# Filtrations and Growth of $\mathbb{G}$-modules

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

We investigate rational representations ($\mathbb{G}$-modules) of affine group
schemes $\mathbb{G}$ over a field of positive characteristic $p$. For any
subspace $X\subset\mathcal{O}(\mathbb{G})$, we consider the abelian
subcategory $Mod(\mathbb{G},X)\subset Mod(\mathbb{G})$ of “$X$-comodules” and
the left exact functor $(-)_{X}:Mod(\mathbb{G})\to Mod(\mathbb{G},X)$ which is
right adjoint to the inclusion functor. We employ “ascending converging
sequences” $\\{X_{i}\\}$ of subspaces of $\mathcal{O}(\mathbb{G})$ to provide
functorial filtrations $\\{M_{X_{i}}\\}$ of each $\mathbb{G}$-module $M$. For
any subspace $X\subset\mathcal{O}(\mathbb{G})$, $Mod(\mathbb{G},X)\subset
Mod(\mathbb{G})$ is naturally identified with the category
$CoMod(\mathcal{O}(\mathbb{G})_{X})$ of comodules for the coalgebra
$\mathcal{O}(\mathbb{G})_{X}\subset X$.

This construction gives a corrected version of the filtration of exponential
degree for a linear algebraic group of exponential type. We provide two types
of ascending, converging sequences of finite dimensional sub-coalgebras for an
arbitrary affine group scheme, one is a generalization to all affine group
schemes of Jantzen’s truncated subcategories for reductive algebraic groups
and the other is a generalization to all affine group schemes of the grading
of polynomial functions for classical groups.

Any ascending converging sequence $\\{X_{i}\\}$ of subspaces of
$\mathcal{O}(\mathbb{G})$ provides a test for injectivity for
$\mathbb{G}$-modules. When each $X_{i}$ is a sub-coalgebra of
$\mathcal{O}(\mathbb{G})$, $\\{X_{i}\\}$ determine a filtration of Hochschild
cohomology. The subcategories $CoHt(\mathbb{G})\subset Mod(\mathbb{G}),\
CoFin(\mathbb{G})\subset Mod(\mathbb{G})$ of $\mathbb{G}$-modules of finite
coheight and of cofinite $\mathbb{G}$-modules do not depend upon the choice of
$\\{X_{i}\\}$ provided that each $X_{i}$ finite dimensional.

###### Key words and phrases:

filtrations, coalgebras, representations

###### 2010 Mathematics Subject Classification:

20G05, 20C20, 20G10

∗ partially supported by the Simons Foundation

##  0\. Introduction

One 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 $\mathbb{G}_{(r)},r>0$
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
especially when combined with the technique of reducing representations in
characteristic 0 to characteristic $p$ (see, for example, [21]). In recent
years, the consideration of support varieties for elementary abelian
$p$-groups (see [3]) has been vastly generalized to modules for finite group
schemes (see, for example, [17]), linear algebraic groups (see, for example,
[10]), and various finite dimensional algebras (see, for example, [1], [16],
[24]).

However, some aspects of the representation theory of $\mathbb{G}$ are not
seen by restrictions to $\mathbb{G}_{(r)}$. Of particular importance is the
existence of mock injective modules which are not injective as
$\mathbb{G}$-modules but whose restrictions to every Frobenius kernel
$\mathbb{G}_{(r)}$ are injective. Using the lens of “stable categories”, we
showed in [14] that localizing with respect to the (triangulated) category of
mock injectives enables an analogue for linear algebraic groups of stable
module categories for finite group schemes.

In [12], we considered linear algebraic groups of exponential type and
utilized the filtration of a $\mathbb{G}$-module $M$ determined by the
filtration of $M$ by exponential degree. This suggested that studying
$\mathbb{G}$-modules for more general affine group schemes using restrictions
of their coactions to ($k$-linear) subspaces of $\mathcal{O}(\mathbb{G})$
should constructively complement the more familiar technique of studying a
$\mathbb{G}$-module $M$ by considering its restriction to
$\\{\mathbb{G}_{a(r)}\\}$. A classic example of this approach is the
consideration of the subspace
$\mathcal{O}(\mathbb{M}_{N,N})_{d}\hookrightarrow\mathcal{O}(GL_{N})$
consisting of polynomials of degree $d$ in the coordinate functions
$x_{i,j}\in\mathcal{O}(GL_{N})$; indeed,
$\mathcal{O}(\mathbb{M}_{N,N})_{d}\subset\mathcal{O}(GL_{N})$ is a sub-
coalgebra whose dual is the classical Schur algebra (see [19]).

In Theorem 1.5, 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
$(-)_{X}:Mod(\mathbb{G})\ \to\ Mod(\mathbb{G},X)$ which 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 $M\otimes X\subset
M\otimes\mathcal{O}(\mathbb{G})$. In Theorem 1.13, we show for any ascending
converging sequence $\\{X_{n}\\}$ of subspaces of $\mathcal{O}(\mathbb{G})$,
$X_{0}\subset X_{1}\subset\cdots X_{n}\subset\cdots\subset\
\bigcup_{d}X_{n}=\mathcal{O}(\mathbb{G}),$ and any $\mathbb{G}$-module $M$
that $M$ is injective if and only if each $M_{X_{n}}$ is injective in
$Mod(\mathbb{G},X_{n})$.

We are particularly interested in choices of such ascending converging
sequences $\\{X_{i}\\}$ of subspaces of $\mathcal{O}(\mathbb{G})$ with the
property that each $X_{i}$ is finite dimensional. However, in Section 2 we
utilize the ascending, converging sequence of (infinite dimensional) subspaces
$\\{\mathcal{E}(d)\\}$ of $\mathcal{O}(\mathbb{G})$ in order to correct the
formulation given in [12] of the filtration by exponential degree for
$\mathbb{G}$-module a linear algebraic group of exponential type. The result
of this correction is that it adds the condition that the filtration be of
$\mathbb{G}$-modules.

We recall a construction by Jantzen in Theorem 3.2 which associates
“generalized Schur algebras” to “truncated subcategories” of $Mod(\mathbb{G})$
for a reductive linear algebraic group $\mathbb{G}$. For any affine group
scheme $\mathbb{G}$ and any ascending converging subsequence $\\{X_{i}\\}$ of
subspaces of $\mathcal{O}(\mathbb{G})$, Theorem 3.4 extends Jantzen’s
construction by constructing an ascending converging sequence
$\\{\mathcal{O}(\mathbb{G})_{X_{i}}\\}$ of sub-coalgebras of
$\mathcal{O}(\mathbb{G})$. The sub-coalgebra $\mathcal{O}(\mathbb{G})_{X}$ is
finite dimensional whenever $X$ is finite dimensional. Moreover, the abelian
category of comodules for $\mathcal{O}(\mathbb{G})_{X}$ is naturally
identified with the full abelian subcategory $Mod(\mathbb{G},X)\subset
Mod(\mathbb{G})$ of Theorem 1.5.

In Theorem 3.8, we provide another construction of ascending converging finite
dimensional sub-coalgebras $\\{\mathcal{O}(\mathbb{G})_{\leq d}\\}$ of
$\mathcal{O}(\mathbb{G})$ by first explicitly defining
$\\{\mathcal{O}(GL_{N})_{\leq d}\\}$ and then using a closed embedding
$\mathbb{G}\hookrightarrow GL_{N}$. Proposition 3.13 compares such sequences
of sub-coalgebras of $\mathcal{O}(\mathbb{G})$ determined by different choices
of embeddings of $\mathbb{G}$ in general linear groups.

Filtrations of $\mathcal{O}(\mathbb{G})$ by subcoalgebras determine
filtrations of the “Hochschild cochain complex”
$\mathcal{C}^{H*}(\mathbb{G},M)$ whose cohomology is the (Hochschild
cohomology) $H^{*}(\mathbb{G},M)$ for a finite dimensional $\mathbb{G}$-module
$M$ as stated Theorem 4.2. An application is given in Corollary 4.3.

Section 5 introduces the full abelian subcategory $CoHt(\mathbb{G})\subset
Mod(\mathbb{G})$ consisting of modules of finite coheight (“generalizing”
modules of finite exponential degree) and the additive subcategory
$CoFin(\mathbb{G})\subset Mod(\mathbb{G})$ consisting of cofinite modules. A
$\mathbb{G}$-module $M$ has finite coheight if and only if it is an
“$X_{i}$-comodule” for $i\gg 0$ for some (equivalently, for all) ascending
converging sequence of finite dimensional subspaces $\\{X_{i}\\}$ of
$\mathcal{O}(\mathbb{G})$, and a $\mathbb{G}$-module $M$ is cofinite if and
only if each $M_{X_{i}}$ is finite dimensional for some (equivalently, for
all) ascending converging sequences of finite dimensional subspaces
$\\{X_{i}\\}$ of $O(\mathbb{G})$. Proposition 5.3 gives some pleasing
properties of $CoHt(\mathbb{G})$ whereas Proposition 5.6 gives a somewhat
different list of properties for $CoFin(\mathbb{G})$.

In Section 6, we introduce the growth of a cofinite $\mathbb{G}$-module with
respect to an ascending converging sequence of finite dimensional sub-
coalgebras $\\{C_{d}\\}$ of $\mathcal{O}(\mathbb{G})$. We give examples of
cofinite $\mathbb{G}$-modules with polynomial, logarithmic, and exponential
growth with respect to the converging, ascending sequence
$\\{\mathcal{O}(\mathbb{G})_{\leq d}\\}$ of sub-coalgebras. We conclude with a
list of questions in Section 7.

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

We thank Bob Guralnick, Julia Pevtsova, Paul Sobaje, and especially Cris
Negron for various 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 $\Delta_{M}:M\ \to\ M\otimes\mathcal{O}(\mathbb{G})$ that is a
$k$-linear map determining natural (with respect to commutative $k$-algebras
$A$) $A$-linear group actions $\mathbb{G}(A)\times(A\otimes M)\to(A\otimes
M)$.

######  Definition 1.1.

Let $i_{X}:X\subset\mathcal{O}(\mathbb{G})$ be a subspace. 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”.

For future reference, we recall the following consequence of the fact that
$(id_{M}\otimes\epsilon)\circ\Delta_{M}=id_{M}$.

######  Lemma 1.2.

Write $\mathcal{O}(G)$ as $k\oplus I(\mathbb{G})$, where
$k\subset\mathcal{O}(\mathbb{G})$ is the span of $1\in\mathcal{O}(\mathbb{G})$
and where $I(\mathbb{G})$ is the kernel of the counit
$\epsilon:\mathcal{O}(\mathbb{G})\to k$. Then for any $\mathbb{G}$-module $M$,
the coaction $\Delta_{M}\to M\otimes\mathcal{O}(\mathbb{G})$ has the form
$m\mapsto m\otimes 1+\sum m_{i}\otimes f_{i},\ f_{i}\in I(\mathbb{G})$.

In particular, if $1\notin X$, then $Mod(\mathbb{G},X)$ consists only of the
0-module.

The next lemma, simple as it may be, will provide structure to
$Mod(\mathbb{G},X)$.

######  Lemma 1.3.

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

  1. (1)

If $j:K\ \to\ M$ is an injective map of $\mathbb{G}$-modules, then $K$ 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 $N$ also 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 $K$, 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 $K$, then
$\Delta(m)=\sum_{\beta,j}a_{\beta,\alpha}m_{\beta}\otimes f_{\alpha}$ lies
both in $K\otimes k[\mathbb{G}]$ so that $a_{\beta,\alpha}=0$ unless
$m_{\beta}\in K$ and lies in $M\otimes X$ so that $a_{\beta,\alpha}=0$ unless
$f_{\alpha}\in X$. Thus $\Delta(m)\in K\otimes X$ if $m\in K$.

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

(1.3.1)
$\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}$$\textstyle{Q\otimes
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id_{Q}\otimes
i_{X}}$$\textstyle{Q\otimes\mathcal{O}(\mathbb{G}).}$

Choose a basis $\\{m_{\beta},\beta\in I\\}$ of $M$ and a subset
$I^{\prime}\subset I$ such that $\\{q(m_{\beta}),\beta\in I^{\prime}\\}$ is a
basis for $Q$ and such that $q(m_{\beta})=0,\beta\notin I^{\prime}$. We define
$\Delta_{Q,X}$ by sending $q(m_{\beta})$ to $((q\otimes
Id_{X})\circ\Delta_{M,X})(m_{\beta})$ for $\beta\in I^{\prime}$. A simple
diagram chase shows that $\Delta_{Q}$ factors through $\Delta_{Q,X}:Q\to
Q\otimes X$ completing the diagram (1.3.1) to a commutative diagram, thus
showing that $Q$ is also an $X$-comodule.

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 $M_{1}\oplus M_{2}\to M$.

######  Lemma 1.4.

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.

