# Infinite dimensional modules for linear algebraic groups

Eric M. Friedlander∗ Department of Mathematics, University of Southern
California, Los Angeles, CA 90089<EMAIL_ADDRESS>
###### Abstract.

We investigate infinite dimensional modules for a linear algebraic group
$\mathbb{G}$ over a field of positive characteristic $p$. For any subcoalgebra
$C\subset\mathcal{O}(\mathbb{G})$ of the coordinate algebra of $\mathbb{G}$,
we consider the abelian subcategory $CoMod(C)\subset Mod(\mathbb{G})$ and the
left exact functor $(-)_{C}:Mod(\mathbb{G})\to CoMod(C)$ that is right adjoint
to the inclusion functor. The class of cofinite $\mathbb{G}$-modules is
formulated using finite dimensional subcoalgebras of $\mathcal{O}(\mathbb{G})$
and the new invariant of “cofinite type” is introduced.

We are particularly interested in mock injective $\mathbb{G}$-modules,
$\mathbb{G}$-modules which are not seen by earlier support theories. Various
properties of these ghostly $\mathbb{G}$-modules are established. The stable
category $StMock(\mathbb{G})$ is introduced, enabling mock injective
$\mathbb{G}$-modules to fit into the framework of tensor triangulated
categories.

###### Key words and phrases:

filtrations, coalgebras, mock injectives

###### 2020 Mathematics Subject Classification:

20G05, 20G10

∗ partially supported by the Simons Foundation

##  0\. Introduction

An approach to studying a $\mathbb{G}$-module $M$ for a connected affine group
scheme $\mathbb{G}$ over a field $k$ of characteristic $p>0$ is to investigate
the restriction of $M$ to Frobenius kernels $\mathbb{G}_{(r)}$ of
$\mathbb{G}$. From some points of view, the representation theory of finite
group schemes such as $\mathbb{G}_{(r)}$ resembles the representation theory
of finite groups and thus shares many useful properties. The technique of
restricting $\mathbb{G}$-modules to Frobenius kernels has been effective in
studying irreducible modules and standard finite dimensional modules for
reductive groups $\mathbb{G}$ (see, for example, [18]). One method for
studying the representations of finite group schemes $G$ involves constructing
a suitable theory of “supports” for $G$-modules. This method, beginning with
the consideration of support varieties for elementary abelian $p$-groups (see
[2]), has been developed for the representation theory of finite group schemes
(see, for example, [13]), linear algebraic groups, various finite dimensional
algebras (see, for example, [1], [12], [22]), and linear algebraic groups (see
[8]).

Some aspects of the representation theory of $\mathbb{G}$ are not seen by
support varieties or by restriction of $\mathbb{G}$-modules to Frobenius
kernels. Of particular importance are proper mock injective modules, modules
which are not injective as $\mathbb{G}$-modules but whose restrictions to each
Frobenius kernel $\mathbb{G}_{(r)}$ is an injective $\mathbb{G}_{(r)}$-module
[9]. Using the lens of “stable categories”, we showed in [11] that localizing
the category of bounded cochain complexes $\mathcal{K}^{b}(Mod(\mathbb{G}))$
of $\mathbb{G}$-modules with respect to the bounded derived category of mock
injective $\mathbb{G}$-modules yields a category (which we denote by
$\overline{StMod}(\mathbb{G})$ in this paper) which serves as a good analogue
for linear algebraic groups of stable module categories for finite group
schemes. Indeed, our motivation for this work has been to further our
understanding of mock injective $\mathbb{G}$-modules. In Section 6, we
formulate some of this understanding in terms of the thick subcategory
$StMock(\mathbb{G})\hookrightarrow StMod(\mathbb{G})$ of bounded complexes
associated to mock injective $\mathbb{G}$-modules.

In [9], we considered linear algebraic groups of exponential type and utilized
the filtration by “exponential degree” of a $\mathbb{G}$-module $M$. This
suggested that a useful approach to studying a $\mathbb{G}$-module $M$ is to
consider filtrations by $\mathbb{G}$-submodules whose coaction is constrained
to specified subspaces of $\mathcal{O}(\mathbb{G})$. In this paper, we
continue this analysis, providing numerous functorial filtrations of
$\mathbb{G}$-modules for an arbitrary linear algebraic group $\mathbb{G}$ each
of which is a filtration by $\mathbb{G}$-modules (not necessarily the case for
the filtration of [9]).

In Theorem 1.4, we associate to an arbitrary subspace (i.e., an arbitrary
$k$-vector subspace) $X\subset\mathcal{O}(\mathbb{G})$ an abelian subcategory
$i_{X}:Mod(\mathbb{G},X)\hookrightarrow Mod(\mathbb{G})$ of the abelian
category of $\mathbb{G}$-modules together with a left exact functor
$(-)_{X}:Mod(\mathbb{G})\ \to\ Mod(\mathbb{G},X)$ that is right adjoint to
$i_{X}$. Essentially by construction, $M_{X}$ is the largest
$\mathbb{G}$-submodule of $M$ whose coaction $\Delta_{M}:M\to
M\otimes\mathcal{O}(\mathbb{G})$ factors through $M\otimes X\subset
M\otimes\mathcal{O}(\mathbb{G})$. We belatedly realized that our construction
is an abstraction of J. Jantzen’s “truncated categories” for reductive groups
(see [18, Chap A]).

In Proposition 1.9, we show for any ascending, converging sequence
$\\{X_{d}\\}$ of subspaces of $\mathcal{O}(\mathbb{G})$ that a
$\mathbb{G}$-module $M$ is injective if and only if each $M_{X_{d}}$ is
injective in $Mod(\mathbb{G},X_{d})$. For $\mathbb{G}$ of exponential type
equipped with the subspaces $\mathcal{E}(d)\subset\mathcal{O}(\mathbb{G})$
considered in [9], Proposition 1.11 establishes that the filtration
$\\{M_{\mathcal{E}(d)}\\}$ of a $\mathbb{G}$-module $M$ is the coarsest
filtration by $\mathbb{G}$-submodules subordinate to the filtration by
exponential degree of $M$ as considered in [9].

If a given subspace $X\subset\mathcal{O}(\mathbb{G})$ is the underlying
subspace of a subcoalgebra $C$, then $Mod(\mathbb{G},X)\ \subset\
Mod(\mathbb{G})$ is naturally identified with $CoMod(C)$, the abelian category
of comodules for $C$. For a given subspace $X\subset\mathcal{O}(\mathbb{G})$,
we consider the smallest subcoalgebra $\mathcal{O}(\mathbb{G})_{\langle
X\rangle}\subset\mathcal{O}(\mathbb{G})$ containing $X$; if $X$ is finite
dimensional, then so is $\mathcal{O}(\mathbb{G})_{\langle X\rangle}$. For our
purposes, it is sufficient to consider filtrations of $\mathbb{G}$-modules
provided by an ascending, converging sequence of finite dimensional
subcoalgebras of $\mathcal{O}(\mathbb{G})$ such as $\\{\langle
X_{d}\rangle\\}$ associated to an ascending, converging sequence of finite
dimensional subspaces $\\{X_{d}\\}$ of $\mathcal{O}(\mathbb{G})$. In
Definition 2.7, we provide another construction of ascending, converging
finite dimensional subcoalgebras $\\{\mathcal{O}(\mathbb{G})_{\leq d,\phi}\\}$
of $\mathcal{O}(\mathbb{G})$ by first explicitly defining
$\\{\mathcal{O}(GL_{N})_{\leq d}\\}$ and then using a specified closed
embedding $\phi:\mathbb{G}\hookrightarrow GL_{N}$. As seen in Proposition
2.12, different closed embeddings $\mathbb{G}$ into general linear groups
determine “comparable” filtrations of $\mathbb{G}$-modules. For a given
$\mathbb{G}$-module $M$, Proposition 3.8 compares and contrasts the filtration
$\\{M_{\leq d,\phi}\\}$ with the family of restrictions of
$\\{M_{|\mathbb{G}_{(r)}}\\}$ to Frobenius kernels of $\mathbb{G}$.

We say that a $\mathbb{G}$-module $M$ is “cofinite” if $M_{X}$ is finite
dimensional for every finite dimensional subspace
$X\subset\mathcal{O}(\mathbb{G})$. The full subcategory $CoFin(\mathbb{G})$ of
$Mod(\mathbb{G})$ consisting of cofinite $\mathbb{G}$-modules has various
closures properties (see Proposition 4.2) but does not contain all cokernels
and is not closed under tensor products. In Definition 4.5, we formulate the
“growth” of cofinite modules; a finer invariant, the “cofinite type” of a
$\mathbb{G}$-module $M$ (denoted $\gamma(\mathbb{G},\phi)_{M})$) is introduced
which depends upon a choice of closed embedding
$\phi:\mathbb{G}\hookrightarrow GL_{N}$. Various computations of
$\gamma(\mathbb{G},\phi)_{M}$ are given.

In Section 5, we investigate mock injective $\mathbb{G}$-modules. As observed
in Proposition 5.1, the full subcategory $Mock(\mathbb{G})$ of
$Mod(\mathbb{G})$ whose objects are mock injective $\mathbb{G}$-modules is an
exact subcategory with enough injective objects, arbitrary directed colimits,
and the “two out of three” property. Moreover, restriction along a closed
embedding $\mathbb{H}\hookrightarrow\mathbb{G}$ determines an exact functor
$(-)_{\mathbb{H}}:Mock(\mathbb{G})\to Mock(\mathbb{H})$ and tensor product in
$Mod(\mathbb{G})$ determines $\otimes:Mock(\mathbb{G})\otimes
Mod(\mathbb{G})\to Mock(\mathbb{G})$. On the other hand,
$Hom_{\mathbb{G}}(-,-):Mock(\mathbb{G})\times mod(\mathbb{G})\to
Mod(\mathbb{G})$ is typically the 0-pairing by Theorem 5.6; in other words,
for familiar $\mathbb{G}$, $Hom_{\mathbb{G}}(J,M)=0$ if $J$ is a mock
injective $\mathbb{G}$-module and $M$ is a finite dimensional
$\mathbb{G}$-module. Examples 5.11 and 5.13 give sample computations of the
cofinite type of classes of cofinite proper mock injective
$\mathbb{G}$-modules. Theorem 5.9 presents a construction due to Hardesty,
Nakano, and Sobaje in [16] of familes of mock injective $\mathbb{G}$-modules

Theorem 6.3 presents a “stable” categorification of this construction, one
which considers triangulated categories in which bounded complexes of
injectives are set equal to 0. This formulation involves the tensor
triangulated category $StMock(\mathbb{G})$, the kernel of the quotient functor
from
$StMod(\mathbb{G})\equiv\mathcal{K}^{b}(Mod(\mathbb{G}))/\mathcal{I}nj(\mathbb{G})\to\overline{StMod}(\mathbb{G})$.

Before mentioning some open questions in Section 8, we provide in Theorem 7.4
a necessary and sufficient condition in terms of $\pi$-points for a
$\mathbb{G}_{a}$-module to be injective.

Throughout this paper, the ground field $k$ is assumed to be of characteristic
$p>0$ for some prime $p$. We use $\mathcal{O}(\mathbb{G})$ to denote the
coordinate algebra of $\mathbb{G}$, and $k[\mathbb{G}]^{R}$ (respectively,
$k[\mathbb{G}]^{L}$) the underlying vector space of $\mathcal{O}(\mathbb{G})$
with the right (resp., left) regular representation. For us, an affine group
scheme is an affine group scheme over $k$ which is of finite type over $k$; a
linear algebraic group is a smooth and connected affine group scheme. A closed
embedding of affine group schemes will always mean a closed immersion which is
a morphism of group schemes.

We thank Paul Balmer, Bob Guralnick, Julia Pevtsova, Paul Sobaje, and
especially Cris Negron for helpful insights.

##  1\. Filtrations by subspaces $X\subset\mathcal{O}(\mathbb{G})$

For an affine group scheme $\mathbb{G}$, we denote by $Mod(\mathbb{G})$ the
abelian category of $\mathbb{G}$-modules; more precisely, the abelian category
of comodules for $\mathcal{O}(\mathbb{G})$ as a coalgebra over $k$. Thus,
$M\in Mod(\mathbb{G})$ is a vector space over $k$ equipped with a right
coaction $\Delta_{M}:M\ \to\ M\otimes\mathcal{O}(\mathbb{G})$ which determines
natural (with respect to commutative $k$-algebras $A$) $A$-linear group
actions $\mathbb{G}(A)\times(A\otimes M)\to(A\otimes M)$. Unless specified
otherwise, tensor products are implicitly assumed to be tensor products of
$k$-vector spaces.

######  Definition 1.1.

Let $i_{X}:X\subset\mathcal{O}(\mathbb{G})$ be a subspace (that is, a
$k$-subspace of $\mathcal{O}(\mathbb{G})$ viewed as a $k$-vector space). We
define $Mod(\mathbb{G},X)$ to be the full subcategory of $Mod(\mathbb{G})$
whose objects are those $\mathbb{G}$-modules $M$ whose coaction $\Delta_{M}$
factors as $(id_{M}\otimes i_{X})\circ\Delta_{M,X}:M\to M\otimes X\to
M\otimes\mathcal{O}(\mathbb{G})$.

We refer to such $\mathbb{G}$-modules as “$X$-comodules”.

We utilize the following lemma investigating the closure properties of
$Mod(\mathbb{G},X)\ \subset Mod(\mathbb{G})$.

######  Lemma 1.2.

Let $X\subset\mathcal{O}(\mathbb{G})$ be a subspace and let $M$ be an
$X$-comodule.

  1. (1)

If $j:N\ \to\ M$ is an injective map of $\mathbb{G}$-modules, then $N$ is also
an $X$-comodule.

  2. (2)

If $q:M\to Q$ is a surjective map of $\mathbb{G}$-modules, then $Q$ is also an
$X$-comodule.

  3. (3)

If $f:M\to N$ is a map of $\mathbb{G}$-modules with $N$ an $X$-comodule, then
the kernel and cokernel of $f$ are $X$-comodules.

###### Proof.

To prove (1), choose a basis $\\{m_{\beta},\beta\in I\\}$ of $M$ such that a
subset of this basis is a basis for $N$, and choose a basis
$\\{f_{\alpha},\alpha\in A\\}$ of $\mathcal{O}(\mathbb{G})$ such that a subset
of this basis is a basis for $X$. If $m\in M$ is an element of $N$, then
$\Delta(m)=\sum_{\beta,\alpha}a_{\beta,\alpha}m_{\beta}\otimes f_{\alpha}$
lies both in $N\otimes k[\mathbb{G}]$ so that $a_{\beta,\alpha}=0$ unless
$m_{\beta}\in N$ and lies in $M\otimes X$ so that $a_{\beta,\alpha}=0$ unless
$f_{\alpha}\in X$. Thus $\Delta(m)\in N\otimes X$ if $m\in N$.

To prove assertion (2), let $j:K\to M$ be the kernel of the surjective map
$q:M\to Q$. Using assertion (1), we have the commutative diagram

(1.2.1)
$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\scriptstyle{\Delta_{K,X}}$$\textstyle{K\otimes
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id_{K}\otimes
i_{X}}$$\scriptstyle{j\otimes
id}$$\textstyle{K\otimes\mathcal{O}(\mathbb{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j\otimes
id_{\mathcal{O}(\mathbb{G}]}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{\Delta_{M,X}}$$\textstyle{M\otimes
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id_{M}\otimes
i_{X}}$$\scriptstyle{q\otimes
id_{X}}$$\textstyle{M\otimes\mathcal{O}(\mathbb{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q\otimes
id_{\mathcal{O}(\mathbb{G})}}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta_{Q}}$$\textstyle{Q\otimes
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id_{Q}\otimes
i_{X}}$$\textstyle{Q\otimes\mathcal{O}(\mathbb{G}).}$

A simple diagram chase for (1.2.1) implies that $\Delta_{Q}$ factors uniquely
through $Q\otimes X$.

To prove (3), observe that the kernel $ker\\{f\\}\subset M$ is an $X$-comodule
by (1) and that the quotient $N\twoheadrightarrow coker\\{f\\}$ is an
$X$-comodule by (2). ∎

We recall that the sum $M_{1}+M_{2}\ \subset\ M$ of $\mathbb{G}$-submodules
$M_{1},M_{2}$ of $M$ is the image of $M_{1}\oplus M_{2}\to M$.

######  Proposition 1.3.

Let $M$ be a $\mathbb{G}$-module and $X\subset\mathcal{O}(\mathbb{G})$ be a
subspace. If $M_{1}\subset M,\ M_{2}\subset M$ are $\mathbb{G}$-submodules
which are $X$-comodules, then $M_{1}+M_{2}\subset M$ is also an $X$-comodule.
Thus, the category $\chi(M)$ whose objects are $X$-comodules of $M$ and whose
maps are inclusions of $\mathbb{G}$-submodules of $M$ is a filtering
subcategory of $Mod(\mathbb{G})$.

Consequently,

(1.3.1) $M_{X}\ \equiv\ \varinjlim_{N\in\chi(M)}N\quad=\
\bigcup_{N\in\chi(M)}N\ \subset\ M$

is well defined as a $\mathbb{G}$-submodule. Moreover, $M_{X}\subset M$ is an
$X$-comodule, the largest $X$-comodule contained in $M$.

###### Proof.

Recall that $M_{1}+M_{2}\subset M$ fits in a short exact sequence of
$\mathbb{G}$-modules $0\to\ M_{1}\cap\ M_{2}\ \to\ M_{1}\oplus M_{2}\ \to\
M_{1}+M_{2}\to 0$. Since $M_{1}\oplus M_{2}$ is clearly an $X$-comodule
whenever $M_{1},M_{2}$ are $X$-comodules, the first assertion follows from
Lemma 1.2.

This implies that the category $\chi(M)$ is filtering; given two objects
$N_{1}\subset M$ and $N_{2}\subset M$ of $\chi(M)$, both map to
$N_{1}+N_{2}\subset M$ which is an object of $\chi(M)$. Thus,
$\varinjlim_{N\in\chi(M)}N\ \to M$ equals the union $\bigcup_{N\in\chi(M)}N\
\subset\ M$. Recall that $(-)\otimes V$ for a given vector space $V$ commutes
with filtered colimits of $k$-vector spaces. Consequently,

$\varinjlim_{N\in\chi(M)}\Delta_{N}:\varinjlim_{N\in\chi(M)}N\ \to\
\varinjlim_{N\in\chi(M)}(N\otimes\mathcal{O}(\mathbb{G}])\ =\
(\varinjlim_{N\in\chi(M)}N)\otimes\mathcal{O}(\mathbb{G})$

factors through
$\varinjlim_{N\in\chi(M)}N\to(\varinjlim_{N\in\chi(M)}N)\otimes X$. In other
words, $M_{X}$ is an $X$-comodule, the largest $X$-comodule contained in $M$.
∎

Theorem 1.4 introduces the functor $(-)_{X}:Mod(\mathbb{G})\quad\to\quad
Mod(\mathbb{G},X)$ right adjoint to the natural embedding.

######  Theorem 1.4.

Let $\mathbb{G}$ be an affine group scheme of finite type over $k$ and let
$i_{X}:X\subset\mathcal{O}(\mathbb{G})$ be a subspace. Denote by

$i_{X*}:Mod(\mathbb{G},X)\quad\hookrightarrow\quad Mod(\mathbb{G})$

the full subcategory of $Mod(\mathbb{G})$ whose objects are $X$-comodules.

  1. (1)

$Mod(\mathbb{G},X)$ is an abelian subcategory which is closed under filtering
colimits.

  2. (2)

Sending a $\mathbb{G}$-module $M$ to the $\mathbb{G}$-submodule $M_{X}$ of $M$
as in (1.3.1) determines a functor

$(-)_{X}:Mod(\mathbb{G})\quad\to\quad Mod(\mathbb{G},X).$

  3. (3)

$(-)_{X}$ is left exact and is right adjoint to the embedding functor
$i_{X*}:Mod(\mathbb{G},X)\to Mod(\mathbb{G})$.

###### Proof.

The fact that $Mod(\mathbb{G},X)$ is an abelian subcategory of
$Mod(\mathbb{G})$ follows directly from Lemma 1.2. To prove that
$Mod(\mathbb{G},X)$ is closed under colimits indexed by a filtering category
$I$, observe that the natural map $\varinjlim_{i}(M_{i}\otimes X)\ \to\
\varinjlim_{i}(M_{i})\otimes X$ is an isomorphism. Thus, if each $M_{i}$ is an
$X$-comodule, so is $\varinjlim_{i}(M_{i})$.

To prove functoriality of $(-)_{X}$, observe that if $f:M\to N$ is a map in
$Mod(\mathbb{G})$ then $f(M_{X})\ \subset\ N$ is contained in $N_{X}$ by Lemma
1.2(2) and the equality $N_{X}\ =\ \bigcup_{N^{\prime}\in\chi_{M}}N^{\prime}\
\subset N$ of (1.3.1). This equality also shows that $(-)_{X}$ is left exact.

