M^{\vee},\mathcal{F}_{\mathfrak{C}}(k))$ to
$M\otimes\mathcal{F}_{\mathfrak{C}}(k)$ (and similarly for
$\mathcal{E}_{\mathfrak{C}}(k))$) provide a test for whether or not $M\in
stmod(\mathbb{G}_{(r)})$ belongs to $\mathfrak{C}$. This argument requires the
compactness of $M$ and the tensor ideal condition on $\mathfrak{C}$.

We complement Theorem 6.5 with the following extension of Rickard’s test in
terms of Rickard’s idempotents, applying now to $C^{\bullet}$ an object of
$stmod(\mathbb{G})$ and $\mathfrak{C}\subset stmod(\mathbb{G})$ an
$(r)$-complete, thick tensor ideal.

######  Theorem 6.6.

Let $\mathbb{G}$ be a linear algebraic group and let $\mathfrak{C}\ \subset\
stmod(\mathbb{G})$ be an thick tensor ideal which is $(r)$-complete. For each
$r>0$, we consider the Rickard idempotent functors

$E_{\mathfrak{C}_{(r)}}(-),\