Consequently, 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
category. Thus

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

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

###### 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 1.3.

This implies that the category $\chi(M)$ is filtering; given two objects
$N_{1}\subset M$ and $N_{2}\subset M$ of $\chi(M)$, then 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 $\bigcup_{N\in\chi(M)}N\
\subset\ M$. 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
$\varinjlim_{N\in\chi(M)}N\to(\varinjlim_{N\in\chi(M)}N)\otimes X$, so that
$\varinjlim_{N\in\chi(M)}N$ is an $X$-comodule. ∎

Theorem 1.5 presents our key construction.

######  Theorem 1.5.

Let $\mathbb{G}$ be an affine group scheme of finite type over $k$ and let
$X\subset\mathcal{O}(\mathbb{G})$ 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 (1.4.1) 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 $i_{X}$.

  4. (4)

$(M\otimes N)_{X}\quad\subset\quad M_{X}\otimes N_{X}$ for any
$\mathbb{G}$-modules $M,N$.

###### Proof.

The fact that $Mod(\mathbb{G},X)$ is an abelian subcategory of
$Mod(\mathbb{G})$ follows directly from Lemma 1.3. 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
1.3(2) and the equality $N_{X}\ =\ \bigcup_{N^{\prime}\in\chi_{M}}N^{\prime}\
\subset N$ of (1.4.1). The fact that $(-)_{X}$ is left exact is implied by
Lemma 1.3.

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 required
isomorphism for the asserted adjunction.

Finally, consider the coaction

$\Delta_{M\otimes N}:M\otimes
N\stackrel{{\scriptstyle\Delta_{M}\otimes\Delta_{N}}}{{\to}}M\otimes
N\otimes\mathcal{O}(\mathbb{G})\otimes\mathcal{O}(\mathbb{G})\stackrel{{\scriptstyle
id_{M}\otimes id_{N}\otimes\mu}}{{\to}}M\otimes
N\otimes\mathcal{O}(\mathbb{G}).$

Write $\Delta_{M}(m)=m\otimes 1+\sum_{i}m_{i}\otimes f_{i}$,
$\Delta_{N}(n)=n\otimes 1+\sum_{j}n_{j}\otimes g_{j}$. Then

$\Delta_{M\otimes N}(m\otimes n)\ =\ m\otimes n\otimes 1+\sum_{j}m\otimes
n_{j}\otimes g_{j}+\sum_{i}m_{i}\otimes n\otimes f_{i}\
+\sum_{i,j}m_{i}\otimes n_{j}\otimes f_{i}g_{j},$

so that if all “coefficients” of $\Delta_{M\otimes N}(m\otimes n)$ lie in $X$,
then all coefficients of $\Delta_{M}(m)$ and $\Delta_{N}(n)$ must also lie in
$X$. ∎

######  Example 1.6.

Perhaps the simplest example of the functor
$(-)_{X}:Mod(\mathbb{G})\quad\to\quad Mod(\mathbb{G},X)$ is the case in which
$X=k$, 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.

######  Proposition 1.7.

Given $r>0$, let $X\subset\mathcal{O}(\mathbb{G})$ be a subspace such that the
composition with the restriction
$\mathcal{O}(\mathbb{G})\twoheadrightarrow\mathcal{O}(\mathbb{G}_{(r)})$ is
injective. Then the composition

(1.7.1) $Mod(\mathbb{G},X)\quad\hookrightarrow\quad Mod(\mathbb{G})\quad\to
Mod(\mathbb{G}_{(r)})$

is fully faithful.

In particular, if $\mathbb{G}$ is a linear algebraic group and $X$ is finite
dimensional, the $Mod(\mathbb{G},X)$ is embedded as a full subcategory of
$Mod(\mathbb{G}_{(r)})$ for $r\gg 0$.

###### Proof.

If $\phi\not=0:M\to N$ in $Hom_{Mod}(\mathbb{G},X)$, then clearly $\phi$ is
not the 0-map from $M$ to $N$ as a $\mathbb{G}_{(r)}$-module.

Assume the composition
$X\subset\mathcal{O}(\mathbb{G})\twoheadrightarrow\mathcal{O}(\mathbb{G}_{(r)})$
is injective and consider two $X$-comodules $M,N$. Let $\phi:M\to N$ be a map
of $\mathbb{G}_{(r)}$-modules (i.e.,
$\mathcal{O}(\mathbb{G}_{(r)})$-comodules). Consider the commutative diagram

(1.7.2)
$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\Delta_{M}}$$\textstyle{M\otimes
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\otimes\mathcal{O}(\mathbb{G}_{(r)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi\otimes
id}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta_{N}}$$\textstyle{N\otimes
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N\otimes\mathcal{O}(\mathbb{G}_{(r)}).}$

The injectivity of the left horizontal arrows and an easy diagram chase shows
that there is a (unique) map $M\otimes X\to N\otimes X$ completing the
diagram. In other words, $\phi$ is the image of a map in $Mod(\mathbb{G},X)$.
∎

The corollary below is an immediate consequence of the left exactness of
$(-)_{X}$ and the exactness of its left adjoint $i_{X}$.

######  Corollary 1.8.

(see [12, Prop 4.12]) If $j:M\hookrightarrow I$ is an embedding of a
$\mathbb{G}$-module $M$ in an injective $\mathbb{G}$-module $I$, then
$j_{X}:M_{X}\hookrightarrow I_{X}$ is an embedding of $M_{X}$ in an injective
object of $Mod(\mathbb{G},X)$. Thus, every $X$-comodule admits an embedding
into an injective $X$-comodule.

Assume $1\in X$ and conasider the right derived functors of
$H^{0}(\mathbb{G},-)$ restricted to $Mod(\mathbb{G},X)$,
$H^{i}((\mathbb{G},X),-):Mod(\mathbb{G},X)\ \to\ k$ sending an $X$-comodule
$M$ to $H^{i}((\mathbb{G},X),M)$. Then
$H^{0}(\mathbb{G},-)=H^{0}((\mathbb{G},X),-)\circ(-)_{X}:Mod(\mathbb{G})\ \to\
k$, so that there is a Grothendieck spectral sequence

(1.8.1) $E_{2}^{s,t}\ =\
H^{s}((\mathbb{G},X),R^{t}(-)_{X}(M))\quad\Rightarrow\quad
H^{s+t}(\mathbb{G},M),$

where $R^{t}(-)_{X}:Mod(\mathbb{G})\to Mod(\mathbb{G},X)$ is the $t$-th right
derived functor of $(-)_{X}$.

######  Remark 1.9.

It is interesting to compare cohomology groups associated to the functors of
(1.7.1). Choose an injective resolution $M\to I^{\bullet}$ of $M$ in
$Mod(\mathbb{G})$. Then $(I^{\bullet})_{X}$ is a complex of injective
$Mod(\mathbb{G},X)$-modules, but not necessarily exact. The restriction to
$\mathbb{G}_{(r)}$ of $I^{\bullet}$ is an injective resolution in
$Mod(\mathbb{G}_{(r)})$ of the restriction of $M$, but the functor
$H^{0}(\mathbb{G}_{(r)},-)$ is not the restriction of $H^{0}(\mathbb{G},-)$
even when restricted to $Mod(\mathbb{G},X)$.

######  Definition 1.10.

We say a sequence of subspaces $\\{X_{i}\\}$ of $\mathcal{O}(\mathbb{G})$ is
an ascending converging sequence of subspaces of $\mathcal{O}(\mathbb{G})$ if
it is indexed by the non-negative numbers and if

  1. (1)

The unit $1\in\mathcal{O}(\mathbb{G})$ is an element of $X_{0}$.

  2. (2)

$X_{i}\subset X_{i+1}$ for all $i\geq 0$;

  3. (3)

$\bigcup_{i\geq 0}X_{i}\ =\ \mathcal{O}(\mathbb{G})$.

We now introduce the filtration of $\mathbb{G}$-modules associated to an
ascending converging sequence of subspaces of $\mathcal{O}(\mathbb{G})$.

######  Proposition 1.11.

Let $\mathbb{G}$ be an affine group scheme of finite type over $k$ and let
$\\{X_{i}\\}$ be an ascending converging subspaces of
$\mathcal{O}(\mathbb{G})$.

  1. (1)

If $M$ is a finite dimensional $\mathbb{G}$-module, then $M$ is an
$X_{i}$-comodule for all $i\gg 0.$

  2. (2)

More generally, for any $\mathbb{G}$-module $M$, $M\ =\ \bigcup_{i\geq
0}M_{X_{i}}.$

###### Proof.

If $M$ 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 some $M\otimes
X_{i}$. 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. ∎

We present our general construction of a filtration by $\mathbb{G}$-submodules
of a given $\mathbb{G}$-module $M$, functorial once one makes a choice of an
ascending converging sequence of subspaces of $\mathcal{O}(\mathbb{G})$.

######  Definition 1.12.

Choose an ascending converging sequence of subspaces $\\{X_{i}\\}$ of
$k[\mathbb{G}]$. Then sending a $\mathbb{G}$-module $M$ to the sequence of
$\mathbb{G}$-submodules

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

is a filtration of $M$ with the property that $M_{X_{i}}$ is an
$X_{i}$-comodule.

Moreover, for any choice of an ascendin, converging sequence of subspaces
$\\{X_{i}\\}$ of $\mathcal{O}(\mathbb{G})$, this filtration is functorial with
respect to $M$.

We remind the reader that a $\mathbb{G}$-module $M$ for a linear algebraic
group $\mathbb{G}$ is said to be mock injective if the restriction of $M$ to
each Frobenius kernel $\mathbb{G}_{(r)}\hookrightarrow\mathbb{G}$ is an
injective $\mathbb{G}_{(r)}$-module. A concrete example of a mock injective
$\mathbb{G}_{a}$-module which is not injective is the submodule
$\\{f\in\mathcal{O}(\mathbb{G}_{a}),f(t)=f(t+a)\ \forall
a\in\mathbb{F}_{p}\\}$ of $\mathcal{O}(\mathbb{G}_{a})$ equipped with the
right = left regular representation (see [20]).

We argue exactly as in the proof of [12, Prop 4.2] to conclude the following
detection of rational injectivity of a $\mathbb{G}$-module.

######  Theorem 1.13.

Consider an affine group scheme $\mathbb{G}$ of finite type over $k$ and
choose an ascending converging sequence $\\{X_{i}\\}$ 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
$i\geq 0$.

In particular, if $L$ is a mock injective $\mathbb{G}$-module which is not
injective, then there exists some $i$ such that $L_{X_{i}}$ is not an
injective $X_{i}$-comodule.

###### Proof.

If $L$ is injective, then the left exactness of $(-)_{X}$ and the exactness of
its left adjoint $i_{X}$ tells us that $L_{X_{i}}\subset L$ is an injective
object of $Mod(\mathbb{G},X_{i})$ for each $i\geq 0$.

Assume now the $\mathbb{G}$-module $L$ has the property that $L_{X_{i}}\subset
L$ is an injective object of $Mod(\mathbb{G},X_{i})$ for all $i\geq 0$. 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 proceed to construct an extension of $f:M\to L$ of $f^{\prime}$. Denote by
$f^{\prime}_{d}:M^{\prime}_{X_{d}}\to L_{X_{d}}$ the restriction of
$f^{\prime}$ to $M^{\prime}_{X_{d}}$. We inductively 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 $f:M\to L$ extending $f^{\prime}$ to be
$\varinjlim_{i}f_{i}:M=\varinjlim_{i}((M^{\prime})_{X_{i}}+M_{X_{i-1}})\to L$.
∎

As an immediate consequence of the functoriality of $(-)_{X}$, we obtain the
following proposition providing invariants for $\mathbb{G}$-modules.

######  Proposition 1.14.

Choose an ascending converging sequence of subspaces $\\{X_{i}\\}$ 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 $i$.

The following corollary of Proposition 1.14 can be applied to distinguish non-
isomorphic mock injective $\mathbb{G}$-modules such as those constructed in
[20].

######  Corollary 1.15.

Choose an ascending converging sequence of subspaces $\\{X_{i}\\}$ of
$\mathcal{O}(\mathbb{G})$ and let $J,\ J^{\prime}$ be mock injective
$\mathbb{G}$-modules. Assume that the socles of $J,\ J^{\prime}$ are equal to
the finite dimensional $\mathbb{G}$-module $M$, determining embeddings of $J,\
J^{\prime}$ in $I=M\otimes k[\mathbb{G}]^{R}$. If $J$ is isomorphic to
$J^{\prime}$, then the images of $J_{X_{i}},\ dimJ^{\prime}_{X_{i}}$ are equal
in $I$ for all $i\geq 0$.

##  2\. Filtrations by exponential degree

Linear algebraic groups of exponential type permit a relatively concrete
description of the cohomology and support varieties of their Frobenius kernels
$\mathbb{G}_{(r)}$. They also were the first examples of smooth, connected
group schemes with a reasonable formulation of support varieties for their
representations. In this section, we provide a simple “fix” to an error of
[12] in order to guarantee that the “filtration by exponential degree” of of a
$\mathbb{G}$-module for a linear algebraic group $\mathbb{G}$ of exponential
type consists of $\mathbb{G}$-submodules.