Functoriality together with (1.3.1) determines the natural inclusion

$Hom_{Mod(\mathbb{G})}(i_{X*}(M),N)\quad\hookrightarrow\quad
Hom_{Mod(\mathbb{G},X)}(M,N_{X})$

inverse to the inclusion $Hom_{Mod(\mathbb{G},X)}(M,N_{X})\ \hookrightarrow\
Hom_{Mod(\mathbb{G})}(i_{X*}(M),N)$ and thus a bijection. This is the natural
isomorphism of the asserted adjunction. ∎

Perhaps the simplest example of the functor
$(-)_{X}:Mod(\mathbb{G})\quad\to\quad Mod(\mathbb{G},X)$ is the case in which
$X=k\cdot 1$, the span of $1\in\mathcal{O}(\mathbb{G})$. In this case,
$(-)_{X}\ =\ H^{0}(\mathbb{G},-)$. Observe that $H^{0}(\mathbb{G},-)$ is left
exact for any $\mathbb{G}$, but is not always exact.

######  Remark 1.5.

The full abelian subcategory $Mod(\mathbb{G},X)\ \hookrightarrow\
Mod(\mathbb{G})$ is typically not closed under extensions. For example, if
$Ext_{\mathbb{G}}^{1}(k,k)\not=0$, then $Mod(\mathbb{G},k\cdot 1)$ is not
closed under extensions.

######  Corollary 1.6.

Retain the hypotheses and notation of Theorem 1.4.

  1. (1)

If $I$ is an injective $\mathbb{G}$-module, then $I_{X}$ is an injective
object of $Mod(\mathbb{G},X)$.

  2. (2)

Similarly, if $J$ is an injective object of $Mod(\mathbb{G},X)$ and $Y\subset
X$ is a subspace, then $J_{Y}$ is an injective object of $Mod(\mathbb{G},Y)$.

  3. (3)

If $M$ is an $X$-comodule with a given embedding $M\hookrightarrow I$ into an
injective $\mathbb{G}$-module, then $M\hookrightarrow I_{X}$ in
$Mod(\mathbb{G},X)$ is an embedding of $M$ in an injective object of
$Mod(\mathbb{G},X)$.

In particular, the abelian category $Mod(\mathbb{G},X)$ has “enough
injectives”.

###### Proof.

If $I$ is an injective $\mathbb{G}$-module, then the fact that $(-)_{X}$ has a
left adjoint (namely, $i_{X*}(-)$) which is left exact implies that $I_{X}$ is
an injective object of $Mod(\mathbb{G},X)$. To prove (2), observe that the
embedding $i_{X,Y,*}:Mod(\mathbb{G},Y)\hookrightarrow Mod(\mathbb{G},X)$ is
left adjoint to the restriction of $(-)_{Y}$ to $Mod(\mathbb{G},X)$ so that
the argument using the existence of a left exact left adjoint applies to prove
(2). Assertion (3) follows from (1) together with the left exactness of
$(-)_{X}$ and the fact that $(-)_{X}$ restricts to the identify on
$Mod(\mathbb{G},X)$. ∎

We say a sequence of subspaces $\\{X_{i}\\}$ of $\mathcal{O}(\mathbb{G})$
indexed by the non-negative numbers $i\geq 0$ is an ascending, converging
sequence of subspaces of $\mathcal{O}(\mathbb{G})$ if $X_{i}\subset X_{i+1}$
for all $i\geq 0$ and if $\bigcup_{i\geq 0}X_{i}\ =\ \mathcal{O}(\mathbb{G})$.

######  Proposition 1.7.

Let $\mathbb{G}$ be an affine group scheme of finite type over $k$ and let
$\\{X_{i}\\}$ be an ascending, converging sequence of subspaces of
$\mathcal{O}(\mathbb{G})$. Sending a $\mathbb{G}$-module $M$ to the sequence
of $\mathbb{G}$-submodules

(1.7.1) $M_{X_{0}}\ \subset\ M_{X_{1}}\ \subset M_{X_{2}}\subset\cdots\
\subset\ \bigcup_{i\geq 0}M_{X_{i}}\ =\ M$

is a filtration of $M$, functorial in $M$, with the property that each
$M_{X_{i}}$ is an $X_{i}$-comodule. We say that $\\{M_{X_{i}}\\}$ is an
ascending, converging sequence of $\mathbb{G}$-submodues of $M$.

If $M$ is a finite dimensional $\mathbb{G}$-module, then $M$ is an
$X_{i}$-comodule for all $i$ sufficiently large.

###### Proof.

If the $\mathbb{G}$-module $M$ is finite dimensional, then $\Delta_{M}:M\to
M\otimes\mathcal{O}(\mathbb{G})$ must have image in some finite dimensional
subspace of $M\otimes X$ and thus must have image contained in $M\otimes
X_{i}$ for $i$ sufficiently large. If $M$ is an arbitrary $\mathbb{G}$-module,
then $M$ is locally finite so that every $m\in M$ lies in some finite
dimensional $\mathbb{G}$-submodule $M^{\prime}\subset M$ and thus must be
contained in some $M_{X_{i}}$ as required. ∎

The following corollary is an immediate consequence of the functoriality of
the filtration $M\ \mapsto\ \\{M_{X_{i}}\\}$ and the fact that $\bigcup_{i\geq
0}M_{X_{i}}\ =\ M$ for any $\mathbb{G}$-module $M$.

######  Corollary 1.8.

For any ascending, converging sequence $\\{X_{i}\\}$ of subspaces of
$\mathcal{O}(\mathbb{G})$ a map $\phi:M\to N$ of $\mathbb{G}$-modules is an
isomorphism if and only if $(\phi)_{X_{i}}:M_{X_{i}}\to N_{X_{i}}$ is an
isomorphism of $\mathbb{G}$-modules for all $i$.

We argue exactly as in the proof of [9, Prop 4.2] to conclude the following
detection of rational injectivity of a $\mathbb{G}$-module.

######  Proposition 1.9.

Consider an affine group scheme $\mathbb{G}$ of finite type over $k$ and let
$\\{X_{i}\\}$ be an ascending, converging sequence of subspaces of
$\mathcal{O}(\mathbb{G})$. Then a $\mathbb{G}$-module $L$ is injective if and
only if $L_{X_{i}}$ is an injective object of $Mod(\mathbb{G},X_{i})$ for all
$i\geq 0$.

###### Proof.

Granted Corollary 1.6, it suffices to show that if a $\mathbb{G}$-module $L$
has the property that $L_{X_{i}}\subset L$ is an injective object of
$Mod(\mathbb{G},X_{i})$ for all $i\geq 0$, then $L$ is an injective
$\mathbb{G}$-module. Let $M^{\prime}\hookrightarrow M$ be an inclusion of
$G$-modules and consider a map $f^{\prime}:M^{\prime}\to L$ of
$\mathbb{G}$-modules. We inductively construct an extension
$f_{d}:M_{X_{d}}\to L_{X_{d}}$ of $f^{\prime}_{d}:M^{\prime}_{X_{d}}\to
L_{X_{d}}$, the restriction of $f^{\prime}$ to $M^{\prime}_{X_{d}}$. Choose
$f_{d}:M_{X_{d}}\to L_{X_{d}}$ extending
$f_{d}^{\prime}+f_{d-1}:(M^{\prime})_{X_{d}}+M_{X_{d-1}}\to L_{X_{d}}$ using
the injectivity of $L_{X_{d}}$ as an object of $Mod(\mathbb{G},X_{d})$ (and
taking $M_{X_{-1}}=0$). We define $f:M\to L$ to be
$\varinjlim_{d}f_{d}:M_{X_{d}}\to L_{X_{d}}$, thereby extending $f^{\prime}$.
∎

We proceed to give in Proposition 1.11 a simple fix for the “filtration by
exponential degree” of a $\mathbb{G}$-module $M$ for a linear algebraic group
$\mathbb{G}$ of exponential type as formulated in [9]. Our modification
provides the coarsest filtration which is a filtration by
$\mathbb{G}$-submodules subordinate to that of [9].

We recall the definition of a linear algebraic group $\mathbb{G}$ of
exponential type, a class of linear algebraic groups for which there is a
somewhat explicit geometric description of the support varieties of its
representations. Let $\mathcal{N}_{p}(\mathfrak{g})$ denote the $p$-nilpotent
cone of the Lie algebra $\mathfrak{g}=Lie(\mathbb{G})$. Thus,
$\mathcal{N}_{p}(\mathfrak{g})\ \subset\ \mathfrak{g}$ is the reduced affine
variety whose set of $K$-points is identified with the elements
$X\in\mathfrak{g}_{K}$ such that $X^{[p]}=0$. We utilize the notation
$\mathcal{C}_{r}(N_{p}(\mathfrak{g}))$ to denote the commuting variety of
$r$-tuples of pair-wise commuting, $p$-nilpotent elements of $\mathfrak{g}$.
The condition of Definition 1.10(5) for a linear algebraic group $\mathbb{G}$
requires that this affine variety has the same $K$-points as the scheme
$V(\mathbb{G}_{(r)})$ introduced in [26] which represents the functor of
1-parameter subgroups of the infinitesimal group scheme $\mathbb{G}_{(r)}$.

######  Definition 1.10.

[8, Defn1.6] Let $\mathbb{G}$ be a linear algebraic group with Lie algebra
$\mathfrak{g}$. A structure of exponential type on $\mathbb{G}$ is a
$\mathbb{G}$-equivariant morphism of $k$-schemes (with respect to adjoint
actions)

(1.10.1) $\mathcal{E}:\mathcal{N}_{p}(\mathfrak{g})\times\mathbb{G}_{a}\
\to\mathbb{G},\quad(B,s)\mapsto\mathcal{E}_{B}(s)$

satisfying the following conditions for all field extensions $K/k$:

  1. (1)

For each $B\in\mathcal{N}_{p}(\mathfrak{g})(K)$,
$\mathcal{E}_{B}:\mathbb{G}_{a,K}\to\mathbb{G}_{K}$ is a 1-parameter subgroup.

  2. (2)

For any pair of commuting $p$-nilpotent elements
$B,B^{\prime}\in\mathfrak{g}_{K}$, the maps
$\mathcal{E}_{B},\mathcal{E}_{B^{\prime}}:\mathbb{G}_{a,K}\to\mathbb{G}_{K}$
commute.

  3. (3)

For any $\alpha\in K$, and any $s\in\mathbb{G}_{a}(K)$,
$\mathcal{E}_{\alpha\cdot B}(s)=\mathcal{E}_{B}(\alpha\cdot s)$.

  4. (4)

Every 1-parameter subgroup $\psi:\mathbb{G}_{a,K}\to\mathbb{G}_{K}$ is of the
form

$\mathcal{E}_{\underline{B}}\ \equiv\
\prod_{s=0}^{r-1}(\mathcal{E}_{B_{s}}\circ F^{s})$

for some $r>0$, some
$\underline{B}\in\mathcal{C}_{r}(\mathcal{N}_{p}(\mathfrak{g}_{K}))$.

  5. (5)

The natural map $\mathcal{C}_{r}(\mathcal{N}_{p}(\mathfrak{g}))\to
V(\mathbb{G}_{(r)})$ induces a bijection on $K$-points sending $\underline{B}$
to the infinitesimal 1-parameter subgroup
$\mathbb{G}_{a(r),K}\to\mathbb{G}_{(r),K}$ which factors
$\mathcal{E}_{\underline{B}}\circ
i_{r}:\mathbb{G}_{a(r),K}\to\mathbb{G}_{a,K}\to\mathbb{G}_{K}$.

Various examples of $\mathbb{G}$ of exponential type are considered in [25];
these include simple classical groups, their standard parabolic subgroups, and
the unipotent radicals of these parabolic subgroups.

######  Proposition 1.11.

([8, Defn 4.5]) Let $(\mathbb{G},\mathcal{E})$ be a linear algebraic group of
exponential type. We define $\mathcal{E}(\mathbb{G})_{d}\ \hookrightarrow\
\mathcal{O}(\mathbb{G})$ to be the subspace

(1.11.1) $\mathcal{E}(\mathbb{G})_{d}\ \equiv\
\mathcal{E}^{*-1}(k[(\mathcal{N}_{p}(\mathfrak{g})][t]_{\leq
d})\quad\subset\quad\mathcal{O}(\mathbb{G})$

where $k[\mathcal{N}_{p}(\mathfrak{g})][t]_{\leq d}\subset
k[\mathcal{N}_{p}(\mathfrak{g})\times\mathbb{G}_{a}]$ is the subspace of
polynomials in $k[\mathcal{N}_{p}(\mathfrak{g})][t]$ of degree $\leq d$. So
defined, $\\{M_{\mathcal{E}(\mathbb{G})_{d}}\\}$ is the coarsest filtration of
$M$ by $\mathbb{G}$-modules subordinate to the “filtration by exponential
degree” of [9].

Moreover, $M_{\mathcal{E}(d)}\otimes N_{\mathcal{E}(e)}\ \subset\ (M\otimes
N)_{\mathcal{E}(d+e)}$ for every pair of $\mathbb{G}$-modules $M,N$.

###### Proof.

The “filtration by exponential degree” of [9, Defn 3.10] associates to the
$\mathbb{G}$-module $M$ and a positive integer $d$ the subspace
$M_{[d]}\subset M$ consisting of elements $m\in M$ with the property that
$\Delta_{M}(m)\in M\otimes\mathcal{E}(\mathbb{G})_{d}$. By Proposition 1.3,
$M_{\mathcal{E}(\mathbb{G})_{d}}$ is the largest $\mathbb{G}$-submodule of $M$
such that $M_{\mathcal{E}(\mathbb{G})_{d}}\subset M_{[d]}$.

The second statement follows easily from the observation the coaction map
$\Delta_{M\otimes N}:M\otimes N\ \to\ (M\otimes
N)\otimes\mathcal{O}(\mathbb{G})$ arises by composing
$\Delta_{M}\otimes\Delta_{N}$ with the product map
$\mathcal{O}(\mathbb{G})\otimes\mathcal{O}(\mathbb{G})\ \to\
\mathcal{O}(\mathbb{G})$. ∎

In the following proposition,
$\mathcal{E}_{B}:\mathbb{G}_{a,K}\to\mathbb{G}_{K}$ is the exponential map
determined by a $K$-point $B$ of $\mathcal{N}_{p}(\mathfrak{g})$ (for some
field extension $K/k$) and the exponential structure
$\mathcal{E}:\mathcal{N}_{p}(\mathbb{G})\times\mathbb{G}_{a}\to\mathbb{G}$.
For any $s\geq 0$, $u_{s}:k[t]\to k$ is the $k$-linear map sending $t^{i}$ to
0 if $i\not=p^{s}$ and sending $t^{p^{s}}$ to 1. We denote by
$(\mathcal{E}_{B})_{*}(u_{s}):\mathcal{O}(\mathbb{G}_{K})\to K$ the $K$-linear
map given by the composition
$u_{s}\circ(\mathcal{E}_{B})^{*}:\mathcal{O}(\mathbb{G}_{K})\to K[t]\to K$.

We utilize the (“$\pi$-point”) support variety $M\mapsto\Pi(\mathbb{G})_{M}$
of [11] extending the construction for finite group schemes given in [13].

We justify saying that $\Pi(\mathbb{G})_{M}$ is the “inverse image under the
projection” of $\Pi(\mathbb{G}_{(r)})_{M|\mathbb{G}_{(r)}}$ by recalling that
$\Pi(\mathbb{G})$ (respectively, $\Pi(\mathbb{G}_{r})$) for $\mathbb{G}$ of
exponential type can be identified with the projectivization of
$\varinjlim_{s}\mathcal{C}_{s}(N_{p}(\mathfrak{g}))$ (resp.,
$\mathcal{C}_{r}(N_{p}(\mathfrak{g}))$) and that there is a natural projection
$\varinjlim_{s}\mathcal{C}_{s}(N_{p}(\mathfrak{g}))\twoheadrightarrow\mathcal{C}_{r}(N_{p}(\mathfrak{g}))$.

######  Proposition 1.12.

Let $(\mathbb{G},\mathcal{E})$ be a linear algebraic group of exponential type
and let $M$ be a $\mathbb{G}$-module with the property that the coaction
$\Delta_{M}:M\to M\otimes\mathcal{O}(\mathbb{G})$ factors through
$M\otimes\mathcal{E}(\mathbb{G})_{p^{r}-1}\hookrightarrow
M\otimes\mathcal{O}(\mathbb{G})$; in other words, assume that
$M=M_{\mathcal{E}(\mathbb{G})_{p^{r}-1}}$. Then, for any $K$-point $B$ of
$\mathcal{N}_{p}(\mathfrak{g})$, $(\mathcal{E}_{B})_{*}(u_{s})$ acts trivially
on $M_{K}$ provided that $s\geq r$.

Consequently, if $M=M_{\mathcal{E}(\mathbb{G})_{p^{r}-1}}$, then the support
variety $\Pi(\mathbb{G})_{M}$ of $M$ is the “inverse image under the
projection” of $\Pi(\mathbb{G}_{(r)})_{M_{|\mathbb{G}_{(r)}}}$ (containing the
center of the “projection” $\Pi(\mathbb{G})\to\Pi(\mathbb{G}_{(r)})$, where
$M_{|\mathbb{G}_{(r)}}$ denotes the restriction of $M$ to the Frobenius kernel
$\mathbb{G}_{(r)}\hookrightarrow\mathbb{G}$.

###### Proof.

The proposition follows immediately from Proposition 1.11 and [9, Prop 3.17].
∎

##  2\. Finite dimensional subcoalgebras of $\mathcal{O}(\mathbb{G})$

We begin with a proposition which indicates some of the advantages of
specializing the discussion of $(-)_{X}:Mod(\mathbb{G})\mapsto
Mod(\mathbb{G},X)$ in Section 1 by requiring $X\subset\mathcal{O}(\mathbb{G})$
to be a subcoalgebra.

######  Proposition 2.1.

Let $\mathbb{G}$ be an affine group scheme and let
$C\subset\mathcal{O}(\mathbb{G})$ be a subcoalgebra; thus, $C$ is a subspace
of $\mathcal{O}(\mathbb{G})$ with the property that the coproduct
$\Delta:\mathcal{O}(\mathbb{G})\to\mathcal{O}(\mathbb{G})\otimes\mathcal{O}(\mathbb{G})$
restricts to $C\to C\otimes C$.

  1. (1)

The full abelian subcategory $Mod(\mathbb{G},C)$ of $Mod(\mathbb{G})$ (as in
Definition 1.1) is naturally identified with the category of comodules for
$C$,
$Mod(\mathbb{G},C)\ \simeq\ CoMod(C)$.

  2. (2)

For any $\mathbb{G}$-module $M$, the $\mathbb{G}$-submodule $M_{C}\subset M$
(as in Proposition 1.3) has underlying vector space $\\{m\in
M:\Delta_{M}(m)\in M\otimes C\\}\ \subset\ M$.

  3. (3)

If $C$ contains $1\in\mathcal{O}(\mathbb{G})$, then
$(k[\mathbb{G}]^{R})_{C}\quad=\quad C$.

###### Proof.

The identification $Mod(\mathbb{G},C)\ \simeq\ CoMod(C)$ follows immediately
from the definitions of these categories.

Denote the subspace $\\{m\in M:\Delta_{M}(m)\in M\otimes C\\}\ \subset\ M$ by
$M_{C}^{\prime}\subset M$. We choose a basis $\\{f_{s}\\}$ of
$\mathcal{O}(\mathbb{G})$ with the property that $\\{f_{s}\\}\cap C$ is a
basis for $C$. The definition of $M_{C}$ immediately implies that
$M_{C}\subset M_{C}^{\prime}$. To prove assertion (2), consider an arbitrary
element $m\in M_{C}^{\prime}$ and write $\Delta(m)=\sum m_{i}\otimes f_{i}$
with each $f_{i}\in\\{f_{s}\\}\cap C$. As argued in [18, I.2.13], it suffices
to show that each $m_{i}$ satisfies $\Delta_{M}(m_{i})\in M\otimes C$ (i.e.,
is an element of $M_{C}^{\prime}$).