In Definition 2.8, we introduce a necessary modification of the definition of
the filtration $M_{[0]}\subset M_{[1]}\subset M_{[2]}\subset\cdots\ M$ of the
filtration given in [12, Defn 3.10] by requiring that this is a filtration of
$M$ by $\mathbb{G}$-submodules. With our revised definition of the filtration
by exponential degree of a $\mathbb{G}$-module $M$, the assertions in [12]
apply with essentially no change. In particular, we obtain the test of
injectivity in Corollary 2.9

We first recall from [10, 1.6] the definition of a structure
$(\mathbb{G},\mathcal{E})$ of exponential type on a linear algebraic group
$\mathbb{G}$, a definition which extends the formulation in [26] of an
embedding $\mathbb{G}\subset GL_{n}$ of exponential type. Up to isomorphism
(as made explicit in [10, 1.7]), if such a structure exists then it is unique.
For this reason, we often drop explicit mention of the exponential map
$\mathcal{E}:\mathcal{N}_{p}(\mathfrak{g})\times\mathbb{G}_{a}\ \to
G,\quad(B,s)\mapsto\mathcal{E}_{B}(s).$

We utilize the notation $\mathcal{N}_{p}(\mathfrak{g})$ for the $p$-nilpotent
cone of the Lie algebra $\mathfrak{g}=Lie(\mathbb{G})$. Thus,
$\mathcal{N}_{p}(\mathfrak{g})\ \subset\ \mathfrak{g}$ consists of those
$X\in\mathfrak{g}$ such that $X^{[p]}=0$. We also 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 $\mathfrak{g}$.
We recall from [26, Thm1.5] the scheme $V(\mathbb{G}_{(r)})$ representing the
functor of 1-parameter subgroups of the infinitesimal group scheme
$\mathbb{G}_{(r)}$.

######  Definition 2.1.

[10, 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)

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

such that 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 [14];
these include simple classical groups, their standard parabolic subgroups, and
the unipotent radicals of these parabolic subgroups.

######  Remark 2.2.

In certain examples, the condition that $\mathcal{E}$ be a morphism of schemes
should be relaxed to require that $\mathcal{E}$ be a “continuous algebraic
map” over $k$ satisfying the properties listed in Definition 2.1 (see [14,
Defn1.9]). This relaxation of the condition that $\mathcal{E}$ be a morphism
has no effect on the arguments we use to consider filtrations.

The consideration of field extensions $K/k$ is employed when investigating
support varieties of $\mathbb{G}$-modules and does not appear to be necessary
when considering coactions. On the other hand, we require considerations of
field extensions $K/k$ in order to verify that our formulation in Definition
2.3 is equivalent to that given in [12].

######  Definition 2.3.

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

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

where $\mathcal{O}(\mathcal{N}_{p}(\mathfrak{g}))[t]_{\leq
d}\subset\mathcal{O}(\mathcal{N}_{p}(\mathfrak{g})\times\mathbb{G}_{a})$ is
the subspace of polynomials in $R[t]$ of degree $\leq d$ for
$R=\mathcal{O}(\mathcal{N}_{p}(\mathfrak{g}))$.

We refer to $Mod(\mathbb{G},\mathcal{E}(d))\ \subset\ Mod(\mathbb{G})$ as the
abelian subcategory of $\mathbb{G}$-modules of exponential degree less than or
equal to $d$.

The following lemma, which follows directly from the definitions, rephrases
the condition that $M$ have exponential degree $\leq d$ as the condition
considered in [12, Defn3.4] for a linear algebraic group $\mathbb{G}$ of
exponential type and $k$ algebraically closed. We utilize
$v_{i}:k[\mathbb{G}_{a}]=k[t]\to k$ to denote the $k$-linear map sending
$t^{j}$ to 0 if $j\not=i$ and sending $t^{i}$ to 1.

######  Lemma 2.4.

Let $(\mathbb{G},\mathcal{E})$ be a linear algebraic group of exponential type
and consider a $\mathbb{G}$-module $M$. Then $M$ is an
$\mathcal{E}(d)$-comodule if and only if for every field extension $K/k$ and
every $K$-point $\underline{B}$ of
$\mathcal{C}_{r}(\mathcal{N}_{p}(\mathfrak{g}))$ the composition (which we
denote by $(\mathcal{E}_{\underline{B}})_{*}(v_{i})$)

$\mathcal{O}(\mathbb{G}_{K})\ \stackrel{{\scriptstyle\mathcal{E}^{*}}}{{\to}}\
\mathcal{O}(\mathcal{N}_{p}(\mathfrak{g}_{K}))\otimes\mathcal{O}(\mathbb{G}_{a,K})\
\stackrel{{\scriptstyle\epsilon_{\underline{B}}\cdot id}}{{\to}}\
\mathcal{O}(\mathbb{G}_{a,K})\stackrel{{\scriptstyle v_{i}}}{{\to}}K$

acts trivially on $M_{K}$ (as
$(1\otimes(\mathcal{E}_{\underline{B}})_{*}(v_{i}))\circ\Delta_{M_{K}}$) for
all $i>d$.

Consequently, for $(\mathbb{G},\mathcal{E})$ be a linear algebraic group of
exponential type and $k$ algebraically closed, the subcategory
$Mod(\mathbb{G},\mathcal{E}(d))\subset Mod(\mathbb{G})$ is the subcategory of
$\mathbb{G}$-modules of exponential degree $\leq d$ in the sense of [12].

######  Remark 2.5.

The subspace $\mathcal{E}(d)$ of (2.3.1) was denoted $(k[\mathbb{G}]))_{\leq
d}$ in [12, Defn3.4]. The error in [12] is the “proof” of [12, Prop3.5]
seeming to verify that $(k[\mathbb{G}]))_{\leq d}$ is a sub-coalgebra of
$\mathcal{O}(\mathbb{G})$. (This subspace should not be confused with the sub-
coalgebra $\mathcal{O}(\mathbb{G})_{\leq d}$ introduced in Theorem 3.8.) Our
consideration of $X$-comodules $M$ for an arbitrary subspace
$X\subset\mathcal{O}(\mathbb{G})$ avoids the condition that $X$ be a sub-
coalgebra by requiring in the definition that $M$ be a $\mathbb{G}$-module.

The formulation of $\mathcal{E}(d)$ in Definition 2.3 is “$k$-rational” and
does not require base change to field extensions $K/k$ in contrast to the
equivalent formulation in Lemma 2.4. This is an advantage of working with
coactions of $\mathcal{O}(\mathbb{G})$ rather than with actions of the
distribution algebra of $\mathbb{G}$.

One disadvantage of the subspace
$\mathcal{E}(d)\subset\mathcal{O}(\mathbb{G})$ is that it is not finite
dimensional in many cases of interests, as we see in the following example.

######  Example 2.6.

If $\mathbb{G}\ =\ \mathbb{T}$ is a torus, then there are no non-trivial
1-parameter subgroups $\mathcal{E}_{B}:\mathbb{G}_{a}\to\mathbb{T}$. In
particular, $\mathbb{T}$ is not of exponential type.

If $(\mathbb{G},\mathcal{E})$ is a reductive algebraic group of exponential
type and if $k$ is algebraically closed, then [12, Prop3.9] implies that
$\mathcal{E}(0)\subset\mathcal{O}(\mathbb{G})$ is the ideal of consisting of
$f\in\mathcal{O}(\mathbb{G})$ which vanish on the closed subscheme
$\mathcal{U}_{p}\subset\mathbb{G}$ of $p$-unipotent elements; in particular,
$\mathcal{E}(0)$ is infinite dimensional for such $\mathbb{G}$. We point out
that $\mathcal{E}(0)\subset k[\mathbb{G}]^{R}$ is not a
$\mathbb{G}$-submodule, much less a sub-coalgebra of
$\mathcal{O}(\mathbb{G})$.

######  Lemma 2.7.

Let $(\mathbb{G},\mathcal{E})$ be a linear algebraic group of exponential
type. Then $\\{\mathcal{E}(d)\\}$ is an ascending converging sequence of
subspaces of $\mathcal{O}(\mathbb{G})$.

Consequently, for any $\mathbb{G}$-module $M$, $M\ =\ \bigcup_{d\geq
0}M_{\mathcal{E}(d)}.$

###### Proof.

Since $\mathcal{O}(\mathcal{N}_{p}(\mathfrak{g})\times\mathbb{G}_{a})\ =\
\bigcup_{d}\mathcal{O}(\mathcal{N}_{p}(\mathfrak{g}))[t]_{\leq d}$,
$\\{\mathcal{E}(d)\\}$ is an ascendin, converging sequence of subspaces of
$\mathcal{O}(\mathbb{G})$.

By [10, Prop2.6], for any finite dimensional $\mathbb{G}$-module $M$ there
exists some $d\geq 0$ such that $M$ has exponential degree $\leq d$ (so that
$M=M_{\mathcal{E}(d)}$ for $d\gg 0$). More generally, any $\mathbb{G}$-module
is locally finite so that $M\ =\ \bigcup_{d\geq 0}M_{\mathcal{E}(d)}.$ ∎

The filtration by exponential degree $\\{M_{\mathcal{E}(d)}\\}$ given in
Definition 2.8 is “intrinsic” in the sense that it requires no choices be made
for its formulation.

######  Definition 2.8.

Let $(\mathbb{G},\mathcal{E})$ be a linear algebraic group of exponential
type. We define the filtration by exponential degree of a $\mathbb{G}$-module
$M$ to be the filtration associated to the ascending converging subspaces
$\\{\mathcal{E}(d)\\}$ of $\mathcal{O}(\mathbb{G})$:

(2.8.1) $M_{\mathcal{E}(0)}\subset M_{\mathcal{E}(1)}\subset
M_{\mathcal{E}(2)}\subset\cdots\ \subset\ \bigcup_{d}M_{\mathcal{E}(d)}=M.$

So defined, $M_{\mathcal{E}(d)}\ \subset\ M$ is the largest
$\mathbb{G}$-submodule of $M$ which has exponential degree $\leq d$.

We emphasize a special case of Theorem 1.13 (as stated in [12]).

######  Corollary 2.9.

Let $(\mathbb{G},\mathcal{E})$ be a linear algebraic group of exponential
type. Then a $\mathbb{G}$-module $L$ is injective if and only if
$L_{\mathcal{E}(d)}$ is an injective object of
$Mod(\mathbb{G},\mathcal{E}(d))$ for all $d\geq 0$.

One attractive feature of this filtration by exponential degree is the
following multiplicative property.

######  Proposition 2.10.

Let $(\mathbb{G},\mathcal{E})$ be a linear algebraic group of exponential
type. For any pair of non-negative integers $(d,e)$, the product map
$\mathcal{O}(\mathbb{G})\otimes\mathcal{O}(\mathbb{G})\ \to\
\mathcal{O}(\mathbb{G})$ restricts to

$\mathcal{E}(d)\otimes\mathcal{E}(e)\quad\to\quad\mathcal{E}(d+e).$

Thus, if $M=M_{\mathcal{E}(d)}$ and if $N=N_{\mathcal{E}(e)}$, then $M\otimes
N\ =\ (M\otimes N)_{\mathcal{E}(d+e)}$.

###### Proof.

The first statement follows immediately from the fact that
$\mathcal{E}^{*}:\mathcal{O}(\mathbb{G})\to\mathcal{O}(\mathcal{N}_{p}(\mathfrak{g}))[t]$
is a map of $k$-algebras, in particular, it is multiplicative. The second then
follows 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
$\Delta_{M}\otimes\Delta_{N}:M\otimes
N\to(M\otimes\mathcal{O}(\mathbb{G}))\otimes(M\otimes\mathcal{O}(\mathbb{G}))$
with the product map $\mathcal{O}(\mathbb{G})\otimes\mathcal{O}(\mathbb{G})\
\to\ \mathcal{O}(\mathbb{G})$. ∎

######  Remark 2.11.

A closed embedding $i:\mathbb{G}\hookrightarrow GL_{N}$ of linear algebraic
groups is an embedding of exponential type if the exponential map
$\mathcal{E}xp:\mathcal{N}_{p}(gl_{N})\times\mathbb{G}_{a}\to GL_{N}$
restricts to a map
$\mathcal{E}:\mathcal{N}_{p}(\mathfrak{g})\times\mathbb{G}_{a}\to\mathbb{G}$.
For future reference, we state explicitly the naturality of $\mathcal{E}(d)$
of Definition 2.3 for such embeddings, whose proof follows directly from this
definition. Namely, for any $d\geq 0$, the following square commutes

(2.11.1)
$\textstyle{\mathcal{E}xp(d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}}$$\textstyle{\mathcal{O}(GL_{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}}$$\textstyle{\mathcal{E}(d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(\mathbb{G}),}$

where the horizontal maps are the (defining) inclusions of Definition 2.3.