Write $\Delta(m_{i})=\sum m_{i_{j}}\otimes f_{i,j}\in
M\otimes\mathcal{O}(\mathbb{G})$ with each $f_{i_{j}}$ in $\\{f_{s}\\}$ and
$f_{i_{j}}\not=f_{i_{j^{\prime}}}$ whenever $j\not=j^{\prime}$. Then

$\Delta(\Delta(m))\ =\ \sum_{i}(\sum m_{i_{j}}\otimes f_{i_{j}}\otimes f_{i})\
=\ \sum_{i}m_{i}\otimes\Delta(f_{i})\ =\ \sum_{i}\sum m_{i}\otimes
f_{i_{k}}\otimes f_{i_{k}^{\prime}}.$

Here, $\Delta(f_{i})=\sum f_{i_{k}}\otimes f_{i_{k}^{\prime}}$. Comparing
terms in $M\otimes\mathcal{O}(\mathbb{G})\otimes\langle f_{i}\rangle$ where
$\langle f_{i}\rangle$ is the 1-dimensional span of $f_{i}$, we conclude that
$\sum_{i}m_{i_{j}}\otimes f_{i,j}\ =\
\sum_{\ell,k:f_{\ell_{k}}^{\prime}=f_{i}}a_{\ell}m_{\ell}\otimes f_{\ell,k}$
for a suitable choice of elements $a_{\ell}\in k$. Since each $f_{\ell,k}$ is
in $C$ because each $f_{i}$ is in the coalgebra $C$, we conclude that each
$f_{i,j}\in C$ as required to show that $m_{i}$ is an element of
$M_{C}^{\prime}$.

To prove assertion (3), we must show $C\subset(k[\mathbb{G}]^{R})_{C}$.
Consider a basis for $\\{f_{s}\\}$ for $\mathcal{O}(\mathbb{G})$ which
includes $1\in\mathcal{O}(\mathbb{G})$. Let $I$ be the augmentation ideal of
$\mathcal{O}(\mathbb{G})$. Consider an arbitrary element $f\in I$. As in [18,
I.2.4], $\Delta(f)$ \- $(f\otimes 1+1\otimes f)\ \in I\otimes I$. Thus,
$\Delta(f)=1\otimes f+f\otimes 1+\sum c_{s,s^{\prime}}f_{s}\otimes
f_{s^{\prime}}$ with $c_{s,s^{\prime}}=0$ if either $f_{s}$ or
$f_{s^{\prime}}$ equals 1 and with each $f_{s^{\prime}}\in C$. Consequently,
if $f\in C\cap I$ and $1\in C$, then

$\Delta(f)\ =\ 1\otimes f-f\otimes 1-\sum c_{s,s^{\prime}}f_{s}\otimes
f_{s^{\prime}}\ \in\ \mathcal{O}(\mathbb{G})\otimes C,$

so that $f\in(k[\mathbb{G}]^{R})_{C}$. ∎

We utilize the constructions of Definition 2.2 below to construct the
subalgebra $\mathcal{O}(\mathbb{G})_{\langle
X\rangle}\subset\mathcal{O}(\mathbb{G})$ “generated” by a subspace
$X\subset\mathcal{O}(\mathbb{G})$.

######  Definition 2.2.

Let $\mathbb{G}$ be an affine group scheme and
$X\subset\mathcal{O}(\mathbb{G})$ be a subspace. Following [18], we denote by
$k\mathbb{G}\cdot X$ the smallest $\mathbb{G}$-submodule of
$\mathcal{O}(\mathbb{G})$ containing $X$ equipped with the right regular
regular representation (i.e., $k[\mathbb{G}]^{R}$); thus,
$\Delta:\mathcal{O}(\mathbb{G})\to\mathcal{O}(\mathbb{G})\otimes\mathcal{O}(\mathbb{G})$
restricts to $k\mathbb{G}\cdot X\ \to\ k\mathbb{G}\cdot
X\otimes\mathcal{O}(\mathbb{G})$.

Similarly, we consider the right regular representation of $\mathbb{G}^{op}$
on $\mathcal{O}(\mathbb{G})=\mathcal{O}(\mathbb{G}^{op})$, so that
$(g,f(x))\in\mathbb{G}^{op}\times\mathcal{O}(\mathbb{G})$ maps to
$f(gx)\in\mathcal{O}(\mathbb{G})$; this is given by the coproduct
$\tau\circ\Delta:\mathcal{O}(\mathbb{G})\to\mathcal{O}(\mathbb{G})\otimes\mathcal{O}(\mathbb{G})\to\mathcal{O}(\mathbb{G})\otimes\mathcal{O}(\mathbb{G})$
where $\tau$ switches tensor factors. For any subspace
$Y\subset\mathcal{O}(\mathbb{G})$, we denote by $k\mathbb{G}^{op}\cdot Y$ the
smallest $\mathbb{G}^{op}$-submodule of $\mathcal{O}(\mathbb{G})$ containing
$Y$.

The following theorem, called the “Fundamental Theorem of Coalgebras” in [28]
and the “Finiteness Theorem” in [21] (when stated for arbitrary coalgebras)
has the following appealing form when specialized to the Hopf algebra
$\mathcal{O}(\mathbb{G})$.

######  Proposition 2.3.

Let $\mathbb{G}$ be an affine group scheme and
$X\subset\mathcal{O}(\mathbb{G})$ be a subspace. Then there is a smallest
subcoalgebra of $\mathcal{O}(\mathbb{G})$ containing $X$,
$\mathcal{O}(\mathbb{G})_{\langle X\rangle}\subset\mathcal{O}(\mathbb{G})$,
given by

$\mathcal{O}(\mathbb{G})_{\langle X\rangle}\quad\equiv\quad
k\mathbb{G}^{op}\cdot(k\mathbb{G}\cdot X).$

If $X$ is finite dimensional, then $\mathcal{O}(\mathbb{G})_{\langle
X\rangle}$ is also finite dimensional.

Consequently, for any ascending, converging sequence $\\{X_{i}\\}$ of finite
dimensional subspaces of $\mathcal{O}(\mathbb{G})$, there is a smallest
ascending, converging sequence $\\{\mathcal{O}(\mathbb{G})_{\langle
X_{i}\rangle}\\}$ of finite dimensional subcoalgebras of
$\mathcal{O}(\mathbb{G})$ satisfying the condition that
$X_{i}\subset\mathcal{O}(\mathbb{G})_{\langle X_{i}\rangle}$.

###### Proof.

Observe that $(\mathcal{O}(\mathbb{G})\otimes\mathcal{O}(\mathbb{G})_{\langle
X\rangle})\cap(\mathcal{O}(\mathbb{G})_{\langle
X\rangle}\otimes\mathcal{O}(\mathbb{G}))$ equals
$\mathcal{O}(\mathbb{G})_{\langle
X\rangle}\otimes\mathcal{O}(\mathbb{G})_{\langle X\rangle}$, Thus, to show
that $\mathcal{O}(\mathbb{G})_{\langle
X\rangle}\subset\mathcal{O}(\mathbb{G})$ is a subcoalgebra, it suffices to
prove that
$\Delta:\mathcal{O}(\mathbb{G})\to\mathcal{O}(\mathbb{G})\otimes\mathcal{O}(\mathbb{G})$
restricts to

(2.3.1) $\mathcal{O}(\mathbb{G})_{\langle
X\rangle}\to\mathcal{O}(\mathbb{G})\otimes\mathcal{O}(\mathbb{G})_{\langle
X\rangle},\quad\mathcal{O}(\mathbb{G})_{\langle
X\rangle}\to\mathcal{O}(\mathbb{G})_{\langle
X\rangle}\otimes\mathcal{O}(\mathbb{G}).$

The second restriction of (2.3.1 follows from the definition of $G^{op}(-)$
acting on $k\mathbb{G}\cdot X$ via the restriction of its right regular action
of $\mathcal{O}(\mathbb{G})$. Since the right regular actions of $\mathbb{G}$
and $\mathbb{G}^{op}$ on $\mathcal{O}(\mathbb{G})$ commute,
$\mathcal{O}(\mathbb{G})_{\langle X\rangle}\ \equiv\
k\mathbb{G}\cdot(k\mathbb{G}^{op}\cdot X)$. This, together with the definition
of $k\mathbb{G}\cdot(-)$ applied to $k\mathbb{G}^{op}\cdot X$, implies the
first restriction of (2.3.1

The second assertion of the proposition follows immediately from the fact that
if $X$ and $Y$ are finite dimensional vector subspaces of
$\mathcal{O}(\mathbb{G})$, then both $k\mathbb{G}\cdot X$ and
$k\mathbb{G}^{op}\cdot Y$ are finite dimensional. (See [18, I.2.13].) ∎

In the following proposition, we relate full subcategories of
$Mod(\mathbb{G})$ to subcoalgebras $CoMod(\mathcal{O}(\mathbb{G})_{\langle
X\rangle})$ of $\mathcal{O}(\mathbb{G})$.

######  Corollary 2.4.

Consider an affine group scheme $\mathbb{G}$ and a full subcategory
$\mathcal{M}\subset Mod(\mathbb{G})$. Let
$X_{\mathcal{M}}\subset\mathcal{O}(\mathbb{G})$ be the smallest subspace
$X\subset\mathcal{O}(\mathbb{G})$ such that $M\in Mod(\mathbb{G},X)$ for all
$M\in\mathcal{M}$. Then $\mathcal{O}(\mathbb{G})_{\langle
X_{\mathcal{M}}\rangle}$ is the smallest subcoalgebra
$C\subset\mathcal{O}(\mathbb{G})$ with the property that $\mathcal{M}\
\hookrightarrow\ CoMod(C)$.

###### Proof.

Assume that $C\subset\mathcal{O}(\mathbb{G})$ is a subcoalgebra such that
$\mathcal{M}\ \hookrightarrow\ Mod(\mathbb{G},C)=CoMod(C)$. The defining
property of $X_{\mathcal{M}}$ implies that $X_{\mathcal{M}}\subset C$.
Consequently, the defining property of $X\mapsto\langle X\rangle$ given in
Proposition 2.3 tells us that $\mathcal{O}(\mathbb{G})_{\langle
X_{\mathcal{M}}\rangle}\ \subset\ C$. ∎

The tensor product of $\mathcal{O}(\mathbb{G})$-comodules involves the product
structure
$\mu:\mathcal{O}(\mathbb{G})\otimes\mathcal{O}(\mathbb{G})\to\mathcal{O}(\mathbb{G})$
induced by the diagonal
$diag:\mathbb{G}\hookrightarrow\mathbb{G}\times\mathbb{G}$. Thus, unless the
subcoalgebra $C\subset\mathcal{O}(\mathbb{G})$ is also a subalgebra, this
tensor product does not induce a tensor product structure on $CoMod(C)$.

The following suggests a useful condition on ascending, converging sequences
of subcoalgebras of $\mathcal{O}(\mathbb{G})$.

######  Proposition 2.5.

Let $\mathbb{G}$ be an affine group scheme. Let $X,\ Y\
\subset\mathcal{O}(\mathbb{G})$ be subspaces and define $X\cdot
Y\subset\mathcal{O}(\mathbb{G})$ to be the subspace spanned by products
$x\cdot y$ with $x\in X,\ y\in Y$. Consider two $\mathbb{G}$-modules $M$ and
$N$.

  1. (1)

The $\mathcal{O}(\mathbb{G})$-module $M_{X}\otimes M_{Y}$ is contained in
$(M\otimes N)_{X\cdot Y}$

  2. (2)

For subcoalgebras $C,\ C^{\prime},\ C^{\prime\prime}$ such that the
multiplication map
$\mu:\mathcal{O}(\mathbb{G})\otimes\mathcal{O}(\mathbb{G})\to\mathcal{O}(\mathbb{G})$
restricts to $C\otimes C^{\prime}\to C^{\prime\prime}$, there is a natural map
of $\mathbb{G}$-modules $M_{C}\otimes N_{C^{\prime}}\ \to\ (M\otimes
N)_{C^{\prime\prime}}.$

###### Proof.

Assertion (1) follows directly from the fact that the coaction of
$\mathcal{O}(\mathbb{G})$ on $M\otimes N$ is given by the composition of the
binary operation of multiplication $\mu$ on the algebra
$\mathcal{O}(\mathbb{G})$ and the tensor product $\Delta_{M}\otimes\Delta_{N}$
of the coactions of $\mathcal{O}(\mathbb{G})$ on $M$ and $N$:

(2.5.1) $\mu\circ(\Delta_{M}\otimes\Delta_{N}):M\otimes N\ \to\
M\otimes\mathcal{O}(\mathbb{G})\otimes N\otimes\mathcal{O}(\mathbb{G})\ \to\
M\otimes N\otimes\mathcal{O}(\mathbb{G}).$

Assertion (2) is a special case of assertion (1). ∎

Let $\mathcal{O}(\mathbb{M}_{N,N})$ be the bialgebra given as the coordinate
algebra of the affine variety of $N\times N$ matrices with monoid structure
given by matrix multiplication. For any $d\geq 0$, we consider the subspace
$\mathcal{O}(\mathbb{M}_{N,N})_{\leq d}\subset\mathcal{O}(\mathbb{M}_{N,N})$
consisting of polynomials in $\\{x_{i,j},1\leq i,j\leq N\\}$ of total degree
$\leq d$. Since $\Delta_{\mathbb{M}_{N,N}}(x_{i,j})=\sum_{s}x_{i,s}\otimes
x_{s,j}$, $\mathcal{O}(\mathbb{M}_{N,N})_{\leq
d}\subset\mathcal{O}(\mathbb{M}_{N,N})$ is a subcoalgebra.

Equip the coordinate algebra $\mathcal{O}(\mathbb{G}_{m})=k[t,t^{-1}]$ of the
linear algebraic group $\mathbb{G}_{m}$ with the filtration
$\\{\mathcal{O}(\mathbb{G}_{m})_{\leq d}\\}$, where
$\mathcal{O}(\mathbb{G}_{m})_{\leq d}\subset\mathcal{O}(\mathbb{G}_{m}$ is the
subspace spanned by $\\{t^{i},\ N\cdot|i|\leq d\\}$; this is a subcoalgebra of
$\mathcal{O}(\mathbb{G}_{m})$.

We shall utilize the following elementary lemma. Its proof is a diagram chase
involving the square of maps of (2.6.1) mapping via coproducts to the similar
square involving tensor squares.

######  Lemma 2.6.

Consider a surjective map $\phi:C\twoheadrightarrow C^{\prime}$ of coalgebras
and an injective map $j:D\hookrightarrow C$ of coalgebras, and denote by
$\overline{D}$ the subspace $\overline{j}:(\phi\circ j)(D)\subset C^{\prime}$.
Then the coproduct $\Delta_{D}:D\to D\otimes D$ induces a coproduct
$\Delta_{\overline{D}}:\overline{D}\to\overline{D}\otimes\overline{D}$ which
equals the restriction of $\Delta_{C^{\prime}}:C^{\prime}\to C^{\prime}\otimes
C^{\prime}$. In other words, $\phi$ and $j$ determine a commutative square of
coalgebras

(2.6.1)
$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hookrightarrow}$$\scriptstyle{j}$$\scriptstyle{\overline{(-)}}$$\scriptstyle{\twoheadrightarrow}$$\textstyle{\overline{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{j}}$$\scriptstyle{\hookrightarrow}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\twoheadrightarrow}$$\scriptstyle{\phi}$$\textstyle{C^{\prime}.}$

###### Proof.

For $d\in D$, write $\Delta_{D}(d)\ =\ \sum d_{i}\otimes d_{i}^{\prime}$, so
that $\Delta_{C}(j(d))=\sum j(d_{i}))\otimes j(d_{i}^{\prime})$ and

$\Delta_{C^{\prime}}(\phi(j(d)))\ =\
\sum\phi(j(d_{i}))\otimes\phi(j(d_{i}^{\prime}))\ =\
\sum\overline{j}(\overline{d_{i}})\otimes\overline{j}(\overline{d_{i}^{\prime}})\in
C^{\prime}\otimes C^{\prime}.$

Thus, $\Delta_{\overline{D}}(\overline{d})\ \equiv\
\sum\overline{d_{i}}\otimes\overline{d_{i}^{\prime}}\in\overline{D}\otimes\overline{D}$
is well defined since $\overline{j}$ is injective. ∎

We apply the above lemma in the special case of the surjective map of
coalgebras
$\eta^{*}:\mathcal{O}(\mathbb{M}_{N,N})\otimes\mathcal{O}(\mathbb{G}_{m})\
\twoheadrightarrow\ \mathcal{O}(GL_{N})$ and the subcoalgebra
$(\mathcal{O}(\mathbb{M}_{N,N})\otimes\mathcal{O}(\mathbb{G}_{m}))_{\leq d}\
\hookrightarrow\
\mathcal{O}(\mathbb{M}_{N,N})\otimes\mathcal{O}(\mathbb{G}_{m})$.

######  Definition 2.7.

Consider the closed immersion of monoid schemes

(2.7.1) $\eta:GL_{N}\ \hookrightarrow\
\mathbb{M}_{N,N}\times\mathbb{G}_{m},\quad A\mapsto(A,det(A)^{-1}),$

identifying $GL_{N}$ as the zero locus of the function
$det(\underline{x})\otimes
t^{-1}\in\mathcal{O}(\mathbb{M}_{N,N}\times\mathbb{G}_{m}),$ where

(2.7.2)
$det(\underline{x})\quad=\quad\sum_{\sigma\in\Sigma_{N}}(-1)^{sgn(\sigma)}\prod_{1\leq
i\leq N}x_{i,\sigma(i)}.$

Using Lemma 2.6, we conclude that

(2.7.3) $\mathcal{O}(GL_{N})_{\leq
d}\quad\equiv\quad\eta^{*}((\mathcal{O}(\mathbb{M}_{N,N})\otimes\mathcal{O}(\mathbb{G}_{m}))_{\leq
d})$

is a finite dimensional subcoalgebra of $\mathcal{O}(GL_{N})$.

######  Proposition 2.8.

Let $(\mathcal{O}(GL_{N})_{\leq d})^{c}$ denote the complement of
$\mathcal{O}(GL_{N})_{\leq d-1}$ in $\mathcal{O}(GL_{N})_{\leq d}$.

  1. (1)

For any non-zero element $f\in\mathcal{O}(GL_{N})$, there is uniquely
associated $d(f)\geq 0$ such that $f\in(\mathcal{O}(GL_{N})_{\leq d(f)})^{c}$.

  2. (2)

For any $f\in(\mathcal{O}(GL_{N})_{\leq d})^{c}$, let $e$ be the minimal non-
negative integer such that $f\cdot
det(\underline{x})^{e}\in\mathcal{O}(\mathbb{M}_{N,N})$. Then
$d(f)=d^{\prime}+e\cdot N$, where $d^{\prime}$ is the minimal non-negative
integer such that $f\cdot
det(\underline{x})^{e}\in\mathcal{O}(\mathbb{M}_{N,N})_{\leq d^{\prime}}$.

  3. (3)

The function sending $d$ to $dim(\mathcal{O}(GL_{N})_{\leq d})$ for a fixed
$N$ differs from the function $d\mapsto\frac{d^{N^{2}}}{(N^{2})!}$ by a
function bounded by a polynomial in $d$ of degree less that $N^{2}$.

  4. (4)

For any $d,e\geq 0$, multiplication in $\mathcal{O}(GL_{N})$ restricts to a
map

$\mu:\mathcal{O}(GL_{N})_{\leq d}\otimes\mathcal{O}(GL_{N})_{\leq
e}\quad\to\quad\mathcal{O}(GL_{N})_{\leq d+e}.$

###### Proof.

The proof of assertion (1) follows from the observation that
$\\{\mathcal{O}(\operatorname{GL}\nolimits_{N})_{\leq d}\\}$ is an ascending,
converging sequence of subspaces (in fact, of subcoalgebras). Assertion (2)
follows from the fact that $\mathcal{O}(GL_{N})$ is the localization of the
unique factorization domain $\mathcal{O}(M_{N,N})$ obtained by inverting
$det(\underline{x})$.

We identify the underlying vector space of $\mathcal{O}(\mathbb{M}_{N\times
N})_{\leq d}$ with the space of polynomials of total degree $\leq d$ in the
polynomial algebra $k[x_{i,j}]$ in $N^{2}$ variables Using induction, one
easily verifies that $dim(\mathcal{O}(\mathbb{M}_{N\times N})_{\leq
d})\quad=\quad{d+N^{2}\choose N^{2}}$, which is a polynomial in $d$ of degree
$N^{2}$ with leading term $((N^{2})!)^{-1}$. Assertion (2) implies that there
is a surjective map

(2.8.1) $\bigoplus_{i=0}^{[d/N]}\mathcal{O}(\mathbb{M}_{N,N})_{\leq
d-iN}\quad\twoheadrightarrow\quad\mathcal{O}(GL_{N})_{\leq d}$

given by sending $f$ in the summand $\mathcal{O}(\mathbb{M}_{N,N})_{\leq
d-iN}$ to $f\cdot det(\underline{x}_{i,j})^{-i}$. There is also an evident
injective map $\mathcal{O}(\mathbb{M}_{N,N})_{\leq
d}\hookrightarrow\mathcal{O}(GL_{N})_{\leq d}$, so that

(2.8.2) ${d+N^{2}\choose N^{2}}\quad\leq\quad dim(\mathcal{O}(GL_{N})_{\leq
d})\quad\leq\quad\sum_{i=0}^{[d/N]}{d-iN+N^{2}\choose N^{2}}.$

This readily assertion (3).

Observe that multiplication restricts to

$(\mathcal{O}(\mathbb{M}_{N,N})\otimes\mathcal{O}(\mathbb{G}_{m}))_{\leq
d}\otimes(\mathcal{O}(\mathbb{M}_{N,N})\otimes\mathcal{O}(\mathbb{G}_{m}))_{\leq
e}\quad\to\quad(\mathcal{O}(\mathbb{M}_{N,N})\otimes\mathcal{O}(\mathbb{G}_{m}))_{\leq
d+e}.$

Granted the definition of $\mathcal{O}(GL_{N})_{\leq d}$ in (2.7.3), this
immediately implies assertion (4). ∎

Justified by Lemma 2.6, we introduce the ascending, converging sequence of
subcoalgebras which we shall primarily use.

######  Definition 2.9.

Consider an affine group scheme $\mathbb{G}$ equipped with a closed embedding
$\phi:\mathbb{G}\hookrightarrow GL_{N}$ for some $N$. We define the ascending,
converging filtration $\\{\mathcal{O}(\mathbb{G})_{\leq d,\phi}\\}$ of finite
dimensional subcoalgebras of $\mathcal{O}(\mathbb{G})$ by setting
$\mathcal{O}(\mathbb{G})_{\leq d,\phi}$ equal to
$\phi^{*}(\mathcal{O}(GL_{N})_{\leq d})\subset\mathcal{O}(\mathbb{G})$.

For any $d,e\geq 0$, multiplication in $\mathcal{O}(\mathbb{G})$ restricts to
a map

(2.9.1) $\mu:\mathcal{O}(\mathbb{G})_{\leq
d,\phi}\otimes\mathcal{O}(\mathbb{G})_{\leq
e,\phi}\quad\to\quad\mathcal{O}(\mathbb{G})_{\leq d+e,\phi}.$

######  Example 2.10.

Consider the closed embedding $\phi:SL_{N}\hookrightarrow GL_{N}$ of matrices
of determinant 1. Then $\mathcal{O}(SL_{N})$ is the quotient of
$\mathcal{O}(GL_{N})$ by the principal ideal $(det(\underline{x})-1)$. Thus,
$det(\underline{x}),\ (det(\underline{x})^{-1}\in\mathcal{O}(GL_{N})_{\leq N}$
both map to $1\in\mathcal{O}(SL_{N})_{\leq 0,\phi}$, whereas the coordinate
functions $x_{i,j}\in\mathcal{O}(GL_{N})_{\leq 1}$ remain coordinate functions
of filtration degree 1 for $\mathcal{O}(SL_{N})$.

For various unipotent linear algebraic groups $\phi:\mathbb{U}\hookrightarrow
GL_{N}$, we give a familiar description of $\\{\mathcal{O}(\mathbb{U})_{\leq
d,\phi}\\}$.

######  Example 2.11.

[9, Ex 2.5] Let $\phi:\mathbb{U}_{N}\hookrightarrow GL_{N}$ be the unipotent
radical of the Borel subgroup of upper triangular matrices of $GL_{N}$. Then
$\phi^{*}:\mathcal{O}(GL_{N})\ \twoheadrightarrow\
\mathcal{O}(\mathbb{U}_{N})\simeq k[y_{i,j};i<j]$ is given by $x_{i,j}\mapsto
y_{i,j},\ i<j$, $x_{i,i}\mapsto 1$, and $x_{i,j}\mapsto 0,\ i>j$. The
coproduct on $\mathcal{O}(\mathbb{U}_{N})\simeq k[y_{i,j};i<j]$ is given by

$\Delta_{\mathbb{U}_{N}}(y_{i,j})\quad=\quad(y_{i,j}\otimes
1)+(\sum_{i<t<j}(y_{i}\otimes y_{t}+y_{t}\otimes y_{j}))+(1\otimes y_{i,j}).$

We identify the subcoalgebra $\mathcal{O}(\mathbb{U}_{N})_{\leq
d,\phi}\subset\mathcal{O}(\mathbb{U}_{N})$ with the subspace of
$k[y_{i,j};i<j]$ consisting of polynomials of total degree $\leq d$ and with
coproduct the restriction of $\Delta_{\mathbb{U}_{N}}$ as above. Thus,

$dim(\mathcal{O}(\mathbb{U}_{N})_{\leq
d,\phi})\quad=\quad\binom{N^{\prime}+d}{N^{\prime}},\quad
N^{\prime}=\frac{N^{2}-N}{2}$

is a polynomial of degree $N^{\prime}$ with leading coefficient
$1/(N^{\prime})!$.

The above discussion for $\mathcal{O}(\mathbb{U}_{N})_{\leq
d,\phi}\subset\mathcal{O}(\mathbb{U}_{N})$ applies (with minor modification)
to $\mathcal{O}(\mathbb{U})_{\leq d,\phi}\subset\mathcal{O}(GL_{N})$ whenever
$\mathbb{U}\hookrightarrow GL_{N}$ is the unipotent radical of a parabolic
subgroup of $GL_{N}$ (defined and split over $\mathbb{F}_{p}$).

We show that changing the embedding $\phi:\mathbb{G}\hookrightarrow GL_{N}$
has limited effect upon the associated ascending, converging sequences
$\\{\mathcal{O}(\mathbb{G})_{\leq d,\phi}\\}$.

######  Proposition 2.12.

Let $\mathbb{G}$ be an affine group scheme and consider two closed embeddings
$\phi:\mathbb{G}\hookrightarrow GL_{N},\
\phi^{\prime}:\mathbb{G}\hookrightarrow GL_{N^{\prime}}.$ There exist positive
numbers $c,c^{\prime}$ such that

$\mathcal{O}(\mathbb{G})_{\leq d,\phi}\ \subset\mathcal{O}(\mathbb{G})_{\leq
c\cdot d,\phi^{\prime}},\quad\mathcal{O}(\mathbb{G})_{\leq d,\phi^{\prime}}\
\subset\mathcal{O}(\mathbb{G})_{\leq c^{\prime}\cdot d,\phi}$

for all $d\geq 0$.

###### Proof.

We define

(2.12.1) $c\ =\ min\\{e:\phi^{*}(\mathcal{O}(\mathbb{M}_{N,_{N}})_{\leq
1})\subset\mathcal{O}(\mathbb{G})_{\leq e,\phi^{\prime}}),\
\phi^{*}(det(x_{i,j}))\in\ \mathcal{O}(\mathbb{G})_{\leq e,\phi^{\prime}}\\}.$

Applying (2.8.1), we conclude that $\mathcal{O}(\mathbb{G})_{\leq d,\phi}\
\subset\ \mathcal{O}(\mathbb{G})_{\leq cd,\phi^{\prime}}$. We similarly define
$c^{\prime}$ such that $\mathcal{O}(\mathbb{G})_{\leq d,\phi^{\prime}}\
\subset\mathcal{O}(\mathbb{G})_{\leq c^{\prime}\cdot d,\phi}.$ ∎

##  3\. Filtering $\mathbb{G}$-modules using subcoalgebras of
$\mathcal{O}(\mathbb{G})$

We explicitly formulate the filtration by $\mathbb{G}$-submodules on a
$\mathbb{G}$-module $M$ given by the ascending, converging sequences
$\\{\mathcal{O}(\mathbb{G})_{\leq d,\phi}\\}$ of subcoalgebras of
$\mathcal{O}(\mathbb{G})$ given in Definition 2.9.

######  Definition 3.1.

Consider an affine group scheme $\mathbb{G}$ equipped with a closed embedding
$\phi:\mathbb{G}\hookrightarrow GL_{N}$ for some $N$. For any
$\mathbb{G}$-module $M$, we denote by $M_{\leq d,\phi}\subset M$ the largest
$\mathcal{O}(\mathbb{G})_{\leq d,\phi}$-subcomodule of $M$. The ascending,
converging sequence $\\{M_{\leq d,\phi}\\}$ of $\mathbb{G}$-submodules of $M$
equals the filtration of $M$ given in Proposition 1.7 for $\\{X_{i}\\}$ equal
to $\\{\mathcal{O}(\mathbb{G})_{\leq d,\phi}\\}$. In particular, by
Proposition 2.1(3), $(k[\mathbb{G}]^{R})_{\leq d,\phi}$ equals
$\mathcal{O}(\mathbb{G})_{\leq d,\phi}$ as $\mathbb{G}$-modules.

If $\phi$ is the identity and $M$ is a $GL_{N}$-module, then we use $M_{\leq
d}$ rather than $M_{\leq d,id}$.

######  Remark 3.2.

We shall be considering the filtration of Definition 2.9 for this seems
somewhat accessible to computations. However, one could consider other
ascending, converging sequences of finite dimensional subcoalgebras such as
$\\{\mathcal{O}(\mathbb{G})_{\langle X_{i}\rangle}\\}$ determined by an
ascending, converging sequence of finite dimensional subspaces $\\{X_{i}\\}$
of $\mathcal{O}(\mathbb{G})$.

Recall the Schur algebra $S(N,d)$, the dual of the subcoalgebra
$\mathcal{O}(\mathbb{M}_{N,N};d)\subset\mathcal{O}(\mathbb{M}_{N,,N})$
consisting of polynomials in the matrix coefficients $x_{i,j}$ which are
homogeneous of degree $d$. A module for $S(N,d)$ (equivalently, a comodule for
$\mathcal{O}(\mathbb{M}_{N,N};d)$) is called a polynomial representation of
$GL_{N}$ homogeneous of degree $d$.

######  Example 3.3.

Let $M$ be a homogeneous polynomial representation of $GL_{N}$ of degree $d$.
Then $M_{\leq s}$ equals $0$ if $s<d$ whereas $M_{\leq s}=M$ if $s\geq d$.

More generally, let $\phi:\mathbb{G}\ \subset GL_{N}$ be a closed embedding of
a linear algebraic group $\mathbb{G}$ with the property that $A(\mathbb{G})\
\equiv\ \mathcal{O}(\mathbb{G})\cap\mathcal{O}(\mathbb{M}_{N,N})$ can be
written as a direct sum $\bigoplus_{d}A(\mathbb{G})_{d}$, where
$A(\mathbb{G})_{d}=\mathcal{O}(\mathbb{G})\cap\mathcal{O}(\mathbb{M}_{N,N})_{d}$
(for example, the classical orthogonal or symplectic groups). If $M$ is an
object of $CoMod(A(\mathbb{G})_{d})\subset Mod(\mathbb{G})$, then $M_{\leq
s,\phi}=0$ if $s<d$ and $M_{\leq s,\phi}=M$ if $s\geq d$. See [6, 1.2].

######  Example 3.4.

Give $\mathcal{O}(\mathbb{G}_{a})=k[t]$ the evident filtration by degree
(equal to that associated to the embedding of
$\phi:\mathbb{G}_{a}\hookrightarrow GL_{2}$ as the unipotent radical of a
Borel subgroup). The subcoalgebra $\mathcal{O}(\mathbb{G}_{a})_{\leq
p^{r}-1,\phi}\subset\mathcal{O}(\mathbb{G}_{a})$ is isomorphic as a coalgebra
to the coordinate algebra of $\mathbb{G}_{a(r)}$; thus, the abelian category
$CoMod(\mathcal{O}(\mathbb{G}_{a})_{\leq p^{r}-1,\phi})$ is isomorphic to
$Mod(\mathbb{G}_{a(r)})$ which in turn is isomorphic to the category of
modules for the elementary abelian $p$-group $(\mathbb{Z}/p)^{\times r}$; this
category is wild if $r>2$ or if $p>2$ and $r=2$.

To give a vector space $M$ the structure of a $\mathbb{G}_{a}$-module is
equivalent to giving a sequences of $p$-nilpotent operators $\psi_{i}:M\to M,\
i\geq 0$ which pair-wise commute and which satisfy the condition that for each
$m\in M$ there exists some $n_{m}$ such that $\psi_{i}(m)=0,i\geq n_{m}$. For
a $\mathbb{G}_{a}$-module $M$, the $\mathbb{G}$-submodule $M_{\leq
p^{r}-1,\phi}\subset M$ consists of those $m\in M$ such that
$\psi_{i}(m)=0,i\geq r$.

######  Proposition 3.5.

Let $\mathbb{G}$ be an affine group scheme equipped with the closed embedding
$\phi:\mathbb{G}\hookrightarrow GL_{N}$ and consider two $\mathbb{G}$-modules
$M,\ M^{\prime}$. Then

$M_{\leq d,\phi}\otimes M^{\prime}_{\leq
d^{\prime},\phi}\quad\subset\quad(M\otimes M^{\prime})_{\leq
d+d^{\prime},\phi}.$

###### Proof.

Since multiplication
$\mu:\mathcal{O}(GL_{N})\otimes\mathcal{O}(GL_{N})\to\mathcal{O}(GL_{N})$
restricts to
$\mathcal{O}(\mathbb{G})_{\leq d,\phi}\otimes\mathcal{O}(\mathbb{G})_{\leq
d^{\prime},\phi}\ \to\ \mathcal{O}(\mathbb{G})_{\leq d+d^{\prime},\phi}$, the
proposition is a consequence of Proposition 2.5(1). ∎

Assuming that $\phi:\mathbb{G}\hookrightarrow GL_{N}$ is a linear algebraic
group defined over $\mathbb{F}_{p^{r}}$ and that the coaction of the
$\mathbb{G}$-module $M$ is also defined over $\mathbb{F}_{p^{r}}$, we next
relate $(M^{(r)})_{\leq p^{r}\cdot d,\phi}$ and $M_{\leq d,\phi}$. The
hypothesis that $M$ is a $\mathbb{G}$-module defined over $\mathbb{F}_{p^{r}}$
implies that the Frobenius twist $M^{(r)}$ of $M$ as formulated in [14, §1] is
given by the restriction of $M$ along $F^{r}:\mathbb{G}\to\mathbb{G}$ (see
[18, I.9.10]).

######  Proposition 3.6.

Assume that $\phi:\mathbb{G}\hookrightarrow GL_{N}$ is a linear algebraic
group defined over $\mathbb{F}_{p^{r}}$ and that $M$ is a $\mathbb{G}$-module
with coaction $\Delta_{M}:M\otimes O(\mathbb{G})$ also defined over
$\mathbb{F}_{p^{r}}$. Then the $r$-th Frobenius twist $M^{(r)}$ of $M$
satisfies

(3.6.1) $M_{\leq d,\phi}\quad=\quad(M^{(r)})_{\leq p^{r}\cdot d,\phi}.$

###### Proof.

Granted our hypotheses, $M^{(r)}$ has coaction

$\Delta_{M^{(r)}}\ \simeq\ (1_{M}\otimes(F^{r})^{*})\circ\Delta_{M}:M\ \to\
M\otimes\mathcal{O}(\mathbb{G})\ \to\ M\otimes\mathcal{O}(\mathbb{G}).$

Thus, (3.6.1) follows from the fact that
$(F^{r})^{*}:\mathcal{O}(\mathbb{G})\to\mathcal{O}(\mathbb{G})$ sends
$f(x_{i,j})\in\mathcal{O}(GL_{N})$ to $f((x_{i,j})^{p^{r}})$, multiplying the
degree of each monomial by $p^{r}$. ∎

We summarize an alternative construction by J. Jantzen in [18, Chap A].
Jantzen’s filtration of $\mathcal{O}(\mathbb{G})$ for $\mathbb{G}$ a reductive
algebraic group has many useful properties, some of which are not satisfied by
the filtration of Definition 2.9. For $\mathbb{G}=\mathbb{G}_{a}$ and
$X\subset\mathcal{O}(\mathbb{G}_{a})$, Remark 1.5 points out that
$Mod(\mathbb{G}_{a},X)$ is not closed under extensions and that
$R^{1}(-)_{X}(k)\not=0$.

Choose a Borel subgroup $\mathbb{B}\subset\mathbb{G}$ and denote by $X_{+}$
denote the set of dominant weights of $\mathbb{G}$.; choose a subset
$\pi\subset X_{+}$. Jantzen considers the full subcategory
$\mathcal{C}_{\pi}(\mathbb{G})\subset Mod(\mathbb{G})$ consisting of modules
whose objects are colimits of finite dimensional $\mathbb{G}$-submodules with
composition factors of the form $L(\lambda)$ with $\lambda\in\pi$. In the
special case of $GL_{N}$ with $\pi\subset X_{+}$ the set of of dominant
weights $\pi(N,d)\subset X_{+}$ in the notation of [18, A.3.1],
$\mathcal{C}_{\pi(N,d)}$ is the category of modules for the Schur algebra
$S(N,d)$ and $\mathcal{O}_{\pi(N,d)}(GL_{N})$ is the coalgebra
$\mathcal{O}(\mathbb{M}_{N,N})_{d}$.

Jantzen considers the functor
$\mathcal{O}_{\pi}:Mod(\mathbb{G})\to\mathcal{C}_{\pi}(\mathbb{G})$ in a
manner similar to the construction of $(-)_{X}:Mod(\mathbb{G})\to
Mod(\mathbb{G},X)$ in Theorem 1.4. The condition that $\pi\subset X_{+}$ be
saturated is the condition that $\mu\in\pi$ whenever there is some
$\lambda>\mu\in X_{+}$ with $\lambda\in\pi$.

######  Theorem 3.7.

Jantzen, [18, ChapA] As above, let $\mathbb{G}$ be reductive, $\pi\subset
X_{+}$ be saturated, and $\mathcal{C}_{\pi}(\mathbb{G})\subset
Mod(\mathbb{G})$. Consider the left exactor functor
$\mathcal{O}_{\pi}(-):Mod(\mathbb{G})\to\mathcal{C}_{\pi}(\mathbb{G})$ right
adjoint to the inclusion functor.

  1. (1)

$\mathcal{O}_{\pi}(\mathbb{G})\equiv\mathcal{O}_{\pi}(\mathcal{O}(\mathbb{G}))$
is a sub-coalgebra of $\mathcal{O}(\mathbb{G})$.

  2. (2)

If $\pi$ is finite, then $\mathcal{O}_{\pi}(\mathbb{G})$ is finite dimensional
(with dimension equal to $\sum_{\lambda\in\pi}(dimH^{0}(\lambda))^{2}$).

  3. (3)

If $\\{\pi_{n}\\}$ is a nested sequence of finite, saturated subsets of
$X_{+}$ whose union is all of $X_{+}$, then
$\\{\mathcal{O}_{\pi_{n}}(\mathbb{G})\\}$ is an ascending, converging sequence
of finite sub-coalgebras of $\mathcal{O}(\mathbb{G})$.

  4. (4)

The abelian category $\mathcal{C}_{\pi}(\mathbb{G})$ can be naturally
identified with the abelian category of
$\mathcal{O}_{\pi}(\mathbb{G})$-comodules.

  5. (5)

$\mathcal{O}_{\pi(N,d)}(GL_{N})$ is the subcoalgebra
$\mathcal{O}(\mathbb{M}_{N,N})_{d}\subset\mathcal{O}(GL_{N})$.

  6. (6)

$\mathcal{C}_{\pi}(\mathbb{G})$ is closed under extensions.

  7. (7)

The higher right derived functors of $\mathcal{O}_{\pi}$,
$R^{i}\mathcal{O}_{\pi}:Mod(\mathbb{G})\to\mathcal{C}_{\pi}(\mathbb{G})$ with
$i>0$, vanish on finite dimensional $\mathbb{G}$-modules in
$\mathcal{C}_{\pi}$.

We next explore how the functors

$(-)_{\leq d,\phi}:Mod(\mathbb{G})\ \to\
CoMod(\mathcal{O}(\mathbb{G})(-)_{\leq
d,\phi},\quad(-)_{|\mathbb{G}_{(r)}}:Mod(\mathbb{G})\ \to\
Mod(\mathbb{G}_{(r)})$

complement each other.

######  Proposition 3.8.

Let $\mathbb{G}$ be a linear algebraic group with given embedding
$\phi:\mathbb{G}\to GL_{N}$, and let $d,r$ be positive integers. Then the
composition of the natural inclusion with restriction to $\mathbb{G}_{(r)}$,

$(-)_{|\mathbb{G}_{(r)}}\circ i_{\leq
d,\phi}:CoMod(\mathcal{O}(\mathbb{G})_{\leq d,\phi})\quad\hookrightarrow\quad
Mod(\mathbb{G})\quad\to\quad Mod(\mathbb{G}_{(r)}),$

is exact and left adjoint to the left exact functor given as the composition

$(-)_{\leq d,\phi}\circ
ind_{\mathbb{G}_{(r)}}^{\mathbb{G}}(-):Mod(\mathbb{G}_{(r)})\quad\to\quad
Mod(\mathbb{G})\quad\to\quad CoMod(\mathcal{O}(\mathbb{G})_{\leq d,\phi}).$

###### Proof.

The exactness of $(-)_{\mathbb{G}_{(r)}}$ and of $i_{\leq d,\phi}$ is evident.
Since $\mathbb{G}/\mathbb{G}_{(r)}\ =\ \mathbb{G}^{(r)}$ is affine, the
exactness of $ind_{\mathbb{G}_{(r)}}^{\mathbb{G}}(-)$ is given by [4]. Thus,
the left exactness of $(-)_{\leq d,\phi}\circ
ind_{\mathbb{G}_{(r)}}^{\mathbb{G}}(-)$ is given by Theorem 1.4(3).