##  3\. Ascending sequences of sub-coalgebras of $\mathcal{O}(\mathbb{G})$

In this section, we consider various constructions of sub-coalgebras of
$\mathcal{O}(\mathbb{G})$. Of particular interest is a construction by Jantzen
recalled in Theorem 3.2 which associates “generalized Schur algebras” to
“truncated subcategories” of $Mod(\mathbb{G})$ for a reductive linear
algebraic group $\mathbb{G}$. For any affine group scheme $\mathbb{G}$ and any
ascending converging subsequence $\\{X_{i}\\}$ of subspaces of
$\mathcal{O}(\mathbb{G})$, Theorem 3.4 extends Jantzen’s construction by
constructing an ascending converging sequence
$\\{\mathcal{O}(\mathbb{G})_{X_{i}}\\}$ of sub-coalgebras of
$\mathcal{O}(\mathbb{G})$. The sub-coalgebra $\mathcal{O}(\mathbb{G})_{X}$ is
finite dimensional whenever $X$ is finite dimensional. Moreover, the abelian
category of comodules for $\mathcal{O}(\mathbb{G})_{X}$ is naturally
identified with the full abelian subcategory $Mod(\mathbb{G},X)\subset
Mod(\mathbb{G})$ of Theorem 1.5.

In Theorem 3.8, we provide another construction of ascending converging
sequences of finite dimensional sub-coalgebras
$\\{\mathcal{O}(\mathbb{G})_{\leq i}\\}$ of $\mathcal{O}(\mathbb{G})$ for an
affine group scheme by first explicitly defining $\\{\mathcal{O}(GL_{N})_{\leq
i}\\}$ and then using a closed embedding $\mathbb{G}\hookrightarrow GL_{N}$.
Proposition 3.13 compares such sequences of sub-coalgebras of
$\mathcal{O}(\mathbb{G})$ determined by different choices of embeddings of
$\mathbb{G}$ in general linear groups.

We begin by recalling the coalgebras dual to the classical Schur algebras
associated to $GL_{N}$.

######  Example 3.1.

Consider the open embedding of
$GL_{N}\hookrightarrow\mathbb{M}_{N,N}\simeq\mathbb{A}^{N^{2}}$ of schemes
over $k$ given as the complement of the zero locus of the determinant
$det(\underline{x})\in\mathcal{O}(\mathbb{M}_{N,N})$. The associated map on
coordinate algebras $O(\mathbb{M}_{N,N})\ \hookrightarrow\
\mathcal{O}(GL_{N})$ gives $O(\mathbb{M}_{N,N})$ the structure of a sub-
coalgebra; moreover, $O(\mathbb{M}_{N,N})$ is equipped with the grading given
by associating the total degree $d=\sum d_{i,j}$ to the monomial $\prod
x_{i,j}^{d_{i,j}}$. The $d$-th graded summand of $O(\mathbb{M}_{N,N})$,
$\mathcal{O}(\mathbb{M}_{N,N})_{d}$ is a finite dimensional sub-coalgebra of
$\mathcal{O}(GL_{N})$ whose linear dual $S(N,d)$ is the Schur algebra (for
$GL_{N}$) of degree $d$. The subcategory $CoMod(\mathcal{O}(GL_{N}))\subset\
Mod(GL_{N})$ is the full subcategory of polynomial representations homogenous
of degree $d$ (see [19], [8]).

If $i:\mathbb{G}\ \subset GL_{N}$ is 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), then each
$A(\mathbb{G})_{d}$ is a sub-coalgebra of $\mathcal{O}(\mathbb{G})$. See [9,
1.2].

We summarize aspects of Chapter A of [21] (see also [8]) providing ascending
converging filtrations by sub-coalgebras of $\mathcal{O}(\mathbb{G})$ for a
reductive group $\mathbb{G}$. We choose a Borel subgroup
$\mathbb{B}\subset\mathbb{G}$ and denote by $X_{+}$ denote the set of dominant
weights of $\mathbb{G}$. A subset $\pi\subset X_{+}$ is said to be saturated
if $\mu\in\pi$ whenever there is some $\lambda>\mu\in X_{+}$ (according to the
usual partial ordering of dominant weights) with $\lambda\in\pi$. 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 [21, A.3.1], $\mathcal{C}_{\pi(N,d)}$ is the category of
modules for the Schur algebra $S(N,d)$ and $\mathcal{O}_{\pi(N,d)}(GL_{N})$
(as discussed below) 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 1.5.

######  Theorem 3.2.

Jantzen, [21, 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 $\sum_{\lambda\in\pi}dim(H^{0}(\lambda))^{2}$).

  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 a finite ascending sequence of
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 sub-coalgebra
$\mathcal{O}(\mathbb{M}_{N,N})_{d}\subset\mathcal{O}(GL_{N})$.

  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 proceed to establish analogues of Theorem 3.2(1) - (5) in the general
context of Section 1. Theorem 3.2(6),(7) rely upon the understanding of
modules with a “good filtration” for a reductive group.

######  Lemma 3.3.

  1. (1)

A subspace $X\subset\mathcal{O}(\mathbb{G})$ containing the unit
$1\in\mathcal{O}(\mathbb{G})$ is a sub-coalgebra if and only if
$\Delta:\mathcal{O}(\mathbb{G})\otimes\mathcal{O}(\mathbb{G})\to\mathcal{O}(\mathbb{G})$
restricts to $\Delta_{X}:X\otimes X\to X$.

  2. (2)

If $C\subset\mathcal{O}(\mathbb{G})$ is a sub-coalgebra containing
$1\in\mathcal{O}(\mathbb{G})$ which (as a $\mathbb{G}$-module) is an object of
$Mod(\mathbb{G},X)$, then $C\subset X$.

  3. (3)

If $C\ \subset\ \mathcal{O}(\mathbb{G})$ is a sub-coalgebra containing
$1\in\mathcal{O}(\mathbb{G})$, then $Mod(\mathbb{G},C)\subset Mod(\mathbb{G})$
(in the sense of Theorem 1.5) is naturally identified with $CoMod(C)\subset
Mod(\mathbb{G})$, where $CoMod(C)$ is the the abelian category of comodules
for $C$.

###### Proof.

If
$\Delta:\mathcal{O}(\mathbb{G})\otimes\mathcal{O}(\mathbb{G})\to\mathcal{O}(\mathbb{G})$
restricts to $\Delta_{X}:X\otimes X\to X$, then the counit of
$\mathcal{O}(\mathbb{G})$ restricts to give a counit for $X$ and the
coassociativity property of $\Delta_{X}$ is the restriction of the
coassociativity property for $\Delta$; the converse stated in assertion (1) is
part of the definition of a sub-coalgebra.

Write $f\in\mathcal{O}(\mathbb{G})$ as $f(1)+f^{\prime}\in k\oplus I$ as in
Lemma 1.2. As observed in [21, I.2.4], $\Delta(f^{\prime})\in
f^{\prime}\otimes 1+1\otimes f^{\prime}+I\otimes I$. This implies that if
$\Delta:C\to C\otimes X$, then every $f\in C$ is an element of $X$.

The third assertion is essentially immediate from the definitions of these
subcategories of $Mod(\mathbb{G})$. ∎

The correspondence of Theorem 3.4(4) below evolved from a conversation with
Cris Negron inspired by [25, Thm2.2.8]. This theorem led to the author’s
belated discovery of Jantzen’s results recalled above in Theorem 3.2.

######  Theorem 3.4.

Let $\mathbb{G}$ be an affine group scheme. For any subspace
$X\subset\mathcal{O}(\mathbb{G})$ containing $1\in\mathcal{O}(\mathbb{G})$, we
denote the subspace $(k[\mathbb{G}]^{R})_{X}$ by
$\mathcal{O}(\mathbb{G})_{X}$.

  1. (1)

The subspace $\mathcal{O}(\mathbb{G})_{X}\subset\mathcal{O}(\mathbb{G})$ is a
sub-coalgebra.

  2. (2)

$\mathcal{O}(\mathbb{G})_{X}\subset X$; thus, $\mathcal{O}(\mathbb{G})_{X}$ is
finite dimensional if $X$ is finite dimensional.

  3. (3)

The natural embedding $CoMod(\mathcal{O}(\mathbb{G})_{X})\subset
Mod(\mathbb{G})$ can be identified with $i_{X}:Mod(\mathbb{G},X)\subset
Mod(\mathbb{G})$ of Theorem 1.5.

  4. (4)

If $\\{X_{i}\\}$ is an ascending converging sequence of subspaces of
$\mathcal{O}(\mathbb{G})$, then $\\{\mathcal{O}(\mathbb{G})_{X_{i}}\\}$ is an
ascendin, converging sequence of sub-coalgebras of $\mathcal{O}(\mathbb{G})$.

###### Proof.

By definition, $\mathcal{O}(\mathbb{G})_{X}\subset\mathcal{O}(\mathbb{G})$ is
a $\mathbb{G}$-module, with coaction of the form
$\mathcal{O}(\mathbb{G})_{X}\to\mathcal{O}(\mathbb{G})_{X}\otimes\mathcal{O}(\mathbb{G})$.
This coaction respects $\mathbb{G}$-actions if we take the coaction to be the
right regular action on the domain copy of $\mathcal{O}(\mathbb{G})_{X}$ and
the coaction on the
$\mathcal{O}(\mathbb{G})_{X}\otimes\mathcal{O}(\mathbb{G})$ to be the tensor
product of the trivial coaction on the tensor factor
$\mathcal{O}(\mathbb{G})_{X}$ and the right regular coaction on the tensor
factor $\mathcal{O}(\mathbb{G})$. In particular,
$(\mathcal{O}(\mathbb{G})_{X}\otimes\mathcal{O}(\mathbb{G}))_{X}=\mathcal{O}(\mathbb{G})_{X}\otimes\mathcal{O}(\mathbb{G})_{X}$
for this coaction on the range. The adjunction of Theorem 1.5(3) thus implies
that the coaction
$\mathcal{O}(\mathbb{G})_{X}\to\mathcal{O}(\mathbb{G})_{X}\otimes\mathcal{O}(\mathbb{G})$
factors through
$\mathcal{O}(\mathbb{G})_{X}\to\mathcal{O}(\mathbb{G})_{X}\otimes\mathcal{O}(\mathbb{G})_{X}$,
so that $\mathcal{O}(\mathbb{G})_{X}$ is a sub-coalgebra of
$\mathcal{O}(\mathbb{G})$ by Lemma 3.3(1).

By construction, $\mathcal{O}(\mathbb{G})_{X}$ is an $X$-comodule so that
assertion (2) follows by Lemma 3.3(2). By assertion (2) and Lemma 3.3(3),
$CoMod(\mathcal{O}(\mathbb{G})_{X})\ \subset\ Mod(\mathbb{G},X)$. On the other
hand, if $M=M_{X}$ with coaction $M\to M\otimes X\subset
M\otimes\mathcal{O}(\mathbb{G})$, then applying $(-)_{X}$ to this composition
we obtain $M\to M\otimes\mathcal{O}(\mathbb{G})_{X}\subset
M\otimes\mathcal{O}(\mathbb{G})$ as argued above to prove (1). Thus, any
$X$-comodule is a comodule for $\mathcal{O}(\mathbb{G})_{X}$. Assertion (4)
follows from Proposition 1.11(2), taking $M=k[\mathbb{G}]^{R}$. ∎

Starting with the grading on $\mathcal{O}(\mathbb{M}_{N,N})$, we proceed to
give another filtration of $\mathcal{O}(GL_{N})$ by finite dimensional
coalgebras. We employ the fact that $\mathbb{M}_{N,N}$ is an affine monid
scheme whose binary operation is matrix multiplication.

######  Lemma 3.5.

Consider an affine monoid scheme $\mathbb{M}$ of finite type over $k$ and a
closed embedding $i:\mathbb{G}\ \hookrightarrow\ \mathbb{M}$ of monoid
schemes. If $C\ \subset\ \mathcal{O}(\mathbb{M})$ is a sub-coalgebra
containing $1\in\mathcal{O}(\mathbb{M})$, then
$i^{*}(C)\subset\mathcal{O}(\mathbb{G})$ is a sub-coalgebra of
$\mathcal{O}(\mathbb{G})$.

###### Proof.

If $i:\mathbb{G}\ \hookrightarrow\ \mathbb{M}$ is a closed embedding of monoid
schemes over $k$, then $i^{*}:\mathcal{O}(\mathbb{M})\ \to\
\mathcal{O}(\mathbb{G})$ is a surjective map of coalgebras. To prove that
$i^{*}(C)$ is a sub-coalgebra if $\mathcal{O}(\mathbb{G})$ if $C$ is a
subcoalgebra of $\mathcal{O}(\mathbb{M})$, we consider the commutative diagram
whose vertical maps are coproducts

(3.5.1)
$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(\mathbb{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(\mathbb{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
i^{*}(C)}$$\textstyle{C\otimes
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(\mathbb{M})\otimes\mathcal{O}(\mathbb{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(\mathbb{G})\otimes\mathcal{O}(\mathbb{G})}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
i^{*}(C)\otimes i^{*}(C).}$

The lemma follows form Lemma 3.3(1) and a simple diagram chase using the
observations that $i^{*}:C\to i^{*}(C)$ is surjective and that
$i^{*}(C)\otimes
i^{*}(C)\to\mathcal{O}(\mathbb{G})\otimes\mathcal{O}(\mathbb{G})$ is
injective. ∎

Consider the (product) affine monoid scheme
$\mathbb{M}_{N,N}\times\mathbb{G}_{m}$. We give
$\mathcal{O}(\mathbb{M}_{N,N}\times\mathbb{G}_{m})$ the filtration defined by
setting $\mathcal{O}(\mathbb{M}_{N,N}\times\mathbb{G}_{m})_{\leq d}$ equal to
the subspace of those functions $\sum_{n}p_{n}(x_{i,j})t^{n}$ such that for
each $n$ the total degree of $p_{n}(x_{i,j})$ plus the absolute value of $nN$
is less than or equal to $d$; in particular, both $t,\
t^{-1}\in\mathcal{O}(\mathbb{G}_{m})$ are elements of filtration degree $N$.

Since the generators $x_{i,j}\in\mathcal{O}(\mathbb{M}_{N,N})$ are homogeneous
of degree 1 with coproducts
$\Delta_{\mathbb{M}_{N,N}}(x_{i,j})=\sum_{\ell}x_{i,\ell}\otimes
x_{\ell,j}\in\mathcal{O}(\mathbb{M}_{N,N})_{1}\otimes\mathcal{O}(\mathbb{M}_{N,N})_{1}$
and the generators $t,\ t^{-1}\in\mathcal{O}(\mathbb{G}_{m})$ are homogeneous
of degree $N$ with coproducts $\Delta_{\mathbb{G}_{m}}(t)=t\otimes t,\
\Delta_{\mathbb{G}_{m}}(t^{-1})=t^{-1}\otimes t^{-1}$ in
$\mathcal{O}(\mathbb{G}_{m})_{\leq N}\otimes\mathcal{O}(\mathbb{G}_{m})_{\leq
N}$, and since $\Delta_{\mathbb{M}_{N,N}},\ \Delta_{\mathbb{G}_{m}}$ commute
with products while the filtrations respect products, we conclude that each
$\mathcal{O}(\mathbb{M}_{N,N}\times\mathbb{G}_{m})_{\leq d}$ is a sub-
coalgebra of $\mathcal{O}(\mathbb{M}_{N,N}\times\mathbb{G}_{m})$.

######  Theorem 3.6.

Consider the closed embedding of affine monoid schemes

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

associated to the natural quotient map

(3.6.2)
$i^{*}:\mathcal{O}(\mathbb{M}_{N,N})\otimes\mathcal{O}(\mathbb{G}_{m})\
\twoheadrightarrow\ \mathcal{O}(GL_{N}),\quad x_{i,j}\otimes 1\mapsto
x_{i,j},\ 1\otimes t\mapsto det(\underline{x})^{-1}.$

We define $\mathcal{O}(GL_{N})_{\leq d}\hookrightarrow\mathcal{O}(GL_{N})$ to
be $i^{*}(\mathcal{O}(\mathbb{M}_{N,N}\times\mathbb{G}_{m})_{\leq
d})\subset\mathcal{O}(GL_{N})$. Then for for each $d\geq 0$,

  1. (1)

$\mathcal{O}(GL_{N})_{\leq d}\hookrightarrow\mathcal{O}(GL_{N})$ is a sub-
coalgebra.

  2. (2)

The map given by multiplication on each summand

(3.6.3) $\bigoplus_{o\leq i\leq[d/N]}\mathcal{O}(\mathbb{M}_{N,N})_{\leq
d-i\cdot N}\otimes
k\\{det(\underline{x})^{-i}\\}\quad\twoheadrightarrow\quad\mathcal{O}(GL_{N})_{\leq
d}$

is surjective for any $d\geq 0$.

  3. (3)

The kernel of (3.6.3) is isomorphic to $\bigoplus_{2\leq
i\leq[d/N]}\mathcal{O}(\mathbb{M}_{N,N})_{\leq d-i\cdot N}.$

  4. (4)

The dimension of $\mathcal{O}(GL_{N})_{\leq d}$ equals the sum of the
dimensions of $\mathcal{O}(M_{N,N})_{\leq d}$ and $\mathcal{O}(M_{N,N})_{\leq
d-N}$.

  5. (5)

The composition of (3.6.3) with the restriction $\mathcal{O}(GL_{N})_{\leq
d}\subset\mathcal{O}(GL_{N})\to\mathcal{O}(SL_{N})$ has image equal to the
image of $\mathcal{O}(\mathbb{M}_{N,N})_{\leq d}$.

###### Proof.

The surjectivity of (3.6.2) follows from the facts that
$\mathcal{O}(\mathbb{M}_{N,N})$ is a domain and that
$GL_{N}\subset\mathbb{M}_{N,N}$ is the open complement of the zero locus of
the (irreducible) determinant function $det(\underline{x})$ which implies that
$\mathcal{O}(GL_{N})\ =\
\mathcal{O}(\mathbb{M}_{N,N})[det(\underline{x})^{-1}]$. The fact that
$GL_{N}\ \hookrightarrow\ \mathbb{M}_{N,N}\times\mathbb{G}_{m}$ is a map of
monoid schemes can be checked at the level of $A$-points for commutative
$A$-algebras $A$. Thus, Lemma 3.5 tells us that each
$\mathcal{O}(GL_{N})_{\leq d}$ is a sub-coalgebra of $\mathcal{O}(GL_{N})$.

The surjectivity of (3.6.3) follows from the surjectivity of (3.6.2) and the
explicit formulation of the filtration on $\mathcal{O}(GL_{N})$. The map from
$\bigoplus_{2\leq i\leq[d/N]}\mathcal{O}(\mathbb{M}_{N,N})_{\leq d-i\cdot N}$
to the domain of (3.6.3) is the sum of maps which send
$\mathcal{O}(\mathbb{M}_{N,N})_{\leq d-(i+2)\cdot N}$ to

$\mathcal{O}(\mathbb{M}_{N,N})_{\leq d-i\cdot N}\otimes
k\\{det(\underline{x})^{-i}\\}\ \oplus\ \mathcal{O}(\mathbb{M}_{N,N})_{\leq
d-(i+1)\cdot N}\otimes k\\{det(\underline{x})^{-(i+1)}\\},$

$f$ mapping to $(f\otimes k\\{det(\underline{x})^{-i}\\})+(f\cdot
det(\underline{x})\otimes k\\{det(\underline{x})^{-i+1}\\}).$ The
identification of this map with the kernel of (3.6.3) is verified “by
inspection” using the fact that $\mathcal{O}(\mathbb{M}_{N,N})$ is a unique
factorization domain

Assertion (4) follows from the short exact sequence involving (3.6.3) and its
kernel as identified in (3). Finally, assertion (5) follows from (3.6.3) and
the fact that $det(\underline{x})\in\mathcal{O}(GL_{N})$ maps to
$1\in\mathcal{O}(SL_{N})$. ∎

We give perhaps the simplest possible non-trivial example of the filtration on
a $GL_{N}$-module M associated to the ascending converging sequence
$\\{\mathcal{O}(GL_{N})_{\leq d}\\}$.

######  Example 3.7.

Let $M$ denote the regular representation for $\mathbb{G}_{m}$ given by the
coproduct $k[t,t^{-1}]\to k[t,t^{-1}]\otimes k[t,t^{-1}]$ sending
$t^{j}\mapsto t^{j}\otimes t^{j}$. Then $M_{\mathcal{O}(\mathbb{G}_{m})_{\leq
i}}\subset M$ equals $\mathcal{O}(\mathbb{G}_{m})_{\leq
i}\subset\mathcal{O}(\mathbb{G}_{m})$. On the other hand, the filtration by
exponential degree degree for $M$ is not defined.

Theorem 3.6 enables a filtration of $\mathcal{O}(\mathbb{G})$ by finite
dimensional sub-coalgebras for any affine group scheme of finite type over
$k$. Such a filtration is determined by a choice of closed embedding
$\mathbb{G}\ \hookrightarrow\ GL_{N}$ (by which we mean a closed immersion
which is a morphism of group schemes over $k$). We denote by
$A(\mathbb{G})_{\leq d}$ the image of $\mathcal{O}(\mathbb{M}_{N,N})_{\leq d}$
under the restriction map associated to the morphisms
$\mathbb{G}\hookrightarrow GL_{N}\subset\mathbb{M}_{N,N}$. In the special case
$\mathbb{G}\subset GL_{N}$ equals $SL_{N}\subset GL_{N}$, this sets
$A(SL_{N})_{\leq d}$ equal to $\mathcal{O}(SL_{N})_{\leq d}$.

######  Theorem 3.8.

Let $\mathbb{G}$ be an affine group scheme provided with a closed embedding
$i:\mathbb{G}\hookrightarrow GL_{N}$. We define $\mathcal{O}(\mathbb{G})_{\leq
d}\ \subset\ \mathcal{O}(\mathbb{G})$ to be the image of
$\mathcal{O}(GL_{N})_{\leq d}$ under the quotient map
$i^{*}:\mathcal{O}(GL_{N})\twoheadrightarrow\mathcal{O}(\mathbb{G})$. Then

(3.8.1) $\bigoplus_{o\leq j\leq[d/M]}A(\mathbb{G})_{\leq d-j\cdot M}\otimes
k\\{i^{*}(det(\underline{x})^{-j})\\}\quad\twoheadrightarrow\quad\mathcal{O}(\mathbb{G})_{\leq
d}$

is surjective for any $d\geq 0$, where $M$ is the minimal positive integer $m$
such that $i^{*}(det(\underline{x}))\in A(\mathbb{G})_{\leq m}$. Moreover,

  1. (1)

$\\{\mathcal{O}(\mathbb{G})_{\leq d},d\geq 0\\}$ is an ascending converging
sequence of finite dimensional coalgebras of $\mathcal{O}(\mathbb{G})$.

  2. (2)

Sending a $\mathbb{G}$-module $M$ to the sequence of $\mathbb{G}$-submodules

(3.8.2) $M_{\mathcal{O}(\mathbb{G})_{\leq 0}}\ \subset\
M_{\mathcal{O}(\mathbb{G})_{\leq 1}}\ \subset M_{\mathcal{O}(\mathbb{G})_{\leq
2}}\subset\cdots\ \subset\ \bigcup_{i\geq 0}M_{\mathcal{O}(\mathbb{G})_{\leq
i}}\ =\ M$

is a filtration of $M$ with the property that
$M_{\mathcal{O}(\mathbb{G})_{\leq i}}$ is a comodule for the finite
dimensional coalgebra $\mathcal{O}(\mathbb{G})_{\leq i}$.

  3. (3)

$M$ is an injective $\mathbb{G}$-module if and only if
$M_{\mathcal{O}(\mathbb{G})_{\leq i}}$ is an injective
$\mathcal{O}(\mathbb{G})_{\leq i}$-comodule for each $i\geq 0$.

###### Proof.

Granted that $A(\mathbb{G})_{\leq d}\equiv i^{*}(\mathcal{O}(\mathbb{B})_{\leq
d}$, the surjective map of (3.8.1) follows by naturality (3.6.3) and the
surjectivity of $i^{*}:\mathcal{O}(GL_{N})\to\mathcal{O}(\mathbb{G})$. The
fact that $\mathcal{O}(\mathbb{G})_{\leq d}\ \subset\ \mathcal{O}(\mathbb{G})$
is a finite dimensional sub-coalgebra follows from Lemma 3.5 applied to
$\mathbb{G}\hookrightarrow GL_{N}$. Thus assertion (1) follows from the fact
that $\\{\mathcal{O}(\mathbb{M}_{N,N}\times\mathbb{G}_{m})_{\leq d}\\}$ is an
ascending converging sequence of finite dimensional sub-coalgebras of
$\mathcal{O}(\mathbb{M}_{N,N}\times\mathbb{G}_{m}$. The fact that
$M=\bigcup_{i\geq 0}M_{\mathcal{O}(\mathbb{G})_{\leq i}}$ is verified in
Proposition 1.11; the fact that $M_{\mathcal{O}(\mathbb{G})_{\leq i}}$ is a
comodule for $\mathcal{O}(\mathbb{G})_{\leq i}$ is verified in Lemma 3.3. The
test given for (3) for injectivity of a $\mathbb{G}$-module $M$ is a special
case of Theorem 1.13. ∎

######  Remark 3.9.

A variant of this filtration on $\mathcal{O}(\mathbb{G})$ can be defined by
first defining a filtration on $\mathcal{O}(SL_{N+1})$ viewed as a quotient of
$\mathcal{O}(\mathbb{M}_{N+1,N+1})$ and then defining a filtration on
$\mathcal{O}(GL_{N})$ by using the natural embedding $j:GL_{N}\hookrightarrow
SL_{N+1},\ A\mapsto A\oplus det(A)^{-1}$.