The asserted adjunction follows from the adjunction equivalences

$Hom_{\mathcal{O}(\mathbb{G})_{\leq
d,\phi}}(M,(ind_{\mathbb{G}_{(r)}}^{\mathbb{G}}(N))_{\leq d,\phi})\ \simeq\
Hom_{\mathbb{G}}(M,ind_{\mathbb{G}_{(r)}}^{\mathbb{G}}(N))\ \simeq\
Hom_{\mathbb{G}_{(r)}}(M_{|\mathbb{G}_{(r)}},N)$

for any $M\in CoMod(\mathcal{O}(\mathbb{G})_{\leq d,\phi})$ and $N\in
Mod(\mathbb{G}_{(r)})$. ∎

The fact that $(-)_{\leq d,\phi}\circ ind_{\mathbb{G}_{(r)}}^{\mathbb{G}}(-)$
is right adjoint to a left exact functor formally implies the following
corollary.

######  Corollary 3.9.

The functor $(-)_{\leq d,\phi}\circ
ind_{\mathbb{G}_{(r)}}^{\mathbb{G}}(-):Mod(\mathbb{G}_{(r)})\ \to\
CoMod(\mathcal{O}(\mathbb{G})_{\leq d,\phi})$ sends injective/projective
$\mathbb{G}_{(r)}$-modules to injective $\mathcal{O}(\mathbb{G})_{\leq
d,\phi}$-comodules.

We supplement Proposition 3.8 with another categorical property.

######  Proposition 3.10.

Adopt the notation and hypotheses of Proposition 3.8. If the composition of
inclusion and quotient maps of coalgebras, $\mathcal{O}(\mathbb{G})_{\leq
d,\phi}\ \hookrightarrow\
\mathcal{O}(\mathbb{G})\twoheadrightarrow\mathcal{O}(\mathbb{G}_{(r)})$ is an
inclusion, then $(-)_{|\mathbb{G}_{(r)}}\circ i_{\leq
d,\phi}:CoMod(\mathcal{O}(\mathbb{G})_{\leq d,\phi})\to Mod(\mathbb{G}_{(r)})$
is a fully faithful embedding of abelian categories.

###### Proof.

Consider two $\mathcal{O}(\mathbb{G})_{\leq d,\phi}$-comodules $M,\
M^{\prime}$ and a $k$-linear map $f:M\to M^{\prime}$. This data provides the
diagram

(3.10.1)
$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\Delta_{M}}$$\textstyle{M\otimes\mathcal{O}(\mathbb{G})_{\leq
d,\phi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-)_{|\mathbb{G}_{(r)}}}$$\textstyle{M_{|\mathbb{G}_{(r)}}\otimes\mathcal{O}(\mathbb{G}_{(r)})}$$\textstyle{M^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta_{M^{\prime}}}$$\textstyle{M^{\prime}\otimes\mathcal{O}(\mathbb{G})_{\leq
d,\phi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-)_{|\mathbb{G}_{(r)}}}$$\textstyle{(M^{\prime})_{|\mathbb{G}_{(r)}}\otimes\mathcal{O}(\mathbb{G}_{(r)}).}$

Granted that the composition $\mathcal{O}(\mathbb{G})_{\leq d,\phi}\
\hookrightarrow\
\mathcal{O}(\mathbb{G})\twoheadrightarrow\mathcal{O}(\mathbb{G}_{(r)})$ is an
inclusion, one easily verifies using a simple diagram chase that $f$ is a map
of $\mathcal{O}(\mathbb{G})_{\leq d,\phi}$-comodules if and only if its
restriction to $\mathbb{G}_{(r)}$ is a map of $\mathbb{G}_{(r)}$-modules. ∎

##  4\. Cofinite $\mathbb{G}$-modules

In this section, we investigate cofinite $\mathbb{G}$-modules, a class of
(necessarily countable) $\mathbb{G}$-modules which seem somewhat amenable to
study. We restrict our attention to linear algebraic groups although the
formalism might be useful for other affine group schemes of countably infinite
dimension over $k$.

######  Definition 4.1.

Let $\mathbb{G}$ be a linear algebraic group. We define a $\mathbb{G}$-module
$M$ to be cofinite if $M_{X}$ is finite dimensional for every finite
dimensional subspace $X\subset\mathcal{O}(\mathbb{G})$. This condition is
equivalent to the condition that each $M_{X_{i}}$ is a finite dimensional
$\mathbb{G}$-module for some ascending, converging sequence $\\{X_{i}\\}$ of
finite dimensional subspaces of $\mathcal{O}(\mathbb{G})$.

We denote by $CoFin(\mathbb{G})\subset Mod(\mathbb{G})$ the full subcategory
of cofinite $\mathbb{G}$-modules.

We establish various properties of cofinite $\mathbb{G}$-modules. We observe
that $k[\mathbb{G}]^{R}$ is cofinite by Proposition 2.1(3) which asserts that
$(k[\mathbb{G}]^{R})_{C}=C$ for any subcoalgebra
$C\subset\mathcal{O}(\mathbb{G})$ containing $1\in\mathcal{O}(\mathbb{G})$.

######  Proposition 4.2.

Let $\mathbb{G}$ be a linear algebraic group, and let $M,\ E,\ N$ be
$\mathbb{G}$-modules.

  1. (1)

If $M$ is cofinite, then any $\mathbb{G}$-submodule of $M$ is also cofinite.

  2. (2)

If $0\to M\to E\to N\to 0$ is exact and if $M,\ N$ are cofinite, then $E$ is
also cofinite.

  3. (3)

If $M$ is finite dimensional, then $M$ embeds in an injective
$\mathbb{G}$-module which is also cofinite.

  4. (4)

If either $M$ or $N$ is not cofinite, then $M\otimes N$ is not cofinite.

  5. (5)

If $M$ is a finite dimensional $\mathbb{G}$-module, $M\otimes N$ is cofinite
if and only if $N$ is cofinite.

###### Proof.

We recalll that the left exactness of $(-)_{X}$ implies assertions (1) and
(2). Assertion (3) is justified by the natural embedding $M\hookrightarrow
M\otimes k[\mathbb{G}]^{R}$ together with the observation that $M\otimes
k[\mathbb{G}]^{R}$ is isomorphic to $M^{tr}\otimes k[\mathbb{G}]^{R}$.

Choose a closed embedding $\phi:\mathbb{G}\hookrightarrow\mathbb{G}$. To prove
assertion (4), assume that $M_{\leq d,\phi},\ N_{\leq e,\phi}$ are both non-
zero and at least one of them is infinite dimensional. Then $(M\otimes
N)_{\leq d+e,\phi}$ contains $M_{\leq d,\phi}\otimes N_{\leq e,\phi}$ and thus
is infinite dimensional.

To prove assertion (5), we show that

(4.2.1) $(M\otimes N)_{\leq d,\phi}\quad\subset\quad
M\otimes(N_{\leq\eta(d),\phi})$

for some function $\eta:\mathbb{N}\to\mathbb{N}$ (depending upon $\mathbb{G}$
and the finite dimensional module $M$). Choose a basis $\\{m_{i},1\leq i\leq
s\\}$ for $M$, a basis $\\{f_{\alpha}\\}$ for $\mathcal{O}(\mathbb{G})$, a
basis $\\{n_{\beta}\\}$ for $N$, and choose $e$ sufficiently large that
$M=M_{\leq e,\phi}$.

Consider a simple tensor $m_{j}\otimes n_{\beta^{\prime}}\in(M\otimes N)_{\leq
d}$ and write $\Delta(m_{j})=\sum_{i}m_{i}\otimes f_{\alpha_{i,j}}$ with
$f_{\alpha_{i,j}}\in\mathcal{O}(\mathbb{G})_{\leq e,\phi}$,
$\Delta(n_{\beta^{\prime}})=\sum_{\beta}n_{\beta}\otimes
f_{\alpha_{\beta,\beta^{\prime}}}$ where each
$f_{\alpha_{i,j}},f_{\alpha_{\beta,\beta^{\prime}}}$ is a non-zero multiple of
some basis element $f_{\alpha}$. The condition that $m_{j}\otimes
n_{\beta^{\prime}}$ is an element in $(M\otimes N)_{\leq d}$ is equivalent to
the condition that each product $f_{\alpha_{i,j}}\cdot
f_{\alpha_{\beta,\beta^{\prime}}}$ is an element of
$\mathcal{O}(\mathbb{G})_{\leq d,\phi}$ Consequently, a sum of simple tensors
$\sum c_{i,\beta}m_{i}\otimes n_{\beta}$ is an element of $(M\otimes N)_{\leq
d,\phi}$ if and only if each $m_{i}\otimes n_{\beta}$ with $c_{i,\beta}\not=0$
is an element of $(M\otimes N)_{\leq d,\phi}$.

Now, consider
$f_{\alpha_{i,j}}\cdot(-):\mathcal{O}(\mathbb{G})\to\mathcal{O}(\mathbb{G})$.
Since $\mathcal{O}(\mathbb{G})$ is an integral domain, the pre-image
$(f_{\alpha_{i,j}}\cdot(-))^{-1}(\mathcal{O}(\mathbb{G})_{\leq d,\phi})$ must
be finite dimensional and thus lie in some $\mathcal{O}(\mathbb{G})_{\leq
d_{i,j},\phi}$. Thus, (4.2.1) holds if we take $\eta(d)$ to be the maximum of
these $d_{i,j}$. ∎

We observe that a much simpler proof of Proposition 4.2(5) can be given in the
case of $GL_{N}$, establishing the more explicit form of (4.2.1):

(4.2.2) $(M\otimes N)_{\leq d}\quad\subset\quad M\otimes(N_{\leq
d+e}),\quad\mathbb{G}=GL_{N}.$

In this case, every element of $\mathcal{O}(GL_{N})$ is a polynomial in
$det(\underline{x})^{-1}$ with coefficients in
$\mathcal{O}(\mathbb{M}_{N,N})$. This readily implies that if
$f_{\beta}\notin\mathcal{O}(GL_{N})_{\leq d+e}$ then $f_{\alpha}\cdot
f_{\beta}\notin\mathcal{O}(GL_{N})_{\leq d}$ for any
$0\not=f_{\alpha}\in\mathcal{O}(GL_{N})_{\leq e}$.

We caution the reader that $M\otimes k[\mathbb{G}]^{R}$ is not cofinite
whenever $M$ is infinite dimensional since $(M\otimes
k[\mathbb{G}]^{R})_{\mathcal{O}(\mathbb{G})_{\leq 0}}=M^{tr}$.

A more pervasive “failing” of the full subcategory $CoFin(\mathbb{G})\
\hookrightarrow Mod(\mathbb{G})$ of cofinite $\mathbb{G}$-modules is that the
quotient of a cofinite $\mathbb{G}$-module need not be cofinite. Thus,
$CofFin(\mathbb{G})$ is not an abelian subcategory of $Mod(\mathbb{G})$ even
for $\mathbb{G}=\mathbb{G}_{a}$.

######  Example 4.3.

Consider the $\mathbb{G}_{a}$-submodule $Q\subset k[t]^{R}$ spanned by
$\\{t^{p^{i}},i\geq 0\\}$. Then $Q$ is cofinite (see Proposition 4.4 below)
with $soc(Q)=k$. However, the quotient $Q/soc(Q)$ is the trivial
$\mathbb{G}_{a}$-module spanned by the images of $\\{t^{p^{i}},i>0\\}$.

In contrast, if $\mathbb{G}$ is reductive and if $M$ is the direct sum
$\bigoplus_{i>0}S_{p^{i}\cdot\lambda}$ for some dominant (positive) weight
$\lambda$, then $M$ is cofinite but does not have finite dimensional socle.

######  Proposition 4.4.

For any linear algebraic group $\mathbb{G}$ and any $\mathbb{G}$-module $M$,
if $soc(M)$ is finite dimensional then $M$ is cofinite.

If $\mathbb{U}$ is a unipotent linear algebraic group, then a
$\mathbb{U}$-module $M$ is cofinite if and only if $soc(M)\ =\ M^{\mathbb{U}}$
is finite dimensional.

###### Proof.

Observe that any $\mathbb{G}$-module $M$ admits a $\mathbb{G}$-equivariant
embedding into $M^{tr}\otimes k[\mathbb{G}]^{R}$ (where $M^{tr}$ is the
underlying vector space of $M$ with the trivial $\mathbb{G}$-action). In
particular, there is a $\mathbb{G}$-equivariant embedding
$soc(M)\hookrightarrow soc(M)\otimes k[\mathbb{G}]^{R}$ which extends by the
injectivity of $soc(M)\otimes k[\mathbb{G}]^{R}$ to a map $j_{M}:M\to
soc(M)\otimes k[\mathbb{G}]^{R}$. By Proposition 4.2(5), $soc(M)\otimes
k[\mathbb{G}]^{R}$ is cofinite. Since any irreducible submodule of the kernel
of $j_{M}$ must be contained in $soc(M)$ and since $j_{M}$ restricts to an
injection on $soc(M)$, we conclude that $j_{M}$ is an embedding. Thus, $M$ is
cofinite by Proposition 4.2(1).

If $\mathbb{U}$ is unipotent and $M$ is any $\mathbb{U}$-module,
$M^{\mathbb{U}}=soc(M)=M_{\leq 0,\phi}$. Thus, if $soc(M)$ is not finite
dimensional, the $M$ is not cofinite. ∎

The invariant $\gamma(\mathbb{G},\phi)_{M}$ of a cofinite $\mathbb{G}$-module
given in (4.5.1) below is only one of many similar invariants one might
define.

######  Definition 4.5.

Let $\mathbb{G}$ be a linear algebraic group equipped with a closed embedding
$\phi:\mathbb{G}\hookrightarrow GL_{N}$. We say that a cofinite
$\mathbb{G}$-module $M$ has cofinite type $\gamma(\mathbb{G},\phi)_{M}$ equal
to $(e,c)$ if

(4.5.1) $\varinjlim_{d}\frac{dim(M_{\mathcal{O}(\mathbb{G})_{\leq
d,\phi}})}{d^{e}}\quad=\quad c\ >\ 0.$

For such a $\mathbb{G}$-module $M$, we say that $M$ has polynomial growth of
degree $e$ with leading coefficient $c$.

######  Example 4.6.

Proposition 2.8(3) tells us that the $\mathcal{O}(GL_{N})$-module
$k[GL_{N}]^{R}$ has cofinite type $(N^{2},((N^{2})!)^{-1})$.

Example 2.11 tells us that the $\mathcal{O}(\mathbb{U}_{N})$-module
$k[\mathbb{U}_{N}]^{R}$ has cofinite type
$(N^{\prime},((N^{\prime})!)^{-1})$ where $N^{\prime}=\frac{N^{2}-N}{2}$.

The cofinite type of the mock injective $\mathbb{G}_{a}$-module $J_{d}$ of
Example 5.11 equals $(1,p^{-d})$ (where $\phi:\mathbb{G}_{a}\hookrightarrow
GL_{2}$ is the closed embedding of strictly upper triangular matrices).

######  Example 4.7.

The $\mathbb{G}_{a}$-submodule $P\ \equiv\\{1,t^{p^{i}}\\}\subset
k[\mathbb{G}_{a}]^{R}$ of primitive elements satisfies
$dim(P_{\mathcal{O}(\mathbb{G}_{a})_{\leq p^{r}}})\ =\ r+1$, so that one could
say $P$ has logarithmic growth.

Let $V$ be the natural representation of $GL_{N}$ of dimension $N$, a
polynomial representation homogeneous of degree $1$ (see Example 4.8). Set $M\
\equiv\ \bigoplus_{n\geq 0}(V^{(n)})^{\oplus n!}.$ Then

$dim(M_{\leq p^{r}})\ =\ \sum_{0\leq s\leq r}N\cdot s!,$

so that $d\mapsto dim(M_{\leq d})$ grows faster than any polynomial in $d$.

######  Example 4.8.

Let $P$ be a polynomial representation of $GL_{N}$ of dimension $n$ which is
homogeneous of degree $s$ and let $M=S^{*}(P)$ be the symmetric algebra on $P$
viewed as a $GL_{N}$-module. Since the coaction of $\mathcal{O}(GL_{N})$ on
$P$ factors through $\mathcal{O}(\mathbb{M}_{N,N})$, $M$ is a graded
$\mathcal{O}(\mathbb{M}_{N,N})$-module with $M_{\leq d\cdot s}\ =\ S^{d}(P)$.
Thus, $dim(M_{\leq d})\ =\ \binom{([d/s]+n}{n}$ (where $[d/s]$ is the largest
integer $\leq d/s$), which as a function of $d$ differs from
$d\mapsto\frac{d^{n}}{s^{n}\cdot n!}$ by an error term of degree (in $d$) less
than $n-1$.

Thus, $M$ is a cofinite $GL_{N}$-module with
$\gamma(GL_{N})_{P}=(n,(s^{n}\cdot n!)^{-1})$.

As we see below, the polynomial growth of a cofinite $\mathbb{G}$-module $M$
is independent of the choice of closed embedding
$\phi:\mathbb{G}\hookrightarrow GL_{N}$.

######  Proposition 4.9.

Let $\mathbb{G}$ be a linear algebraic group and $M$ a cofinite
$\mathbb{G}$-module. Consider two closed embeddings
$\phi:\mathbb{G}\hookrightarrow GL_{N}\
\phi^{\prime}:\mathbb{G}\hookrightarrow GL_{N^{\prime}}$. If
$\gamma(\mathbb{G},\phi)_{M}\ =\ (e,c)$ and if
$\gamma(\mathbb{G},\phi^{\prime})_{M}\ =\ (e^{\prime},c^{\prime})$, then $e\
=\ e^{\prime}$.

In particular (see Definition 3.1), we conclude that the polynomial growth of
$d\mapsto\mathcal{O}(\mathbb{G})_{\leq d,\phi}$ is independent of the
embedding $\phi$.

###### Proof.

If $\phi,\psi:\mathbb{N}\to\mathbb{N}$ are sequences of polynomial growth
$e,f$ respectively, then $\phi\circ\psi$ has polynomial growth $e\cdot f$. In
particular, given an ascending, converging sequence $n\mapsto\phi(n)$ of
polynomial growth $e$, then a subsequence $n\mapsto\phi(\psi(n))$ with
$\psi(n)$ growing linearly in $n$ also has growth $e$

Thus, the proposition follows by appealing to Proposition 2.12. ∎

We next compute the degree of polynomial growth of the right reqular
representation $k[\mathbb{G}]^{R}$ and $\mathbb{G}$-submodules of the form
$k[\mathbb{G}/\mathbb{H}]^{R}\subset k[\mathbb{G}]^{R}$.

######  Proposition 4.10.

Let $\phi:\mathbb{G}\hookrightarrow GL_{N}$ be a smooth, closed embedding of
linear algebraic groups, and let $\mathfrak{g}$ denote the Lie algebra of
$\mathbb{G}$. Then the polynomial growth of $d\mapsto
dim(\mathcal{O}(\mathbb{G})_{\leq d,\phi})$ is equal $dim(\mathfrak{g})$.

Moreover, consider a closed embedding $\mathbb{H}\hookrightarrow\mathbb{G}$ of
linear algebraic groups with $\mathbb{G}/\mathbb{H}$ affine and view
$\mathcal{O}(\mathbb{G}/\mathbb{H})$ as the $\mathbb{G}$-submodule
$k[\mathbb{G}/\mathbb{H}]^{R}$ of $k[\mathbb{G}]^{R}$. Then the polynomial
growth of $d\mapsto dim(\mathcal{O}(\mathbb{G}/\mathbb{H})_{\leq d,\phi})$ is
equal to $dim(\mathfrak{g})-dim(\mathfrak{h})$.

###### Proof.

Let $n$ denote the dimension of the smooth variety $\mathbb{G}$ and let
$I\subset\mathcal{O}(\mathbb{G})$ denote the augmentation ideal of
$\mathcal{O}(\mathbb{G})$, equal to the maximal ideal at the identity of
$\mathcal{O}(\mathbb{G})$. Since $\mathcal{O}(\mathbb{G})_{I}$ is a regular
local ring of dimension $n$, the dimension of
$\mathcal{O}(\mathbb{G})/I^{d+1}$ equals $\binom{n+d}{n}$; thus,
$d\mapsto\mathcal{O}(\mathbb{G})/I^{d+1}$ has polynomial growth of degree $n$
(in the sense of Definition 4.5).