######  Proposition 3.10.

Consider $\mathcal{O}(GL_{N})_{\leq d}$ as in Theorem 3.6. Then for $d\geq N$

(3.10.1) $dim((\mathcal{O}(GL_{N})_{\leq d})\ =\ {d+N^{2}\choose
N^{2}}+{(d-N)+N^{2}\choose N^{2}}$

which is a polynomial of degree $N^{2}$ in $d$, and

(3.10.2) $dim((\mathcal{O}(SL_{N})_{\leq d})\ =\ {d+N^{2}\choose
N^{2}}-{(d-N)+N^{2}\choose N^{2}}.$

which is a polynomial of degree $N^{2}-1$ in $d$.

###### Proof.

The first assertion follows immediately from Theorem 3.6(4). Provided $d\geq
N$ there is a short exact sequence

$0\to\mathcal{O}(M_{N,N})_{\leq
d-N}\stackrel{{\scriptstyle(det(\underline{x})-1)\cdot}}{{\to}}\mathcal{O}(M_{N,N})_{\leq
d}\to\mathcal{O}(SL_{N})_{\leq d}\to 0$

which implies (3.10.2). Since the leading terms of
$dim(\mathcal{O}(M_{N,N})_{\leq d})$ and
$dim(\mathcal{O}(M_{N,N})_{\leq d-N})$ are equal to $d^{N}/(N^{2})!$ whereas
the coefficient of $d^{N^{2}-1}$ for $dim(\mathcal{O}(M_{N,N})_{\leq d})$ is
greater than that for $dim(\mathcal{O}(M_{N,N})_{\leq d-n})$, we conclude that
the difference is a polynomial in $d$ of degree $N^{2}-1$. ∎

######  Remark 3.11.

Given $i:\mathbb{G}\hookrightarrow GL_{N}$, the sub-coalgebras
$\mathcal{O}(\mathbb{G})_{\leq d}\subset\mathcal{O}(\mathbb{G})$ might appear
somewhat strange. For example, suppose $i:GL_{n}\hookrightarrow GL_{N}$ is
given by some $N$-dimensional faithful representation of $GL_{n}$. Let
$\\{x_{i,j}\\}$ be coordinate functions on $GL_{N}$ and let $\\{y_{s,t}\\}$ be
coordinate functions on $GL_{n}$ and set $p_{i,j}(\underline{y})$ equal to
$i^{*}(x_{i,j})$ expressed as a polynomial in
$\\{y_{s,t},det(\underline{y})^{-1}\\}$. Then each
$p_{i,j}(\underline{y})\in\mathcal{O}(GL_{n})$ is in the image of
$x_{i,j}\in\mathcal{O}(GL_{N})_{\leq 1}$ and thus has filtration degree $\leq
1$ using the filtration determined by $i:GL_{n}\to GL_{N}$. However, if we use
the filtration of $\mathcal{O}(GL_{n})$ by taking $n=N$ in Theorem 3.6,
typically $p_{i,j}(\underline{y})\in\mathcal{O}(GL_{n})$ will have higher
filtration degree.

For $(\mathbb{G},\mathcal{E})$ a linear algebraic group of exponential type,
we relate the filtration by exponential degree given in Definition 2.8 to that
of Theorem 3.8.

######  Proposition 3.12.

Equip $GL_{N}$ with the usual exponential map
$\mathcal{E}xp:\mathbb{G}_{a}\times\mathcal{N}_{p}(\mathfrak{gl}_{N})\to
GL_{N}$. Then the sub-coalgebra $\mathcal{O}(GL_{N})_{\leq d}$ of
$\mathcal{O}(GL_{N})$ is contained in the subspace
$\mathcal{E}xp((p-1)d)\subset\mathcal{O}(GL_{N})$ (see (2.3.1). In other
words, any $\mathcal{O}(GL_{N})_{\leq d}$-comodule has exponential degree less
than or equal to $(p-1)d$.

Furthermore, if $\mathbb{G}\hookrightarrow GL_{N}$ is a closed embedding of
exponential type so that $\mathcal{E}xp$ restricts to
$\mathcal{E}:\mathbb{G}_{a}\times\mathcal{N}_{p}(\mathfrak{g})\to\mathbb{G}$,
then $\mathcal{O}(\mathbb{G})_{\leq d}\ \subset\ \mathcal{O}(\mathbb{G})$ is
contained in $\mathcal{E}((p-1)d)$.

More generally, for any linear algebraic group $(\mathbb{G},\mathcal{E})$ of
exponential type equipped with some closed embedding
$\mathbb{G}\hookrightarrow GL_{N}$ and any $d\geq 0$, there exists some
$n_{d}$ such that $\mathcal{O}(\mathbb{G})_{\leq d}\ \subset\
\mathcal{E}(n_{d})$.

###### Proof.

Consider the truncated exponential map
$\overline{\mathcal{E}xp}:\mathbb{M}_{N,N}\times\mathbb{G}_{a}\to GL_{N}$
given functorially by sending $(Y\in\mathbb{M}_{N,N}(A),t\in A)$ to
$1+tY+\cdots+t^{p-1}Y^{p-1}/(p-1)!$. The map
$\overline{\mathcal{E}xp}^{*}:\mathcal{O}(GL_{N})\to\mathcal{O}(\mathbb{M}_{N,N})[t]$
sends $x_{i,j}\in\mathcal{O}(GL_{N})$ to the $(i,j)$-th coordinate of
$1+tY+\cdots t^{p-1}Y^{p-1}/(p-1)!$ where $Y=(y_{s,t})$ is the “generic”
$N\times N$-matrix whose entries are the $N^{2}$ variables $y_{s,t}$. Then
$\mathcal{E}xp^{*}:\mathcal{O}(GL_{N})\to\mathcal{O}(\mathcal{N}_{p}(\mathfrak{gl}_{N}))[t]$
is the composition of $\overline{\mathcal{E}xp}^{*}$ and the restriction map
$\mathcal{O}(\mathbb{M}_{N,N})[t]\twoheadrightarrow\mathcal{O}(\mathcal{N}_{p}(\mathfrak{gl}_{N}))[t]$.
Since
$\overline{\mathcal{E}xp}^{*}(x_{i,j})\in\mathcal{O}(\mathbb{M}_{N,N})\otimes
k[t]_{\leq p-1}$ and $\mathcal{E}xp^{*}(det(\underline{x}))=1$, the fact that
$\mathcal{E}xp^{*}$ is an algebra map implies that $\mathcal{O}(GL_{N})_{\leq
d}\subset\mathcal{E}xp((p-1)d)$.

By Remark 2.11, the inclusion $\mathcal{O}(GL_{N})_{\leq
d}\subset\mathcal{E}xp((p-1)d)$ implies the inclusion
$\mathcal{O}(\mathbb{G})_{\leq d}\ \subset\ \mathcal{E}((p-1)d)$ whenever
$\mathbb{G}\hookrightarrow GL_{N}$ is a closed embedding of exponential type.

Assume now that $(\mathbb{G},\mathcal{E})$ is an arbitrary linear algebraic
group of exponential type equipped with a closed embedding
$\mathbb{G}\hookrightarrow GL_{N}$. Since $\mathcal{O}(\mathbb{G})_{\leq d}$
is finite dimensional and since $\\{\mathcal{E}(n)\\}$ is an ascending
converging sequence of subspaces of $\mathcal{O}(\mathbb{G})$, we conclude
that for each $d$ there exists some $n_{d}$ such that
$\mathcal{O}(\mathbb{G})_{\leq d}\ \subset\ \mathcal{E}(n_{d})$. ∎

We show that changing the embedding $\mathbb{G}\hookrightarrow GL_{N}$ has
limited effect on the associated ascending converging sequences of sub-
coalgebras of $\mathcal{O}(\mathbb{G})$.

######  Proposition 3.13.

Let $\mathbb{G}$ be an affine group scheme and consider two closed embeddings
$\phi:\mathbb{G}\ \hookrightarrow\ GL_{N},\ \phi^{\prime}:\mathbb{G}\
\hookrightarrow\ GL_{N^{\prime}}.$ Let $\\{\mathcal{O}(\mathbb{G})_{\leq
d,\phi}\\}$ and $\\{\mathcal{O}(\mathbb{G})_{\leq d,\phi^{\prime}}\\}$ be the
ascending converging sequences of sub-coalgebras of $\mathcal{O}(\mathbb{G})$
associated to $\phi,\ \phi^{\prime}$ as in Theorem 3.8. Then 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 consider a closed embedding $GL_{N^{\prime}}\times GL_{N}\ \hookrightarrow\
GL_{M}$ with restrictions $\psi^{\prime}:GL_{N^{\prime}}\hookrightarrow
GL_{M},\ \psi:GL_{N}\hookrightarrow GL_{M}$. The image of the sub-coalgebra
$\mathcal{O}(GL_{M})_{\leq e}$ of $\mathcal{O}(GL_{M})$ in
$\mathcal{O}(GL_{N})\otimes\mathcal{O}(GL_{N^{\prime}})$ is given by
$\bigoplus_{e_{1}+e_{2}=e}\mathcal{O}(GL_{N})_{\leq
e_{1},\psi}\otimes\mathcal{O}(GL_{N^{\prime}})_{\leq e_{2},\psi^{\prime}}$
which maps surjectively to both $\mathcal{O}(GL_{N})_{\leq e,\psi}$ and
$\mathcal{O}(GL_{N^{\prime}})_{\leq e,\psi^{\prime}}$, so that
$\mathcal{O}(\mathbb{G})_{e,\psi\circ\phi}\ =\
\mathcal{O}(\mathbb{G})_{e,\psi^{\prime}\circ\phi^{\prime}}$. We conclude that
it suffices to assume that
$\phi^{\prime}=\psi\circ\phi:\mathbb{G}\hookrightarrow GL_{N}\hookrightarrow
GL_{M}$.

Let $R$ denote the minimum positive integer greater that the filtration
degrees of $\psi^{*}(x_{i,j})\in\mathcal{O}(GL_{N})$ and $1/N$ times the
filtration degree of $\psi^{*}(det(\underline{x}))$. Then
$\phi^{\prime*}(\mathcal{O}(GL_{M})_{\leq
d})\equiv\mathcal{O}(\mathbb{G})_{\leq d,\phi^{\prime}}$ is contained in
$\phi^{*}(\mathcal{O}(GL_{N})_{\leq Rd})\equiv\mathcal{O}(\mathbb{G})_{\leq
Rd,\phi}$. On the other hand, $\phi^{*}(\mathcal{O}(GL_{N})_{\leq d})$ is
contained in $\phi^{\prime^{*}}(\mathcal{O}(GL_{M})_{\leq d})$. ∎

We prove a multiplicative property for the filtrations of (3.8.2), an analogue
of Proposition 2.10.

######  Proposition 3.14.

Let $\mathbb{G}$ be an affine group scheme provided with a closed embedding
$i:\mathbb{G}\subset\operatorname{GL}\nolimits_{N}$ as in Theorem 3.8.

  1. (1)

The product
$\mu:\mathcal{O}(\mathbb{G})\otimes\mathcal{O}(\mathbb{G})\to\mathcal{O}(\mathbb{G})$
of the $k$-algebra $\mathcal{O}(\mathbb{G})$ restricts to pairings for any
$d,e\geq 0$

(3.14.1) $\mathcal{O}(\mathbb{G})_{\leq d}\otimes\mathcal{O}(\mathbb{G})_{\leq
e}\quad\to\quad\mathcal{O}(\mathbb{G})_{\leq d+e}.$

  2. (2)

For any pair of $\mathbb{G}$-modules $M,\ N$ and any $d,e\geq 0$, the natural
embedding $M_{\mathcal{O}(\mathbb{G})_{\leq d}}\otimes
N_{\mathcal{O}(\mathbb{G})_{\leq e}}\ \hookrightarrow\ M\otimes N$ factors
through an embedding of $\mathbb{G}$-modules

(3.14.2) $M_{\mathcal{O}(\mathbb{G})_{\leq d}}\otimes
N_{\mathcal{O}(\mathbb{G})_{\leq e}}\quad\hookrightarrow\quad(M\otimes
N)_{\mathcal{O}(\mathbb{G})_{\leq d+e}},$

and thus of $\mathcal{O}(\mathbb{G})_{\leq d+e}$-comodules.

###### Proof.

To prove that the pairing (3.14.1) is well defined, we first verify by direct
inspection that the analogue of (3.14.1) for
$\mathbb{M}_{N,N}\times\mathbb{G}_{m}$,

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

is well defined. One then verifies that (3.14.1) is well defined for $GL_{N}$
(with its closed embedding in $\mathbb{M}_{N,N}\times\mathbb{G}_{m}$) by
functoriality. This same argument, now applied to $\mathbb{G}\hookrightarrow
GL_{N}$, verifies the pairing (3.14.1).

The embedding (3.14.2) now follows using functoriality (or a diagram chase)
once one recognizes that the coproduct $\Delta_{M\otimes N}:M\otimes
N\to(M\otimes N)\otimes\mathcal{O}(\mathbb{G})$ is given by the tensor product
of the coproducts for $M$ and $N$ followed by the product
$\mu:\mathcal{O}(\mathbb{G})\otimes\mathcal{O}(\mathbb{G})\to\mathcal{O}(\mathbb{G})$,

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

This embedding is an embedding of $\mathcal{O}(\mathbb{G})_{\leq
d+e}$-comodules by Lemma 1.4 ∎

We give in Proposition 3.15 a result complementary to Proposition 3.14(1). To
do so, we impose a strong condition on $\mathbb{G}\hookrightarrow GL_{N}$.
Even for $GL_{N}$ such a complementary result requires caution: let $k_{det},\
k_{det^{-1}}$ be 1-dimensional $GL_{N}$-modules given by the actions of
$GL_{N}$ through the determinant and the inverse of the determinant. Thus,
$(k_{det})_{\mathcal{O}(GL_{N})_{\leq
N-1}}=0=(k_{det^{-1}})_{\mathcal{O}(GL_{N})_{\leq N-1}}$, whereas
$k_{det}\otimes k_{det^{-1}}=k$.

######  Proposition 3.15.

Assume that $\mathbb{G}$ is a linear algebraic group with an embedding in
$GL_{N}$ satisfying the condition that $A(\mathbb{G})=\bigoplus
A(\mathbb{G})_{d}$, where
$A(\mathbb{G})=\mathcal{O}(\mathbb{G})\cap\mathcal{O}(\mathbb{M}_{N,N})$ and
$A(\mathbb{G})_{d}=\mathcal{O}(\mathbb{G})\cap\mathcal{O}(\mathbb{M}_{N,N})_{d}$.
Consider an $\mathcal{O}(\mathbb{G})_{\leq e}$-module $N$ and a
$\mathbb{G}$-module $M$ such that $M\otimes N\ =\ (M\otimes
N)_{\mathcal{O}(\mathbb{G})_{\leq d}}$.

Then $M\ =\ M_{\mathcal{O}(\mathbb{G})_{\leq d+e}}$.

###### Proof.

Observe that multiplication in the integral domain $\mathcal{O}(\mathbb{G})$
restricts to injective maps $A(\mathbb{G})_{s}\otimes A(\mathbb{G})_{t}\to
A(\mathbb{G})_{s+t}$ and thus to injective maps $A(\mathbb{G})_{\leq s}\otimes
A(\mathbb{G})_{\leq t}\to A(\mathbb{G})_{\leq s+t}$. Assume that
$f\notin\mathcal{O}(\mathbb{G})_{\leq d+e}$ and consider some
$0\not=h\in\mathcal{O}(\mathbb{G})_{\leq e}$. Write $f=\sum p_{i}\otimes
det(\underline{x})^{-i}$ and $h=\sum q_{j}\otimes det(\underline{x})^{-j}$
such that $i=0$ whenever $det(\underline{x})$ divides $p_{i}$ and $j=0$
whenever $det(\underline{x})$ divides $q_{j}$ (i.e., in “reduced form”). We
can then write $f\cdot h$ as a non-zero sum of terms of the form $p_{i}\cdot
q_{j}\otimes det(\underline{x})^{-\ell}$ which have filtration degree $>d$
(because at most $e/M$ cancellations of factors $det(\underline{x})\otimes
det(\underline{x})^{-1}$ can occur). We conclude that if
$f\in\mathcal{O}(\mathbb{G})$ satisfies $f\cdot
h\in\mathcal{O}(\mathbb{G})_{\leq d}$ then $f\in\mathcal{O}(\mathbb{G})_{\leq
d+e}$.

We now assume that $M\otimes N=(M\otimes N)_{\mathcal{O}(\mathbb{G})_{\leq
d}}$ and $N=N_{\mathcal{O}(\mathbb{G})_{\leq e}}$. Choose a basis
$\\{m_{i}\\}$ for $M$ and choose some element $m\in M$ with the property that
$\Delta_{M}(m)=\sum m_{i}\otimes f_{i}$ with some
$f_{i}\notin\mathcal{O}(\mathbb{G})_{\leq d+e}$ Similarly, choose a basis
$\\{n_{j}\\}$ for $N$ and write $\Delta_{N}(n)=\sum n_{j}\otimes h_{j}$ for
some $n\not=0$ with each $h_{j}$ an element of $\mathcal{O}(\mathbb{G})_{\leq
e}$. Since $\Delta_{M\otimes N}(m\otimes n)=\sum m_{i}\otimes n_{j}\otimes
f_{i}h_{j}$, the assumption that $M\otimes N=(M\otimes
N)_{\mathcal{O}(\mathbb{G})_{\leq d}}$ implies that each
$f_{i}h_{j}\in\mathcal{O}(\mathbb{G})_{\leq d}$. By the previous paragraph,
this implies that $f_{i}\in\mathcal{O}(\mathbb{G})_{\leq d+e}$. ∎

For various unipotent linear algebraic groups $\mathbb{U}$, we give a familiar
description of $\mathcal{O}(\mathbb{U})_{\mathcal{O}(GL_{N})_{\leq d}}$ (see
[12]).

######  Example 3.16.

Let $i:\mathbb{U}\hookrightarrow\mathbb{U}_{N}\subset GL_{N}$ be a smooth
closed embedding of a unipotent linear algebraic group $\mathbb{U}$ with the
property that the Lie algebra of $\mathbb{U}$ is a sum of root subspaces
$k\cdot X_{i,j}\in\mathfrak{gl}_{N}$. Then $\mathcal{O}(\mathbb{U})$ is the
tensor product of corresponding root subgroups $E_{i,j}$. For any $d\geq 0$,
$\mathcal{O}(\mathbb{U})_{\leq d}\subset\mathcal{O}(\mathbb{U})$ (as in
Theorem 3.6) can be identified as the sub-coalgebra of restrictions of
polynomial functions on $\mathbb{U}_{N}$ of total degree $\leq d$ with respect
to the grading which assigns degree 1 to each coordinate function $x_{i,j}$
with $i<j$.

As pointed out in [12, Ex3.7],
$f=2x_{1,3}-x_{1,2}x_{2,3}\in\mathcal{O}(\mathbb{U}_{3})$ has exponential
degree equal to 1, so that $\mathcal{E}_{\mathbb{U}_{3}}(1)$ is not contained
in $\mathcal{O}(\mathbb{U}_{3})_{\leq 1}$. A similar example shows that
$\mathcal{E}_{\mathbb{U}_{N}}(N-1)$ is not contained in
$\mathcal{O}(\mathbb{U}_{N})_{\leq N-1}$.

In [12, Prop3.8], a somewhat computational argument (using $k$ algebraically
closed) complements Proposition 3.12 by showing for
$\mathbb{G}=\mathbb{U}_{N},\ N\leq p$ that $\mathcal{E}(d)\ \subset\
\mathcal{O}(\mathbb{U}_{N})_{\leq(d+1)(N-1)-1}$; in particular,
$\mathcal{E}(d)$ is finite dimensional. A similar bound for
$\mathcal{E}_{\mathbb{U}}(d)$ might be established for
$\mathbb{U}\subset\mathbb{G}$ the unipotent radical of a “standard parabolic
subgroup” of a reductive algebraic group $\mathbb{G}$ of exponential type
provided that the nilpotence degree of $\mathbb{U}$ is less than $p$.

##  4\. Filtration of Hochschild cohomology

In Theorem 4.2, we utilize filtrations of $\mathcal{O}(\mathbb{G})$ by sub-
coalgebras to filter the Hochschild complex $\mathcal{C}^{H*}(\mathbb{G},M)$
for $\mathbb{G}$ with coefficients in a finite dimensional $\mathbb{G}$-module
$M$. This enables us to express rational cohomology groups
$H^{n}(\mathbb{G},M)$ for a finite dimensional $\mathbb{G}$-module $M$ as
(natural with respect to $M$) colimits of cohomology groups of cochain
complexes finite dimensional in each degree.

We recall that $H^{n}(\mathbb{G},M)$ can be computed as the cohomology of the
Hochschild cochain complex $\mathcal{C}H^{*}(\mathbb{G},M)$ for the coalgebra
$\mathcal{O}(\mathbb{G})$ and the
$(\mathcal{O}(\mathbb{G}),\mathcal{O}(\mathbb{G}))$ bi-comodule $M$ whose
right coaction is that associated to the (left) $\mathbb{G}$-action on $M$ and
whose left coaction is via the counit. In degree $n$,
$\mathcal{C}H^{n}(\mathbb{G},M)=M\otimes\mathcal{O}(\mathbb{G})^{\otimes n}$
and the differential
$\partial^{n}:\mathcal{C}H^{n}(\mathbb{G},M)\to\mathcal{C}H^{n+1}(\mathbb{G},M)$
has the form $\sum_{i=0}^{n+1}(-1)^{i}\partial_{i}^{n}$.

######  Proposition 4.1.

(see [21, I.4.14], [7, 3.1]) Let $C\subset\mathcal{O}(\mathbb{G})$ be a sub-
coalgebra of $\mathcal{O}(\mathbb{G})$ for some affine group scheme
$\mathbb{G}$ of finite type over $k$ and $M$ be a $C$-comodule. Then the
complex $\mathcal{C}H^{*}(\mathbb{G},M)$ restricts to a complex
$\mathcal{C}H^{*}(C,M)$ which in degree $n$ is given by the natural inclusion
$\mathcal{C}H^{n}(C,M)=M\otimes C^{\otimes n}\ \hookrightarrow\
M\otimes\mathcal{O}(\mathbb{G})^{\otimes n}=\mathcal{C}H^{n}(\mathbb{G},M)$.

Moreover, the Hochschild cohomology of $C$ with coefficients in the $C$-module
$M$ is given by

(4.1.1) $H^{n}(C,M)\quad\equiv\quad
H^{n}(\mathcal{C}H^{*}(C,M))\quad\simeq\quad Ext^{n}_{C\otimes
C^{op}}(M^{*},C),$

where $Ext^{n}_{C\otimes C^{op}}(-,-)$ is computed in the abelian category of
$C\otimes C^{op}$-comodules.

The explicit formula for this differential can be found in [21, I.4.14] and is
further discussed in the proof of Theorem 4.2. The fact that the differential
$\partial^{n}$ restricts to $\mathcal{C}H^{n}(C,M)\to\mathcal{C}H^{n+1}(C,M)$
is verified by inspection (using the fact that
$C\subset\mathcal{O}(\mathbb{G})$ is a sub-coalgebra).

One potentially useful role of ascending converging sequences of sub-
coalgebras of $\mathcal{O}(\mathbb{G})$ is given by the following theorem.

######  Theorem 4.2.

Let $\mathbb{G}$ be an affine group scheme of finite type over $k$ and let
$\\{C_{d}\\}$ be an ascending converging sequence of sub-coalgebras. For any
finite dimensional $\mathbb{G}$-module $M$, there is a natural isomorphism

(4.2.1)
$\varinjlim_{d}H^{n}(\mathbb{G},M)\quad\stackrel{{\scriptstyle\sim}}{{\to}}\quad
H^{n}(C_{d},M_{C_{d}})).$

In the special case $M=k$, this natural isomorphism gives an isomorphism of
$k$-algebras

$\varinjlim_{d}H^{n}(C_{d},k))\quad\stackrel{{\scriptstyle\sim}}{{\to}}\quad
H^{n}(\mathbb{G},k).$

###### Proof.

The differential of $\mathcal{C}H^{*}(\mathbb{G},M)$ is defined as
$\partial^{n}=\sum_{i=0}^{n+1}\partial^{n}_{i}$ as in [21, I.4.14]. Recall
that $\partial^{n}_{0}(m\otimes f_{1}\cdots\otimes f_{n})\ =\
\Delta_{M}(m)\otimes f_{1}\otimes\cdots f_{n}$. If we replace $M$ by $M_{C}$
so that $\Delta_{M}$ restricts to $M_{C}\to M_{C}\otimes C$, then
$\partial^{n}_{0}$ so defined restricts to a map
$\mathcal{C}H^{n}(C,M_{C})\to\mathcal{C}H^{n+1}(C,M_{C})$. A similar argument
(replacing $\Delta_{M}$ in the previous argument by $\Delta_{C}:C\to C\otimes
C$) shows that
$\partial^{n}_{i}:\mathcal{C}H^{n}(\mathbb{G},M)\to\mathcal{C}H^{n+1}(\mathbb{G},M)$
for $i>0$ restricts to
$\partial^{n}_{i}:\mathcal{C}H^{n}(C,M)\to\mathcal{C}H^{n+1}(C,M)$.