Choose $e$ such that the composition $\mathcal{O}(\mathbb{G})_{\leq
e,\phi}\hookrightarrow\mathcal{O}(\mathbb{G})\twoheadrightarrow\mathcal{O}(\mathbb{G})/I^{2}$
is surjective and choose $j$ such that $\mathcal{O}(\mathbb{G})_{\leq
1,\phi}\to\mathcal{O}(\mathbb{G})/I^{j}$ is injective. Then

(4.10.1) $dim(\mathcal{O}(\mathbb{G})_{\leq d,\phi})\quad\leq\quad
dim(\mathcal{O}(\mathbb{G})/I^{d\cdot j})\quad\leq\quad
dim(\mathcal{O}(\mathbb{G})_{\leq e\cdot d\cdot j}).$

Thus, as in the proof of Proposition 4.9, $d\mapsto
dim(\mathcal{O}(\mathbb{G})_{\leq d,\phi})$ has polynomial growth equal to
$n$.

Let $\mathcal{M}\subset\mathcal{O}(\mathbb{G}/\mathbb{H})$ denote the maximal
ideal at the coset $[\mathbb{H}]\in\mathbb{G}/\mathbb{H}$. Since
$\mathbb{G}/\mathbb{H}$ is smooth of dimension equal to
$dim(\mathfrak{g})-dim(\mathfrak{h})$, the local ring
$(\mathcal{O}(\mathbb{G}/\mathbb{H}))_{\mathcal{M}}$ is a regular local ring
of the same dimension. Consequently, $d\mapsto
dim(\mathcal{O}(\mathbb{G}/\mathbb{H})/\mathcal{M}^{d+1})$ has polynomial
growth equal to $dim(\mathfrak{g})-dim(\mathfrak{h})$. Choose $e$ such that
the composition $\mathcal{O}(\mathbb{G}/\mathbb{H})_{\leq e,\phi}\ =\
\mathcal{O}(\mathbb{G}/\mathbb{H})\cap\ k[\mathbb{G}]^{R}_{\leq e,\phi}\ \to\
\mathcal{O}(\mathbb{G}/\mathbb{H})/\mathcal{M}^{2}$ is surjective, and choose
$j$ such that $\mathcal{O}(\mathbb{G}/\mathbb{H})_{\leq 1,\phi}\ \to\
\mathcal{O}(\mathbb{G}/\mathbb{H})/\mathcal{M}^{j}$ is injective. Then

(4.10.2) $dim(\mathcal{O}(\mathbb{G}/\mathbb{H})_{\leq d,\phi})\quad\leq\quad
dim(\mathcal{O}(\mathbb{G}/\mathbb{H})/\mathcal{M}^{d\cdot j})\quad\leq\quad
dim(\mathcal{O}(\mathbb{G}/\mathbb{H})_{\leq e\cdot d\cdot j}),$

so that argument of the proof of Proposition 2.12 implies that $d\mapsto
dim(\mathcal{O}(\mathbb{G}/\mathbb{H})_{\leq d,\phi})$ has polynomial growth
equal to $dim(\mathfrak{g})$-$dim(\mathfrak{h})$. ∎

As a consequence of Proposition 4.10, we obtain the following.

######  Proposition 4.11.

Let $\phi:\mathbb{G}\hookrightarrow GL_{N}$ be a smooth, closed embedding of
affine group schemes, and let $\mathfrak{g}$ denote the Lie algebra of
$\mathbb{G}$. The $\mathbb{G}$-modules

$k[\mathbb{G}]^{R},\quad k[\mathbb{G}]^{L},\quad k[\mathbb{G}]^{Ad}$

each have polynomial growth of degree equal to $dim(\mathfrak{g})$ with
respect to $\\{\mathcal{O}(\mathbb{G})_{\leq d,\phi}\\}$.

###### Proof.

The growth of $k[\mathbb{G}]^{R}$ with respect to
$\\{\mathcal{O}(\mathbb{G})_{\leq d,\phi}\\}$ equals $dim(\mathfrak{g})$ by
Proposition 4.10. Since $\sigma_{\mathbb{G}}:k[\mathbb{G}]^{L}\to
k[\mathbb{G}]^{R}$ is an isomorphism of $\mathbb{G}$-modules, we conclude that
the growth of $k[\mathbb{G}]^{L}$ with respect to
$\\{\mathcal{O}(\mathbb{G})_{\leq d,\phi}\\}$ also equals $dim(\mathfrak{g})$.

We consider the effect of the antipode $\sigma_{GL_{N}}$ on filtrations. For
any invertible $N\times N$-matrix $A$, Cramer’s rule tell us that $A$ has
inverse $B=(b_{i,j})$, where $b_{i,j}=(-1)^{i+j}det(A_{j,i})\cdot det(A)^{-1}$
where $A_{j,i}$ is the $N{\text{-}}1\times N{\text{-}}1$ matrix obtained by
eliminating the $j$-th row and $i$-th column of $A$. Thus,
$\sigma_{GL_{N}}(x_{i,j})\in\mathcal{O}(GL_{N})$ is the product of the
$N{\text{-}}1$ degree polynomial
$(-1)^{i+j}det(\\{x_{s,t},s\not=j,t\not=i\\})$ and the function
$det(\underline{x})^{-1}$ (which is given filtration degree $N$); in other
words, $\sigma_{GL_{N}}(x_{i,j})\in\mathcal{O}(GL_{N})_{\leq 2N-1}$ but
$\sigma_{GL_{N}}(x_{i,j})\notin\mathcal{O}(GL_{N})_{\leq 2N-2}$. Since
$\sigma_{GL_{N}}:\mathcal{O}(GL_{N})\to\mathcal{O}(GL_{N})$ is an anti-algebra
morphism, we conclude that $\sigma_{GL_{N}}$ restricts to
$\sigma_{GL_{N}}:\mathcal{O}(GL_{N})_{\leq
d}\to\mathcal{O}(GL_{N})_{\leq(2N-1)d}$.

The coaction determining the comodule structure of $k[GL_{N}]^{Ad}$ is the
composition

$\Delta_{Ad}\ \equiv\ \tau\circ\mu_{1,3}\circ(\sigma_{GL_{N}}\otimes 1\otimes
1)\circ(1\circ\Delta_{GL_{N}})\circ\Delta_{GL_{N}}:$ $\mathcal{O}(GL_{N})\
\to\ \mathcal{O}(GL_{N})^{\otimes 2}\ \to\ \mathcal{O}(GL_{N})^{\otimes 3}\
\to\ \mathcal{O}(GL_{N})^{\otimes 3}\ \to\ \mathcal{O}(GL_{N})^{\otimes 2}\
\to\ \mathcal{O}(GL_{N})^{\otimes 2},$

where $\mu_{1,3}$ multiplies the first and third tensor factors. (See [18,
I.2.8(7].) Writing $\Delta_{Ad}(x_{i,j})$ as $\sum_{s,t}x_{s,t}\otimes
f_{s,t}^{i,j}$, we observe that each $f_{s,t}^{i,j}$ is a product of a
function of filtration degree $\leq 2N-1$ and a function of degree 1, thus
$f_{s,t}^{i,j}$ has filtration degree $\leq 2N$. Thus
$x_{i,j}\in(k[GL_{N}]^{Ad})_{2N}$. We conclude that

$dim(\mathcal{O}(GL_{N})_{\leq d})\quad\leq\quad
dim((k[GL_{N}]^{Ad})_{2Nd})\quad\leq\quad dim(\mathcal{O}(GL_{N})_{\leq
2Nd}),$

so that once again the argument of the proof of Proposition 2.12 implies that
the polynomial growth of $k[GL_{N}]^{Ad}$ is also $N^{2}$.

Observe that
$\sigma_{\mathbb{G}}(\phi^{*}(x_{i,j}))\in\mathcal{O}(\mathbb{G})$ has
filtration degree $\leq 2N-1$. Since the restriction
$\phi^{*}:\mathcal{O}(GL_{N})\to\mathcal{O}(\mathbb{G})$ is a surjective map
of Hopf algebras and since $\mathcal{O}(\mathbb{G})_{\leq d,\phi}$ is defined
to be $\phi^{*}(\mathcal{O}(\mathbb{G})_{\leq d})$, we may apply $\phi^{*}$ to
the above arguments for $GL_{N}$ to conclude the corresponding statements for
$\mathbb{G}$. ∎

##  5\. Mock injective $\mathbb{G}$-modules

A $\mathbb{G}$-module $M$ for a (connected) linear algebraic group is called
mock injective if the restriction $M_{|\mathbb{G}_{(r)}}$ of $M$ to each
Frobenius kernel $\mathbb{G}_{(r)}$ is an injective $\mathbb{G}_{(r)}$-module.
Every injective $\mathbb{G}$-module is mock injective.

The following list of properties of mock injective $\mathbb{G}$-modules
following easily from the exactness of
$(-)_{|\mathbb{G}_{(r)}}:Mod(\mathbb{G})\to Mod(\mathbb{G}_{(r)})$ and the
corresponding properties for support properties for
$\mathbb{G}_{(r)}$-modules. (See [9, Prop 4.6].)

######  Proposition 5.1.

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

  1. (1)

A $\mathbb{G}$-module is mock injective if and only if its support variety
$\Pi(G)_{M}$ (as defined in [11]) is empty.

  2. (2)

A directed colimit $\varinjlim_{i}M_{i}$ of mock injective
$\mathbb{G}$-modules is mock injective.

  3. (3)

Let $0\to M_{1}\to M_{2}\to M_{3}\to 0$ be an exact sequence of
$\mathbb{G}$-modules. If two of $M_{1},M_{2},M_{3}$ are mock injective, then
the third is also mock injective.

  4. (4)

If $\mathbb{H}\hookrightarrow\mathbb{G}$ is a closed embedding of linear
algebraic groups and $M$ is a mock injective $\mathbb{G}$-module, then the
restriction to $\mathbb{H}$ of $M$ is a mock injective $\mathbb{H}$-module.

None of the above properties is valid for all linear algebraic groups if “mock
injective” is replaced by “injective.”

(5) As for injective $\mathbb{G}$-modules, if $M$ is a mock injective
$\mathbb{G}$-module then $M\otimes N$ is also mock injective for all
$\mathbb{G}$-modules $N$.

We continue to find the class of mock injective $\mathbb{G}$-modules
mysterious. The dichotomy between the classes of injective
$\mathbb{G}$-modules and mock injective $\mathbb{G}$-modules is emphasized by
the following proposition.

######  Proposition 5.2.

Let $\mathbb{G}$ be a linear algebraic group with closed embedding
$\phi:\mathbb{G}\hookrightarrow GL_{N}$.

  1. (1)

$M$ is mock injective if and only if the support variety
$\Pi(\mathbb{G}_{(r)})_{M_{|\mathbb{G}_{(r)}}}$ is empty for all $r>0$.

  2. (2)

$M$ is injective if and only if the $\mathbb{G}$-module $M_{\leq d,\phi}$ is
an injective $\mathcal{O}(\mathbb{G})_{\leq d,\phi}$ comodule for every $d\geq
0$.

###### Proof.

Assertion (1) follows from the detection of injectivity property for support
varieties for infinitesimal group schemes ([27], [23]) now expressed in the
notation/terminology of [11].

Assertion (2) is given by Proposition 1.9. ∎

If a mock injective $\mathbb{G}$-module is not injective (as a
$\mathbb{G}$-module), then it is called a proper mock injective
$\mathbb{G}$-module. A $\mathbb{G}$-module $L$ is an injective if and only if
every short exact sequence $0\to L\to M\to N\to 0$ of $\mathbb{G}$-modules
splits. If $J$ is a proper mock injective $\mathbb{G}$-module $J$ with a
embedding $J\hookrightarrow I$ of $J$ into an injective $\mathbb{G}$-module
$I$, then the short exact sequence $0\to J\to I\to I/J\to 0$ does not split.

######  Remark 5.3.

Mock injective modules are necessarily infinite dimensional. They contrast
greatly to “Parshall’s Conjecture” proved by Lin and Nakano [20, Cor 3.5]
which states that for a finite dimensional $\mathbb{G}$-module $M$ that if $M$
is injective over $\mathbb{G}_{(1)}$ then it must be projective over
$\mathbb{G}(\mathbb{F}_{p})$. See Example 5.11 below and extended to the
analogue for the restriction to $\mathbb{G}_{(r)}$ versus restriction to
$\mathbb{G}_{\mathbb{F}_{p^{r}}}$ in [7, Thm 4.5, Prop 5.1].

We recall the first examples of proper mock injective $\mathbb{G}$-modules, an
interpretation of results of Cline, Parshall, and Scott concerning induced
modules. (See [4].)

######  Example 5.4.

[9, Prop 4.54] Let $\mathbb{G}$ be a linear algebraic group and
$j:\mathbb{H}\hookrightarrow\mathbb{G}$ a closed embedding of the linear
algebraic group $\mathbb{H}$. Then the restriction
$(k[\mathbb{G}]^{R})_{|\mathbb{H}}$ to $\mathbb{H}$ of the right regular
representation of $\mathbb{G}$ is a mock injective $\mathbb{H}$-module. On the
other hand, $(k[\mathbb{G}]^{R})_{|\mathbb{H}}$ is an injective
$\mathbb{H}$-module if and only if $\mathbb{G}/\mathbb{H}$ is an affine
variety.

In particular, if $\mathbb{G}$ is a reductive algebraic group and $\mathbb{H}$
is not reductive, then $(k[\mathbb{G}]^{R})_{|\mathbb{H}}$ is a proper mock
injective $\mathbb{H}$-module.

######  Remark 5.5.

The reader may gain some intuition, as we have, by comparing the fixed points
of $k[\mathbb{G}_{a}]^{R}$ under the actions of the finite subgroup schemes
$\mathbb{G}_{a}(\mathbb{F}_{p})$ and $\mathbb{G}_{a(1)}$ of $\mathbb{G}_{a}$.
We identify these actions using

$(k[\mathbb{G}_{a}(\mathbb{F}_{q})])^{*}\otimes
k[\mathbb{G}_{a}(\mathbb{F}_{q})]^{R}\ \to\
k[\mathbb{G}_{a}(\mathbb{F}_{q})]^{R},\quad\alpha\otimes t^{n}\ \mapsto\
\sum_{i\leq n}{n\choose i}\alpha^{n-i}t^{i}$
$(k[\mathbb{G}_{a(r)}])^{*}\otimes k[\mathbb{G}_{a}(\mathbb{F}_{q})]^{R}\ \to\
k[\mathbb{G}_{a}(\mathbb{F}_{q})]^{R},\quad\hat{t^{i}}\otimes t^{n}\ \mapsto\
\sum_{i\leq n}{n\choose i}t^{n-i};$

here, $\alpha$ denotes an element of $\mathbb{G}_{a}(\mathbb{F}_{q})$, a
generator of $(k[\mathbb{G}_{a}(\mathbb{F}_{q})])^{*}$. The fixed point space
$k[t]^{\mathbb{G}(\mathbb{F}_{p})}$ consists of those $f(t)$ which are
polynomials in $t^{p}$-$t$ (see Proposition 5.12); this is a proper mock
injective module by Theorem 5.9. On the other hand, the fixed point space
$k[t]^{\mathbb{G}_{a(1)}}$ consists of those $f(t)$ which are polynomials in
$t^{p}$; the restriction to $\mathbb{G}_{a(1)}$ of this fixed point space has
trivial $\mathbb{G}_{a(1)}$-action and thus is not an injective
$\mathbb{G}_{a(1)}$-module.

See Example 5.11 for an investigation of the role of
$\mathbb{G}_{a}(\mathbb{F}_{q})$ fixed points in constructing mock injective
$\mathbb{G}_{a}$-modules.

We provide a somewhat surprising property of mock injective
$\mathbb{G}$-modules.

######  Theorem 5.6.

Consider a linear algebraic group $\mathbb{G}$, a finite dimensional subspace
$X\subset\mathcal{O}(\mathbb{G})$, and a mock injective $\mathbb{G}$-module
$J$. Then

$Hom_{\mathbb{G}}(J,M_{X})\quad=\quad 0$

for every $\mathbb{G}$-module $M$ provided that either

(i.) $\mathbb{G}$ is unipotent, or

(ii.) $\mathbb{G}$ is semi-simple and simply connected and $p\geq 2h-2$, where
$h$ is the Coxeter number of $\mathbb{G}$.

###### Proof.

First, consider a unipotent linear algebraic group $\mathbb{U}$ with closed
embedding $\phi:\mathbb{U}\hookrightarrow GL_{N}$ and a mock injective
$\mathbb{U}$-module $J$. Assume the existence of a map $\rho:J\to M$ sending
$j\in J$ to $m\not=0\in M$. For all $r>0$, $J_{|\mathbb{U}_{(r)}}$ is a free
$\mathbb{U}_{(r)}$-module so that $j$ lies in some free, rank 1
$\mathbb{U}_{(r)}$-summand $(J_{|\mathbb{U}_{(r)}})_{j}$ of
$J_{|\mathbb{U}_{(r)}}$. Namely, we can take the cyclic $\mathbb{U}_{(r)}$
submodule $k\mathbb{U}_{(r)}\cdot j\subset J_{|\mathbb{U}_{(r)}}$, and extend
this inclusion to the injective hull of $k\mathbb{U}_{(r)}\cdot j$ which is a
free, rank 1 $\mathbb{U}_{(r)}$-summand of $J_{|\mathbb{U}_{(r)}}$. For
notational simplicity, let $J_{j}$ denote $(J_{|\mathbb{U}_{(r)}})_{j}$.

Assume that $M=M_{\leq d,\phi}$ and choose $r$ sufficiently large that the
composition $\mathcal{O}(\mathbb{U})_{\leq
d,\phi}\subset\mathcal{O}(\mathbb{U})\twoheadrightarrow\mathcal{O}(\mathbb{U}_{(r-1})$
is injective. Choose an identification $\mathcal{O}(\mathbb{U}_{r)})\ \simeq
J_{j}$ of $\mathbb{U}_{(r)}$-modules and choose $\psi\in
k\mathbb{U}_{(r)}=(\mathcal{O}(\mathbb{U}_{r)}))^{*}$ such that $\psi$
vanishes on the image $X$ of $\mathcal{O}(\mathbb{U})_{\leq
d,\phi}\subset\mathcal{O}(\mathbb{U})\twoheadrightarrow\mathcal{O}(\mathbb{U}_{(r)})$
but does not vanish on a generator $g_{r}$ of $J_{j}$. Observe that
$\Delta_{J_{j}}(g_{r})\notin J_{j}\otimes X$. This implies that
$\rho(g_{r})\notin M_{\leq d,\phi}$ and thus must equal 0. Since $g_{r}$ is a
generator of $J_{j}$, there exists some $\theta\in k\mathbb{U}_{(r)}$ sending
$g_{r}$ to $j$, so that $\theta(\rho(g_{r}))=\rho(\theta(g_{r}))=\rho(j)=m$.
This contradicts our assumption that $0\not=m\in M$.

Assume now that $\mathbb{G}$ is semi-simple and simple connected and that
$M=M_{X}$ for some finite dimensional subspace
$X\subset\mathcal{O}(\mathbb{G})$. Once again we proceed by contradiction,
assuming the exists a map $\rho:J\to M$ sending $j\in J$ to $m\not=0\in M$.
Let $\\{S_{\lambda}\\}$ be the set of isomorphism classes of irreducible
$\mathbb{G}$-modules, each of finite dimension $d(\lambda)$, and assume that
the injective hull of $S_{\lambda}$ restricted to $\mathbb{G}_{(r)}$ for $r$
sufficiently large admits a $\mathbb{G}$-structure. This implies that the
injective hull of $S_{\lambda}$ as a $\mathbb{G}$-module is the colimit the
injective hulls as $\mathbb{G}_{(r)}$-modules (see [18, II.11.17]). Then
$k[\mathbb{G}]$ is isomorphic as a $\mathbb{G}$-module to a direct sum of
$I_{\lambda}$’s, with $I_{\lambda}$ occurring finitely often depending on
$d_{\lambda}$. Each $I_{\lambda}$ is infinite dimensional (see [18, II.11.4]).

Since $J$ embeds into $J^{tr}\otimes k[\mathbb{G}]^{R}$ as a
$\mathbb{G}$-module and since this embedding splits when restricted to each
$\mathbb{G}_{(r)}$, we conclude that $J_{|\mathbb{G}_{(r)}}$ for each $r>0$ is
a direct sum of cyclic modules, the images of generators of copies of
$\mathcal{O}(\mathbb{G}_{r)})$ constituting $(J^{tr}\otimes
k[\mathbb{G}]^{R})_{\mathbb{G}_{(r)}}$. Moreover, as $r$ increases, the
dimension of these cyclic summands of $J_{|\mathbb{G}_{(r)}}$ also increases.
Thus, if the restriction to $J_{|\mathbb{G}_{(r)}}$ of $g_{r}\in J$ is the
image of a generator of a copy of $\mathcal{O}(\mathbb{G}_{r)})$ in
$\mathcal{O}(\mathbb{G})_{|\mathbb{G}_{(r)}}$, then $\Delta_{J}(g_{r})\in
J\otimes\mathcal{O}(\mathbb{G})$ restricts to a sum of terms in
$J_{\mathbb{G}_{(r)}}\otimes\mathcal{O}(\mathbb{G}_{(r)})$ such that the
number of terms of this sum grows as $r$ increases.

We can thus repeat the argument given for $\mathbb{U}$, now replacing “some
free $\mathbb{U}_{(r)}$-summand $(J_{|\mathbb{U}_{(r)}})_{j}$ $\ \ldots\
\ldots g_{r}\in(J_{|\mathbb{U}_{(r)}})_{j}$” by a cyclic summand of
$(J_{|\mathbb{U}_{(r)}})_{j}$ of $J_{|\mathbb{G}_{(r)}}$ containing the chosen
element $j$ assumed to map to $m\not=0\in M$ and also containing some $g_{r}$
which is the image of a generator of a copy of $\mathcal{O}(\mathbb{G}_{(r)})$
in $\mathcal{O}(\mathbb{G})_{|\mathbb{G}_{(r)}}$. Namely, these conditions
enable us to chose $\psi\in
k\mathbb{G}_{(r)}=(\mathcal{O}(\mathbb{G}_{r)}))^{*}$ such that $\psi$
vanishes on the image $X$ of $\mathcal{O}(\mathbb{G})_{\leq
d,\phi}\subset\mathcal{O}(\mathbb{G})\twoheadrightarrow\mathcal{O}(\mathbb{G}_{(r)})$
but does not vanish on $g_{r}$ of $J_{j}\subset J$. Once again, we obtain a
contradiction because $j\in J$ when restricted to $J_{j}$ is the image under
some $\theta\in k\mathbb{G}_{(r)}$ of $g_{r}$. ∎

We summarize some useful properties of the functor $ind^{\mathbb{G}}_{H}(-)$
following [4]

######  Proposition 5.7.

Let $\mathbb{G}$ be a linear algebraic group and $H\subset\mathbb{G}$ a closed
subgroup scheme with the property that $\mathbb{G}/H$ is affine.

  1. (1)

The induction functor $ind^{\mathbb{G}}_{H}(-):Mod(H)\ \to\ Mod(\mathbb{G})$
is an exact functor between abelian categories

  2. (2)

If $M$ is an injective $H$-module, then $ind_{H}^{\mathbb{G}}(M)$ is an
injective $\mathbb{G}$-module.

  3. (3)

There is a natural identification
$H^{i}(\mathbb{G},ind_{H}^{\mathbb{G}}(M))\simeq H^{i}(H,M)$ for any
$H$-module $M$, any $i\geq 0$.

  4. (4)

If $M$ is an $H$-module such that $H^{n}(H,k)\not=0$ for some $n>0$, then
$Ind_{H}^{\mathbb{G}}(M)$ is not an injective $\mathbb{G}$-module.

  5. (5)

If $M$ is a finite dimensional $H$-module, then $ind_{H}^{\mathbb{G}}(M)$ is a
cofinite $\mathbb{G}$-module.

###### Proof.

The exactness of $ind^{\mathbb{G}}_{H}(-)$ is proved in [4]. Assertion (2)
follows from the fact that $ind^{\mathbb{G}}_{H}(-)$ has an exact left adjoint
(namely, the restriction functor) (see the proof of Corollary 1.6). The
equivalence $H^{0}(\mathbb{G},ind_{H}^{\mathbb{G}}(M))\simeq H^{0}(H,M)$ is
given by the universal property for
$Hom_{\mathbb{G}}(-,ind_{H}^{\mathbb{G}}(M))$ applied to the
$\mathbb{G}$-module $k$. Assertion (3) follows by applying the Grothendieck
spectral sequence for the composite functor $H^{0}(\mathbb{G},-)\circ
ind_{H}^{\mathbb{G}}(-)$ together with this equivalence and the exactness of
$ind_{H}^{\mathbb{G}}(-)$. Assertion (3) immediately implies assertion (4).

Finally, if $M$ is finite dimensional, then we may realize
$ind_{H}^{\mathbb{G}}(M)$ as the fixed point space under the right action of
$H$ on $M^{tr}\otimes k[\mathbb{G}]_{L}$ and thus a $\mathbb{G}$-submodule of
$M^{tr}\otimes k[\mathbb{G}]_{L}$. Hence, assertion (5) follows from the fact
that $M^{tr}\otimes k[\mathbb{G}]_{L}$ is a cofinite $\mathbb{G}$-module. ∎

We supplement Proposition 5.7 with the following proposition concerning the
composition $ind_{H}^{\mathbb{G}}(-)\circ(-)_{H}$.

######  Proposition 5.8.

Let $\mathbb{G}$ be a linear algebraic group and $H\subset\mathbb{G}$ a closed
subgroup scheme with the property that $\mathbb{G}/H$ is affine. Then the
exact functor

$ind_{H}^{\mathbb{G}}(-)\circ(-)_{H}:Mod(\mathbb{G})\quad\to\quad
Mod(\mathbb{G})$

is faithful.

Moreover, if $M,\ N$ are $\mathbb{G}$-modules and if
$\psi:ind_{H}^{\mathbb{G}}(M_{|H})\to ind_{H}^{\mathbb{G}}(N_{|H})$ is an
isomorphism of $\mathbb{G}$-modules, then $\psi$ induces an isomorphism of
$H$-modules $M_{|H}\stackrel{{\scriptstyle\sim}}{{\to}}N_{|H}$.

###### Proof.

To prove that $ind^{\mathbb{G}}_{H}(-)\circ(-)_{H}$ is faithful, we observe
for $\mathbb{G}$-modules $N$ and $M$ that if $f:N_{|H}\to M_{|H}$ is a map of
$H$-modules, then $ind^{\mathbb{G}}_{H}(f)$: $ind^{\mathbb{G}}_{H}(N_{|H})\to
ind^{\mathbb{G}}_{H}(M_{|H})$ has the property that pre-composition with the
canonical $\mathbb{G}$-map $i_{N}:N\to ind^{\mathbb{G}}_{H}(N_{|H})$ and post-
composition with the canonical $H$-map $ev_{H}:ind^{\mathbb{G}}_{H}(M_{|H})\\\
\twoheadrightarrow M_{|H}$ equals $f:N_{|H}\to M_{|H}$.

Let $N,\ M$ be $\mathbb{G}$-modules and let
$\psi:ind^{\mathbb{G}}_{H}(N_{|H})\to ind^{\mathbb{G}}_{H}(M_{|H})$ be an
isomorphism of $\mathbb{G}$-modules. Observe that the identity map $N\to
N_{|H}$ determines $i_{N}:N\to ind^{\mathbb{G}}_{H}(N_{|H})$ which is inverse
to the canonical map $ev_{N}:ind^{\mathbb{G}}_{H}(N_{|H})\to N_{|H}$ of
$H$-modules. The naturality of this splitting implies that the map
$ev_{M}\circ\psi\circ i_{N}:N\to M_{H}$ of $H$-modules when composed with
$ev_{N}\circ\psi^{-1}\circ i_{M}:M\to N_{H}$ equals the identity of $N_{H}$.
This implies that $ev_{M}\circ\psi\circ i_{N}$ is an isomorphism of
$H$-modules. ∎

In [16, §2], Hardesty, Nakano, and Sobaje provide a method for constructing
many proper mock injective modules. We elaborate upon their construction in
the following theorem.

######  Theorem 5.9.

(cf. [16, Prop 2.1.1, Prop 2.2.2] Let $\mathbb{G}$ be a linear algebraic group
defined over $\mathbb{F}_{p}$, let $q=p^{r}$ be some power of $p$, and assume
that $k$ contains $\mathbb{F}_{q}$. Let $H$ be a subgroup of
$\mathbb{G}(\mathbb{F}_{q})$ stable under the action of $F^{r}$, the $r$-th
power of the Frobenius map.

  1. (1)

Then $ind^{\mathbb{G}}_{H}(-):Mod(H)\quad\to\quad Mod(\mathbb{G})$ takes
values in mock injective $\mathbb{G}$-modules.

  2. (2)

If every irreducible $H$ module is the restriction of a $\mathbb{G}$-module,
then $ind^{\mathbb{G}}_{H}(M)$ is a proper mock injective $\mathbb{G}$-module
if and only if the $M$ is not an injective $H$-module.

###### Proof.

The proof that $ind^{\mathbb{G}}_{H}(-)$ takes values in mock injective
$\mathbb{G}$-modules is essentially given in [16]; we give details of the
formulation stated above. We first observe that the restriction
$(k[\mathbb{G}]^{R})_{H}$ of the $\mathbb{G}$-module $k[\mathbb{G}]^{R}$ to
$H$ is an injective $H$-module since $\mathbb{G}/H$ is affine (see [4]).
Moreover, if $I$ is an injective $\mathbb{G}$-module, then the embedding
$Soc(I)\hookrightarrow Soc(I)^{tr}\otimes k[\mathbb{G}]^{R}$ extends to an
embedding $I\hookrightarrow Soc(I)^{tr}\otimes k[\mathbb{G}]^{R}$ which splits
as a map of $\mathbb{G}$-modules, so that $I_{|H}$ must be injective as an
$H$-module. If $F^{r}:\mathbb{G}\to\mathbb{G}$ restricts to an isomorphism on
$H$, then the restriction $L_{|H}$ to $H$ of the $\mathbb{G}$-module
$L\equiv(F^{r})^{*}(k[\mathbb{G}]^{R})$ is an injective $H$-module; this
implies that $M\otimes L_{|H}$ is an injective $H$-module for any $H$-module
$M$.

The tensor identity (see [18, I.3.6]) tells us that for any $H$-module $M$
there is a natural isomorphism

(5.9.1) $ind_{H}^{\mathbb{G}}(M)\otimes L\quad\simeq\quad
ind_{H}^{\mathbb{G}}(M\otimes L_{|H})$

of $\mathbb{G}$-modules. Since $L$ is trivial as a $\mathbb{G}_{(r)}$-module,
(5.9.1) implies that $(ind_{H}^{\mathbb{G}}(M))_{|\mathbb{G}_{(r)}}$ is a
direct summand of $(ind_{H}^{\mathbb{G}}(M\otimes
L_{|H}))_{|\mathbb{G}_{(r)}}$. Since $L_{|H}$ is injective as an $H$-module,
$M\otimes L_{|H}$ is injective as an $H$-module, so that Proposition 5.7(2)
tells us that $ind_{H}^{\mathbb{G}}(M\otimes L_{|H})$ is injective as a
$\mathbb{G}$-module. As seen above, this implies that
$(ind_{H}^{\mathbb{G}}(M\otimes L_{|H}))_{|\mathbb{G}_{(r)}}$ is injective as
a $\mathbb{G}_{(r)}$-module. Thus, the direct summand
$(ind_{H}^{\mathbb{G}}(M))_{|\mathbb{G}_{(r)}}$ is also injective as a
$\mathbb{G}_{(r)}$-module. Since this applies to any $r>0$, we conclude that
$ind_{H}^{\mathbb{G}}(M)$ is a mock injective $\mathbb{G}$-module.

Assertion (2) follows from a simple argument using $Ext$-groups as detailed in
[16]. ∎

The following observation is of interest in view of the challenge of studying
quotients of infinite dimensional $\mathbb{G}$-modules as seen in Theorem 5.6.

######  Corollary 5.10.

As in Theorem 5.9, consider a linear algebraic group $\mathbb{G}$ defined over
$\mathbb{F}_{p}$, assume $\mathbb{F}_{q}\subset k$ with $q=p^{r}$, and
consider a subgroup $H\subset\mathbb{G}(\mathbb{F}_{q})$ stable under the
action of $F^{r}$. If $0\to N\to M\to Q\to 0$ is an exact sequence in
$Mod(H)$, then

(5.10.1) $0\to\ ind^{\mathbb{G}}_{H}(N)\quad\to\quad
ind^{\mathbb{G}}_{H}(M)\quad\to\quad ind^{\mathbb{G}}_{H}(Q)\to 0.$

is a short exact sequence of mock injective $\mathbb{G}$-modules.

We make explicit the preceding discussion in the special case in which
$\mathbb{G}$ equals the additive group $\mathbb{G}_{a}$. Notice that the third
assertion shows how cofinite type can distinguish isomorphism classes of mock
injective modules, whereas the last assertion emphasizes that proper mock
injective $\mathbb{G}$-modules are unlikely to have finite injective
dimension.

######  Example 5.11.

Let $J_{d}$ denote the proper mock injective $\mathbb{G}_{a}$-module
$ind^{\mathbb{G}}_{\mathbb{G}_{a}(\mathbb{F}_{p^{d}})}(k)$, equipped with the
natural $\mathbb{G}_{a}$-equivariant embedding into $k[t]^{L}$ (which we
denote by $\mathcal{O}(\mathbb{G}_{a})\ =\ k[t]$ since $\mathbb{G}_{a}$ is
abelian).

  1. (1)

With respect to this embedding, $J_{d}$ can be identified as the subalgebra of
$k[t]$ consisting of polynomials in $t^{p^{d}}-t$. This is verified in greater
generality in Proposition 5.12 below.

  2. (2)

$\mathcal{O}(\mathbb{G}_{a})/J_{d}$ admits an embedding into
$\mathcal{O}(\mathbb{G}_{a})^{\oplus d}$. Namely, the images of
$\\{t,t^{p},\ldots,t^{p^{d-1}}\\}$ in $\mathcal{O}(\mathbb{G}_{a})/J_{d}$ form
a basis for the socle of $\mathcal{O}(\mathbb{G}_{a})/J_{d}$.

  3. (3)

The cofinite type $\gamma(\mathbb{G}_{a},\phi)_{J_{d}}$ of the
$\mathbb{G}_{a}$-module $J_{d}$ equals $(1,q^{-1})$. This readily follows from
(1) above. On the other hand, the cofinite type of
$\mathcal{O}(\mathbb{G})/J_{d}$ equals $(q-1,q^{-1})$.

  4. (4)

In the special case $d=1$, $J_{1}\ =\
ind^{\mathbb{G}}_{\mathbb{G}_{a}(\mathbb{F}_{p})}(k)$ has an injective
resolution

$J_{1}\ \hookrightarrow\ I^{0}\stackrel{{\scriptstyle
d^{1}}}{{\to}}I^{1}\stackrel{{\scriptstyle d^{2}}}{{\to}}I^{2}\to\cdots\to
I^{n-1}\stackrel{{\scriptstyle d^{n}}}{{\to}}I^{n}\cdots$

with the property that each $I^{n}$ is isomorphic to
$\mathcal{O}(\mathbb{G}_{a})$ (i.e., the injective hull of $k$) and that
$d^{n}=d^{n+2}$.

The identification of $ind_{H}^{\mathbb{G}}(k)$ used in Example 5.11 has the
following elaboration. The key to the proof of this proposition is the Lang
isomorphism $\mathbb{G}/\mathbb{G}(\mathbb{F}_{q})\
\stackrel{{\scriptstyle\sim}}{{\to}}\ \mathbb{G}$, an isomorphism of varieties
over $k$ [19].

######  Proposition 5.12.

Let $q=p^{r}$ and assume that $\mathbb{F}^{q}\subset k$. Consider a linear
algebraic group $\mathbb{G}$ defined over $\mathbb{F}_{p}$ provided with an
embedding $\phi:\mathbb{G}\hookrightarrow GL_{N}$ defined over
$\mathbb{F}_{p}$ and let $H$ denote the finite group
$\mathbb{G}(\mathbb{F}_{q})$. Recall the Lang map

(5.12.1) $F^{r}/id:\mathbb{G}\ \twoheadrightarrow\ \mathbb{G}/H\ \simeq\
\mathbb{G},\quad g\mapsto F^{r}(g)\cdot g^{-1},$

a finite ètale map of $k$-varieties (see [19]).

The map on coordinate algebras

$(F^{r}/id)^{*}=\mu\circ(\sigma_{\mathbb{G}}\otimes(F^{r})^{*})\circ\Delta:\mathcal{O}(\mathbb{G})\
\to\mathcal{O}(\mathbb{G})\otimes\mathcal{O}(\mathbb{G})\to\mathcal{O}(\mathbb{G}),\quad
f\ \mapsto\sigma_{\mathbb{G}}(f)\cdot f^{r}$

has image the $\mathbb{G}$-submodule
$ind_{H}^{\mathbb{G}}(k)=(k[\mathbb{G}]^{L})^{H}\subset
k[\mathbb{G}]^{L}\simeq k[\mathbb{G}]^{R}$. This implies that

(5.12.2)
$(k[\mathbb{G}]^{R})_{\leq\frac{d}{2N-1+q},\phi}\quad\subset\quad(ind_{H}^{G}(k))_{\leq
d,\phi}\quad\subset\quad(k[\mathbb{G}]^{L})_{\leq d,\phi}.$

###### Proof.

As seen in the proof of Proposition 4.11,
$\sigma_{GL_{N}}(\mathcal{O}(GL_{N})_{\leq
d})\subset\mathcal{O}(GL_{N})_{\leq(2N-1)\cdot d}$, so that
$\sigma_{\mathbb{G}}(\mathcal{O}(\mathbb{G})_{\leq d,\phi})\ \subset\
\mathcal{O}(\mathbb{G})_{\leq(2N-1)\cdot d,\phi}$. Moreover, if
$f\in\mathcal{O}(GL_{N})_{\leq d}$, then
$(F^{r})^{*}(f)\in\mathcal{O}(GL_{N})_{\leq qd}.$ Consequently,
$(F^{r}/id)^{*}(\mathcal{O}(\mathbb{G})_{\leq d,\phi})\ \subset\
\mathcal{O}(\mathbb{G})_{\leq(2N-1+q)d,\phi}.$ ∎

In the next example, we determine the cofinite type of the mock injective
$\mathbb{U}_{N}$-modules
$ind_{\mathbb{U}_{N}(\mathbb{F}_{q})}^{\mathbb{U}_{N}}(k)\subset
k[\mathbb{U}_{N}]^{L}$. See Example 2.11 for the description of
$\mathcal{O}(\mathbb{U}_{N})_{\leq d,\phi}$. For notational convenience, we
omit explicit mention of the natural inclusion
$\phi:\mathbb{U}_{N}\hookrightarrow GL_{N}$.

######  Example 5.13.

Consider the antipode
$\sigma_{\mathbb{U}_{N}}:\mathcal{O}(\mathbb{U}_{N})=k[x_{i,j},i<j]\to
k[y_{s,t},s<t]=\mathcal{O}(\mathbb{U}_{N})$. Observe that
$\sigma_{\mathbb{U}_{N}}(\mathcal{O}(\mathbb{U}_{N})_{\leq
d})\subset\mathcal{O}(\mathbb{U}_{N})_{\leq(N-1)d}$, that
$\sigma_{\mathbb{U}_{N}}(x_{1,N})\notin\mathcal{O}(\mathbb{U}_{N})_{\leq
N-2}$, and that
$\sigma_{\mathbb{U}_{N}}(x_{i,j})\in\mathcal{O}(\mathbb{U}_{N})_{\leq N-2}$
and $\sigma_{\mathbb{U}_{N}}(x_{i,j})\in\mathcal{O}(\mathbb{U}_{N})_{\leq
N-23}$for $(i,j)\not=(1,N)$.

To determine the cofinite type of
$ind_{\mathbb{U}_{N}(\mathbb{F}_{q})}^{\mathbb{U}_{N}}(k)$ (which we denote by
$M$), we consider the image of

$(F^{r}/id)^{*}:k[\mathbb{U}_{N}]_{\leq d}\quad\to\quad M_{\leq(p^{r}+N-1)d}\
\subset\ k[\mathbb{U}_{N}]_{\leq(p^{r}+N-1)d},$

and proceed to show that $d\mapsto dim(M_{\leq(p^{r}+N-1)d})$ has the same
polynomial degree and the same leading coefficient as $d\mapsto
dim((F^{r}/id)^{*}(k[\mathbb{U}_{N}]_{\leq d}))=dim(k[\mathbb{U}_{N}]_{\leq
d})$ which was computed explicitly in Example 4.6. Namely, the above
observations about the filtration degree of $\sigma_{\mathbb{U}_{N}}(x_{i,j})$
imply the short exact sequence

$0\to(F^{r}/id)^{*}(k[\mathbb{U}_{N}]_{\leq d})\quad\to\quad
M_{\leq(p^{r}+N-1)d}\quad\to\quad
x_{1,N}\bullet(F^{r}/id)^{*}(k[\mathbb{U}_{N}]_{\leq d-1})\to 0.$

Consequently,

$\gamma(\mathbb{U}_{N})_{M}\ =\
(N^{\prime},((p^{r}+N-1)^{N^{\prime}}N!)^{-1})\quad
N^{\prime}=\frac{N^{2}-N}{2}.$

######  Example 5.14.

One can apply the argument of Example 5.13 to obtain a similar computation for
$GL_{N}$ replacing the structure of $\mathcal{O}(\mathbb{U}_{N})$ as a
polynomial algebra by the “sum expansion” of $\mathcal{O}(GL_{N})$ given in
(2.8.2). We then conclude for $M=ind_{GL_{N}(\mathbb{F}_{q})}^{GL_{N}}(k)$
that

$\gamma(GL_{N})_{M}\ =\ (N^{2},((p^{r}+N-1)^{N^{2}}(N^{2})!)^{-1}).$

##  6\. Stable Module Categories

In Theorem 6.3, we provide a stable module-theoretic version of Theorem 5.9 by
dividing out injective modules, thereby focusing upon proper mock injective
modules. We introduce the tensor triangulated category $StMock(\mathbb{G})$,
complementary to stable categories considered in [11].

We remind the reader that the stable module category $StMod(G)$ of a finite
group scheme $G$ over $k$ is the category whose objects are $G$-modules (i.e.,
$kG$-modules) and whose space of maps $Hom_{StMod(G)}(M,N)$ for any pair $M,\
N$ of $G$-modules is the quotient of $Hom_{G}(M,N)$ by the subspace of maps
$M\to N$ of $G$-modules which factor through an injective $G$-module. Using
the fact that $kG$ is self-injective, one provides $StMod(G)$ with the
structure of a tensor triangulated category. As shown in [11, Thm 4.2], there
is a natural tensor triangulated equivalence

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

natural with respect to finite group schemes $G$. Here,
$\mathcal{I}nj^{b}(Mod(G))$ denotes the homotopy category of bounded cochain
complexes of $G$-modules which are quasi-isomorphic to a bounded complex of
injective $G$-modules.

With this in mind, we make the following definition, changing the use of the
notation $StMod(\mathbb{G})$, notation which was used in [11] for the tensor
triangulated category which we denote below by $\overline{StMod}(\mathbb{G})$.

######  Definition 6.1.

Let $\mathbb{G}$ be a linear algebraic group. Denote by
$\mathcal{K}^{b}(Mod(\mathbb{G}))$ the tensor triangulated category given as
the homotopy category of bounded cochain complexes of $\mathbb{G}$-modules and
denote by
$\mathcal{I}nj^{b}(\mathbb{G})\hookrightarrow\mathcal{K}^{b}(Mod(\mathbb{G}))$
the thick subcategory of bounded cochain complexes $C^{\bullet}$ quasi-
isomorphic to a bounded complex of injective $\mathbb{G}$-modules. We define
$StMod(\mathbb{G})$ to be Verdier quotient

$\mathcal{K}^{b}(Mod(\mathbb{G}))\quad\twoheadrightarrow\quad\mathcal{K}^{b}(Mod(\mathbb{G}))/\mathcal{I}nj^{b}(Mod(\mathbb{G}))\quad\equiv\quad
StMod(\mathbb{G}).$

Denote by
$\mathcal{M}ock^{b}(\mathbb{G})\hookrightarrow\mathcal{K}^{b}(Mod(\mathbb{G}))$
the thick subcategory given as the homotopy category of bounded cochain
complexes $C^{\bullet}$ whose restriction to $Mod(\mathbb{G}_{(r)})$ is quasi-
isomorphic to a bounded complex of injective $\mathbb{G}_{(r)}$-modules for
each $r>0$. We define $StMock(\mathbb{G})$ to be Verdier quotient

$\mathcal{M}ock^{b}(\mathbb{G})\quad\twoheadrightarrow\quad\mathcal{M}ock^{b}(\mathbb{G})/\mathcal{I}nj^{b}(Mod(\mathbb{G}))\quad\equiv\quad
StMock(\mathbb{G}).$

Finally, we define $\overline{StMod}(\mathbb{G})$ to be the Verdier quotient

(6.1.1) $StMock(\mathbb{G})\quad\twoheadrightarrow\quad
StMock(\mathbb{G})/\mathcal{M}ock^{b}(\mathbb{G})\quad\equiv\quad\overline{StMod}(\mathbb{G}).$

The following proposition relates classes of $\mathbb{G}$-modules to their
associated classes in the appropriate Verdier quotient formulated using
bounded cochain complexes of $\mathbb{G}$-modules.

######  Proposition 6.2.

Let $\mathbb{G}$ be a linear algebraic group and $M$ a $\mathbb{G}$-module
determining the chain complex $M[0]\in\mathcal{K}^{b}(Mod(\mathbb{G}))$. Then
the class of $M[0]$ in $\overline{StMod}(\mathbb{G})$ is $0$ if and only if
$M$ is a mock injective.

Moreover, if $M$ is a mock injective $\mathbb{G}$-module, the class in
$StMock(\mathbb{G})$ of $M[0]\in\mathcal{M}ock^{b}(\mathbb{G})$ is $0$ if and
only if $M$ is an injective $\mathbb{G}$-module.

###### Proof.

If $M$ is a mock injective $\mathbb{G}$-module, then essentially by definition
the class of $M[0]$ is 0. Conversely, if the class of $M[0]$ in
$\overline{StMod}(\mathbb{G})$ is $0$, then $M[0]\in StMock(\mathbb{G})$. In
this case, the restriction to each $\mathbb{G}_{(r)}$ of $M[0]$ is $0\in
StMock(\mathbb{G}_{(r)})$. By [11, Cor 4.5], this implies each
$M_{|\mathbb{G}_{(r)}}$ is an injective $\mathbb{G}_{(r)}$ module so that $M$
is a mock injective $\mathbb{G}$-module.

The second assertion follows from the fact that the kernel of
$\mathcal{M}ock^{b}(\mathbb{G})\to StMod(\mathbb{G})$ equals
$\mathcal{K}^{b}(Inj(\mathbb{G}))$, since $\mathcal{K}^{b}(Inj(\mathbb{G}))\
\subset\ \mathcal{M}ock^{b}(\mathbb{G})$ is a thick subcategory. ∎

The following theorem summarizes the relationships between these various
stable categories, providing a form of “categorization” of Theorem 5.9

######  Theorem 6.3.

Let $\mathbb{G}$ be a linear algebraic group defined over $\mathbb{F}_{p}$,
let $q=p^{r}$ be some power of $p$, and assume that $k$ contains
$\mathbb{F}_{q}$. Let $H$ be a subgroup of $\mathbb{G}(\mathbb{F}_{q})$ stable
under the action of $F^{r}$, the $r$-th power of the Frobenius map. Then there
is a commutative diagram of tensor triangulated categories and tensor
triangulated functors

(6.3.1)
$\textstyle{\mathcal{I}nj^{b}(Mod(H))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hookrightarrow}$$\scriptstyle{ind_{H}^{\mathbb{G}}(-)}$$\textstyle{\mathcal{I}nj^{b}(Mod(\mathbb{G}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hookrightarrow}$$\scriptstyle{=}$$\textstyle{\mathcal{I}nj^{b}(Mod(\mathbb{G}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hookrightarrow}$$\textstyle{D^{b}(Mod(H))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\twoheadrightarrow}$$\scriptstyle{ind_{H}^{\mathbb{G}}(-)}$$\textstyle{D^{b}(Mock(\mathbb{G}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\twoheadrightarrow}$$\scriptstyle{\hookrightarrow}$$\textstyle{D^{b}(Mod(\mathbb{G}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\twoheadrightarrow}$$\textstyle{StMod(H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ind_{H}^{\mathbb{G}}(-)}$$\textstyle{StMock(\mathbb{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\scriptstyle{\hookrightarrow}$$\textstyle{StMod(\mathbb{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\twoheadrightarrow}$.$\textstyle{\overline{StMod}(\mathbb{G})}$

where the derived categories are Verdier quotients of corresponding homotopy
categories of cochain complexes “localized” at subcategories of cochain
complexes which are acylic.

The homotopy categories of injective modules in the upper row are the kernels
of the Verdier quotients of the functors from categories in the second row to
categories in the third row of (6.3.1). Moreover, the bottom right functor is
the Verdier quotient with kernel the right horizontal map of the third row of
(6.3.1).

###### Proof.

The upper left horizontal functor is well defined because
$ind_{H}^{\mathbb{G}}(-)$ is exact and sends injective $H$-modules to
injective $\mathbb{G}$-modules. Exactness of $ind_{H}^{\mathbb{G}}(-)$
justifies the middle left horizontal functor. The naturality of the
construction of Verdier quotients justifies the lower left horizontal functor
as well as the commutativity of the left squares of (6.3.1) follows from the
definitions. The commutativity of the right squares of (6.3.1) follows from
Definition 6.1.

The assertion about kernels of Verdier quotients follows from the thickness of
the asserted kernels. ∎

##  7\. Criteria for injectivity of $\mathbb{G}_{a}$-modules

One goal for the study of mock injective $\mathbb{G}$-modules is to refine the
theory of support varieties $M\mapsto\Pi(\mathbb{G})_{M}$ as considered in
[11] so that the support of $M$ is empty if and only if $M$ is an injective
$\mathbb{G}$-module. In the very special case of $\mathbb{G}=\mathbb{G}_{a}$,
Theorem 7.4 provides such a refinement which suggests a possible approach for
more general $\mathbb{G}$.

######  Proposition 7.1.

For any $r>0$, there is a commutative square of commutative coalgebras

(7.1.1) $\textstyle{\mathcal{O}(\mathbb{G}_{a})_{\leq
p^{r+1}-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(\mathbb{G}_{a(r+1)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(k\mathbb{Z}/p^{\times
r+1})^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(\mathbb{G}_{a})_{\leq
p^{r}-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(\mathbb{G}_{a(r)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(k\mathbb{Z}/p^{\times
r})^{*}}$

whose horizontal arrow are isomorphisms, whose left vertical arrow is the
natural inclusion, whose middle vertical arrow is induced by the embedding
$\mathbb{G}_{(r)}\hookrightarrow\mathbb{G}_{(r+1)}$, and whose right vertical
arrow is the dual of the map group algebras induced by the embedding
$\mathbb{Z}/p^{\times r}\to\mathbb{Z}/p^{\times r+1}$ sending
$(a_{1},\ldots,a_{r})$ to $(a_{1},\ldots,a_{r},0)$. The left horizontal arrows
are induced by the natural maps

$\mathcal{O}(\mathbb{G}_{a})_{\leq
p^{r}-1}\to\mathcal{O}(\mathbb{G}_{a})_{\leq
p^{r+1}-1}\to\mathcal{O}(\mathbb{G}_{a})\to\mathcal{O}(\mathbb{G}_{a(r+1)})\to\mathcal{O}(\mathbb{G}_{a(r)})$

and the right horizontal maps are duals of familiar isomorphisms.

  1. (1)

For each $r>0$, there is a coalgebra splitting

(7.1.2) $s_{r}:\mathcal{O}(\mathbb{G}_{a})_{\leq
p^{r+1}-1}\quad\twoheadrightarrow\quad\mathcal{O}(\mathbb{G}_{a})_{\leq
p^{r}-1}$

of the natural inclusion $\mathcal{O}(\mathbb{G}_{a})_{\leq
p^{r}-1}\hookrightarrow\mathcal{O}(\mathbb{G}_{a})_{\leq p^{r+1}-1}$ fitting
in the commutative square

(7.1.3) $\textstyle{\mathcal{O}(\mathbb{G}_{a})_{\leq
p^{r+1}-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{r}}$$\textstyle{\mathcal{O}(\mathbb{G}_{a(r+1)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(\mathbb{G}_{a})_{\leq
p^{r}-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(\mathbb{G}_{a(r)})}$

  2. (2)

For a $\mathbb{G}_{a}$-module $M$ and each $r>0$, let $M_{r}\subset M$ denote
the subset consisting of elements $m\in M$ such that the action of
$\varinjlim_{s>r}k\mathbb{Z}/p^{\times s}$ on $m$ factors through the
projection $\varinjlim_{s>r}k\mathbb{Z}/p^{\times s}\to k\mathbb{Z}/p^{\times
r}$. Then the $\mathcal{O}(\mathbb{G}_{a})_{\leq p^{r}-1}$-subcomodule
$M_{\mathcal{O}(\mathbb{G}_{a})_{\leq p^{r}-1}}\subset M$ is the pull-back of
$M_{r}\subset M$ along the isomorphism $\mathcal{O}(\mathbb{G}_{a})_{\leq
p^{r}-1}\stackrel{{\scriptstyle\sim}}{{\to}}(k\mathbb{Z}/p^{\times r})^{*}$ of
(7.1.1).

###### Proof.

The left commutative square of (7.1.1) together with the fact that the its
horizontal map is an isomorphism determines the coalgebra splitting $s_{r}$ of
(7.1.2) which fits in the commutative diagram (7.1.3).

The distribution algebra $k\mathbb{G}_{a}$ is the polynomial algebra on
countably infinitely many $p$-nilpotent generators
$k[u_{0},u_{1},\ldots,u_{n}\ldots]/(\\{u_{i}^{p}\\})$. The action of $u_{r}$
on an $\mathcal{O}(\mathbb{G}_{a})=k[t]$-comodule $M$ with coproduct
$\Delta_{M}:M\to M\otimes k[t]$ sends $m\in M$ to
$(id_{M}\otimes\gamma_{p^{r}})(\Delta_{M})(m))$, where $\gamma_{p^{r}}:k]t]\to
k$ reads off the coefficient of $t^{p^{r}}$ of each $f(t)\in k[t]$. Thus,
$M_{\leq p^{r}-1}\subset M$ consists of those elements $m\in M$ such that
$u_{n}$ acts trivially for $n\geq r$. Indexing the factors of
$\mathbb{Z}/p^{\times r}$ as $\prod_{s=0}^{r}\mathbb{Z}/p$, we conclude that
the pull-back of $M_{r}\subset M$ along the isomorphism of coalgebras
$\mathcal{O}(\mathbb{G})_{\leq
p^{r}-1}\stackrel{{\scriptstyle\sim}}{{\to}}(k\mathbb{Z}/p^{\times r})^{*}$
equals $M_{\leq p^{r}-1}$. ∎

Proposition 7.1(2) together with the commutativity of (7.1.1) easily implies
the following corollary which gives a concrete description of
$\mathbb{G}_{a}$-modules in terms of a sequence of modules for elementary
abelian $p$-groups.

######  Corollary 7.2.

The isomorphisms of (7.1.1) determine a 1-1 correspondence between the data of
a filtered $\mathbb{G}_{a}$-module $M=\varinjlim_{r}M_{\leq p^{r}-1}$ and the
data of a sequence $M_{1}\subset\cdots\subset M_{r}\subset
M_{r+1}\subset\cdots$ of $k$ vector spaces with each $M_{r}$ equipped with an
action of $k\mathbb{Z}/p^{\times r}$ such that $M_{r}\subset M_{r+1}$ consists
of those $m\in M_{r+1}$ on which the action of $k\mathbb{Z}/p^{\times r+1}$
factors through the projection onto the first $r$ factors,
$k\mathbb{Z}/p^{\times r+1}\twoheadrightarrow k\mathbb{Z}/p^{\times r}$
inducing the action of $k\mathbb{Z}/p^{\times r}$ on $M_{r}$.

Combining Proposition 7.1 and Corollary 7.2, we obtain the following
proposition.

######  Proposition 7.3.

Retain the notation of Proposition 7.1. Then the the map
$\Pi(\mathbb{Z}/p^{\times r})\to\Pi(\mathbb{Z}/p^{\times r+1})$ induced by the
embedding of finite groups $\mathbb{Z}/p^{\times
r}\hookrightarrow\mathbb{Z}/p^{\times r+1}$ which sends $(a_{1},\ldots,a_{r})$
to $(a_{1},\ldots,a_{r},0)$ restricts to

(7.3.1) $\Pi(\mathbb{Z}/p^{\times r})_{M_{r}}\ =\ \Pi(\mathbb{Z}/p^{\times
r+1})_{M_{r+1}}\cap\Pi(\mathbb{Z}/p^{\times r})\ \hookrightarrow\
\Pi(\mathbb{Z}/p^{\times r+1})_{M_{r+1}}.$

Moreover, there is a commutative square

(7.3.2)
$\textstyle{\Pi(\mathbb{G}_{(r)})_{M_{<p^{r}-1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\Pi(\mathbb{Z}/p^{\times
r})_{M_{r}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Pi(\mathbb{G}_{(r+1)})_{M_{<p^{r+1}-1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\Pi(\mathbb{Z}/p^{\times
r+1})_{M_{r+1}}}$

whose horizontal bijections are given by Proposition 7.1(2) and whose left
vertical arrow is the composition of the map
$\Pi(\mathbb{G}_{(r)})_{M_{<p^{r}-1}}\ \to\
\Pi(\mathbb{G}_{(r+1)})_{M_{<p^{r}-1}}$ associated to the splitting $s_{r}$ of
(7.1.2) followed by the map $\Pi(\mathbb{G}_{(r+1)})_{M_{<p^{r}-1}}\ \to\
\Pi(\mathbb{G}_{(r+1)})_{M_{<p^{r+1}-1}}$ enabled by (7.3.1).

###### Proof.

Let $\alpha:K[t]/t^{p}\to K\mathbb{Z}/p^{\times r}$ be a flat map representing
a $\pi$-point in $\Pi(\mathbb{Z}/p^{\times r}).$ Than $\alpha$ represents a
$\pi$ point of $\Pi(\mathbb{Z}/p^{\times r})_{M_{r}}$ if and only if
$\alpha^{*}((M_{r})_{K})$ is not a free $K[t]/t^{p}$-module. The composition
of $\alpha$ with the embedding $i_{r}:K\mathbb{Z}/p^{\times r}\hookrightarrow
K\mathbb{Z}/p^{\times r+1}$ has the property that
$(i_{r}\circ\alpha)^{*}(M_{r+1})_{K})=\alpha^{*}((M_{r})_{K})$. Thus,
$i_{r}\circ\alpha$ represents a $\pi$-point in $\Pi(\mathbb{Z}/p^{\times
r})_{M_{r+1}}$ if and only if $\alpha$ represents a $\pi$ point of
$\Pi(\mathbb{Z}/p^{\times r})_{M_{r}}$

The verification of the commutativity of the square (7.3.2) follows from the
definitions of the maps involved. ∎

The following criterion for injectivity for a $\mathbb{G}_{a}$-module might
possibly permit a generalization linear algebraic groups other than
$\mathbb{G}=\mathbb{G}_{a}$.

######  Theorem 7.4.

Retain the notation of Proposition 7.1. Then the following are equivalent.

  1. (1)

$M$ is an injective $\mathbb{G}_{a}$-module $M$.

  2. (2)

$\varinjlim_{r}\Pi(\mathbb{G}_{(r)})_{M_{\mathcal{O}(\mathbb{G}_{a})_{\leq
p^{r}-1}}}$ is empty.

  3. (3)

$\varinjlim_{r}\Pi(\mathbb{Z}/p^{\times r})_{M_{r}}$ is empty.

###### Proof.

By Proposition 1.9 and Proposition 2.1, $M$ is injective if and only if
$M_{\mathcal{O}(\mathbb{G}_{a})_{\leq p^{r}-1}}$ is an injective object of
$CoMod(\mathcal{O}(\mathbb{G}_{a})_{\leq p^{r}-1})$ for all $r>0$. The
injectivity assertion of Proposition 7.3 together with Proposition (7.1)
implies the equivalence of conditions (1) and (3). The equivalence of (2) and
(3) now follows from (7.3.2). ∎

##  8\. Questions

We mention a few of the many questions which arise when considering mock
injective $\mathbb{G}$-modules.

######  Question 8.1.

Is there a useful “geometric invariant” for mock $\mathbb{G}$-modules which
complements support varieties by detecting injectivity of
$\mathbb{G}$-modules? (See Theorem 7.4 above.)

######  Question 8.2.

For $\mathbb{U}$ a unipotent linear algebraic group, can we utilize ascending,
converging sequences of subcoalgebras of $\mathcal{O}(\mathbb{U})$ to enable
computations of the Hochschild cohomology of $\mathbb{U}$? (See [10].)

######  Question 8.3.

If $M$ is a cofinite mock injective $\mathbb{G}$-modules, does $M$ admit a
filtration by mock injective $\mathbb{G}$-modules which are obtained by
inducing $H$-modules to $\mathbb{G}$ for finite subgroups
$H\hookrightarrow\mathbb{G}$?

######  Question 8.4.

Under what hypotheses does a cofinite mock injective $\mathbb{G}$-module $J$
admit an embedding into an injective $\mathbb{G}$-module $I$ with quotient
$I/J$ which is also cofinite?

######  Question 8.5.

Let $\mathbb{G}$ be semi-simple, simply connected algebraic group defined over
$\mathbb{F}_{p}$. If $J$ is a mock injective $\mathbb{G}$-module whose
restriction to each $\mathbb{G}(\mathbb{F}_{q})$ is injective, then is $J$ an
injective $\mathbb{G}$-module?

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