Whenever $M$ is finite dimensional, $M=M_{C_{d}}$ for $d\gg 0$ by Proposition
1.11. Thus, for $M$ a finite dimensional $\mathbb{G}$-module, the complex
$\mathcal{C}H^{*}(\mathcal{O}(\mathbb{G}),M)$ is the colimit of the cochain
complexes $\mathcal{C}H^{*}(C_{d},M_{C_{d}})$.

The algebra structure of $H^{\bullet}(\mathbb{G},k)$ is given by the evident
tensor product identification
$\mathcal{C}H^{m}(\mathbb{G},k)\otimes\mathcal{C}H^{n}(\mathbb{G},k)\to\mathcal{C}H^{m+n}(\mathbb{G},k)$
(see [21, I.4.19].) This identification restricts to each
$\mathcal{C}H^{*}(C_{d},k)$, inducing the map of (graded commutative)
$k$-algebras $H^{*}(C_{d},k)\to H^{*}(\mathbb{G},k)$ for each $d\geq 0$.

Thus, (4.2.1) follows from (4.1.1). ∎

The following immediate corollary of Theorem 4.2 should be contrasted to our
lack of understanding (even in the case of $\mathbb{G}=\mathbb{U}_{N}$; see
[13]) of whether or not a non-nilpotent class $\zeta\in
H^{\bullet}(\mathbb{G},k)$ always has non-nilpotent restriction to some
$H^{*}(\mathbb{G}_{(r)},k)$.

######  Corollary 4.3.

Adopt the notation and hypotheses of Theorem 4.2. If $\zeta\in
H^{\bullet}(\mathbb{G},k)$ is not nilpotent, then for all $d\gg 0$ the class
$\zeta$ is the image under $H^{*}(C_{d},k)\to H^{*}(\mathbb{G},k)$ of a non-
nilpotent class in $H^{*}(C_{d},k).$

In the following proposition we observe that the Andersen-Jantzen spectral
sequence converging to $H^{*}(\mathbb{G},M)$ (see [21, I.9.14]) can be adapted
to investigate $H^{*}(C,M_{C})$ for a sub-coalgebra
$C\subset\mathcal{O}(\mathbb{G})$.

######  Proposition 4.4.

Let $\mathbb{G}$ be a linear algebraic group,
$C\subset\mathcal{O}(\mathbb{G})$ a sub-coalgebra containing the unit of
$\mathcal{O}(\mathbb{G})$, and $M$ a $C$-comodule. Using powers of the
augmentation ideal of $\mathcal{O}(\mathbb{G})$, we obtain a filtration
$\\{\mathcal{C}H^{*}(C,M)_{i}\\}$ of the Hochschild complex
$\mathcal{C}H^{*}(C,M)$ leading to a convergent spectral sequence of the form

(4.4.1)
$E_{1}^{i,j}=H^{i+j}(\mathcal{C}H^{*}(C,M)_{i}/\mathcal{C}H^{*}(C,M)_{i+1})\
\Rightarrow\ H^{i+j}(C,M).$

In the special case $M=k$ , each
$\mathcal{C}H^{*}(C,k)_{n}/\mathcal{C}H^{*}(C,k)_{n+1}$ is a subcomplex of
$\mathcal{C}H^{*}(\mathbb{G},k)_{n}/\mathcal{C}H^{*}(\mathbb{G},k)_{n+1}$;
this latter complex can be identified with the Hochschild complex of a
commutative group scheme the dual of whose coordinate algebra is isomorphic to
the commutative $k$-algebra $k[u_{s}^{t},0\leq s<n,1\leq t\leq
m]/((u_{s}^{t})^{p})$ where $m=dim(\mathfrak{g})$.

###### Proof.

Let $I\subset\mathcal{O}(\mathbb{G})$ denote the augmentation ideal, the
kernel of the counit $\epsilon:\mathcal{O}(\mathbb{G})\to k$. As in [21,
I.2.4], for any sub-coalgebra $C\subset\mathcal{O}(\mathbb{G})$ (containing
the unit $k\to\mathcal{O}(\mathbb{G}))$, we may write $C=k\oplus I_{C}$ where
$I_{C}=I\cap C$. Powers of $I$ ($I^{0}=\mathcal{O}(\mathbb{G}),I^{1}=I,\cdots
I^{n}\cdots$) yield a decreasing sequence of ideals of
$\mathcal{O}(\mathbb{G})$. Since $\mathbb{G}$ is irreducible,
$\bigcap_{n}I^{n}=0$. We define $I_{C}^{n}=I_{C}\cap I^{n}$, determining a
decreasing filtration $\\{I_{C}^{n}\\}$ of $C$. As in [21, I.9.12], for a
$C$-comodule $M$ this determines a decreasing filtration
$\\{\mathcal{C}H^{*}(C,M)_{n}\\}$ of the Hochschild complex
$\mathcal{C}H^{*}(C,M)$ with $\mathcal{C}H^{*}(C,M)_{n}\ \subset\
\mathcal{C}H^{*}(C,M)$ given in degree $i$ by

$\mathcal{C}H^{i}(C,M)_{n}\ \equiv\ \sum M\otimes
I_{C}^{a(1)}\otimes\cdots\otimes I_{C}^{a(i)},\quad\sum_{s=1}^{i}a(i)\geq n.$

As in [21, I.9.12], the filtration of $\\{\mathcal{C}H^{*}(C,M)_{n}\\}$ on
$\mathcal{C}H^{*}(C,M)$ determines a convergent spectral sequence of the form
(4.4.1).

In the special case $C=\mathcal{O}(\mathbb{G})$ and $M=k$, the $n$-th graded
piece of this filtered complex
$\mathcal{C}H^{*}(\mathbb{G},k)_{n}/\mathcal{C}H^{*}(\mathbb{G},k)_{n+1}$ can
be identified with the Hochschild complex
$\mathcal{C}H^{*}((\underline{\mathfrak{g}})_{(n)},k)$ by [21, I.9.12(3)],
where $(\underline{\mathfrak{g}})_{(n)}\simeq(\mathbb{G}_{a(n)})^{\times m}$.
The group algebra of $(\underline{\mathfrak{g}})_{(n)}$ isomorphic to the
commutative $k$-algebra $k[u_{s}^{t},0\leq s<n,1\leq t\leq
m]/((u_{s}^{t})^{p})$. ∎

##  5\. Categories of finite coheight and cofinite $\mathbb{G}$-modules

In this section, we introduce two subcategories $CoHt(\mathbb{G})$ and
$CoFin(\mathbb{G})$ of $\mathbb{G}$-modules, those which are finite coheight
and those which are cofinite. These classes are complementary: only finite
dimensional $\mathbb{G}$-modules are both finite coheight and cofinite.

######  Definition 5.1.

Let $\mathbb{G}$ be an affine group scheme and let $M$ be a
$\mathbb{G}$-module. Then $M$ is said to have finite coheight if there exists
some finite dimensional subspace $X\ \subset\ \mathcal{O}(\mathbb{G})$ such
that $M$ is an $X$-comodule.

######  Remark 5.2.

Let $(\mathbb{G},\mathcal{E})$ be a linear algebraic group of exponential type
and $M$ a $\mathbb{G}$-module of finite coheight with coproduct
$\Delta_{M}:M\to M\otimes X$ for some finite dimensional subspace
$X\subset\mathcal{O}(\mathbb{G})$. Then $M$ has exponential degree $\leq d$
(see Definition 2.3) provided that the image of the composition $X\ \subset\
\mathcal{O}(\mathbb{G})\stackrel{{\scriptstyle\mathcal{E}}}{{\to}}\mathcal{O}(\mathcal{N}_{p}(\mathfrak{g}))\otimes
k[t]$ lies in $\mathcal{O}(\mathcal{N}_{p}(\mathfrak{g}))\otimes k[t]_{\leq
d}$.

######  Proposition 5.3.

Let $\mathbb{G}$ be an affine group scheme and let $CoHt(\mathbb{G})\ \subset\
Mod(\mathbb{G})$ be the full subcategory of $\mathbb{G}$-modules of finite
coheight.

  1. (1)

Any finite dimensional $\mathbb{G}$-module has finite coheight.

  2. (2)

$M\otimes N$ has finite coheight if the $\mathbb{G}$-modules $M,\ N$ have
finite coheight.

  3. (3)

Any direct summand of $M$ has finite coheight if the $\mathbb{G}$-module $M$
has finite coheight.

  4. (4)

If $X\subset\mathcal{O}(\mathbb{G})$ is a finite dimensional subspace, then
any filtering colimit $\varinjlim_{i}M_{i}$ of $X$-comodules $\\{M_{i},\ i\in
I\\}$ has finite coheight.

  5. (5)

$CoHt(\mathbb{G})$ is an abelian subcategory of $Mod(\mathbb{G})$.

###### Proof.

Assertion (1) follows from Proposition 1.11(1) and assertion (2) is given by
(3.14.2 of Proposition 3.14. Assertion (3) follows directly from Lemma 1.3(1)
and assertion (4) follows from the observation that a directed colimit of
$X$-comodules is itself an $X$-comodule.

By Theorem 1.5, $Mod(\mathbb{G},X)\subset Mod(\mathbb{G})$ is an abelian
subcategory for any subspace $X\subset\mathcal{O}(\mathbb{G})$. Thus, the fact
that $CoHt(\mathbb{G})$ is an abelian subcategory of $Mod(\mathbb{G})$ follows
by identifying $CoHt(\mathbb{G})$ as the colimit of subcategories
$Mod(\mathbb{G},X_{d})$ for some ascending, converging sequence of finite
dimensional subspaces $\\{X_{d}\\}$ of $\mathcal{O}(\mathbb{G})$. ∎

######  Remark 5.4.

The subcategory $CoHt(\mathbb{G})\quad\subset\ Mod(\mathbb{G})$ is typically
not closed under extensions. For example, consider the 2-dimensional
representation $V$ of $\mathbb{G}_{a}$ with $k$-basis $\\{v,w\\}$
$\mathbb{G}_{a}$-action given by the coaction $V\to V\otimes k[t]$ sending $v$
to $v\otimes 1+w\otimes t$ and $w$ to $w\otimes 1$. The $r$-th Frobenius twist
of $V^{(r)}$ of $V$ has basis $\\{v(r),w(r)\\}$ with coaction sending $v(r)$
to $v(r)\otimes 1+w(r)\otimes t^{r}$ and $w(r)$ to $w(r)\otimes 1$. Then $M\
\equiv\ \bigoplus_{r\geq 0}V^{(r)}$ is not of finite coheight. On the other
hand, $M$ is an extension of a $\mathbb{G}_{a}$-module of coheight 0 by
another $\mathbb{G}_{a}$-module of coheight 0.

One can view the following cofinite condition as a strengthening of the
condition that the fixed point subspace $M^{\mathbb{G}}\equiv\\{m\in
M:\Delta_{M}(m)=m\otimes 1\\}\subset M$ be finite dimensional.

######  Definition 5.5.

Let $\mathbb{G}$ be an affine group scheme. We define a $\mathbb{G}$-module
$M$ to be cofinite if there exists some ascending converging sequence
$\\{X_{i}\\}$ of finite dimensional subspaces of $\mathcal{O}(\mathbb{G})$
such that each $M_{X_{i}}$ is a finite dimensional $\mathbb{G}$-module.

This condition is equivalent to the condition that each $M_{X_{i}}$ is a
finite dimensional $\mathbb{G}$-module for any ascending converging sequence
$\\{X_{i}\\}$ of finite dimensional subspaces of $\mathcal{O}(\mathbb{G})$.

To justify the assertion equivalence of Definition 5.5, assume that
$\\{X_{i}\\},\ \\{Y_{j}\\}$ are ascending converging sequence $\\{X_{i}\\}$ of
finite dimensional subspaces of $\mathcal{O}(\mathbb{G})$ and that $M$
satisfies the condition that each $M_{X_{i}}$ is a finite dimensional
$\mathbb{G}$-module. Since each $Y_{j}$ is finite dimensional, there exists
some $X_{i(j)}$ with $Y_{j}\subset X_{i(j)}\subset\mathcal{O}(\mathbb{G})$
Then $M_{Y_{j}}$ is contained in $M_{X_{i(j)}}$ and thus is finite
dimensional.

We establish various properties of cofinite $\mathbb{G}$-modules.

######  Proposition 5.6.

Let $\mathbb{G}$ be an affine group scheme and let $CoFin(\mathbb{G})\
\subset\ Mod(\mathbb{G})$ be the full subcategory of cofinite
$\mathbb{G}$-modules.

  1. (1)

If $M$ is both of finite coheight and cofinite, then $M$ is finite
dimensional.

  2. (2)

If $0\to M\to E\to N\to 0$ is exact in $Mod(\mathbb{G})$ with both $M$ and $N$
cofinite, then $E$ is also cofinite.

  3. (3)

Any direct summand of a cofinite $\mathbb{G}$-module is also cofinite.

  4. (4)

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

Furthermore, if $\mathbb{G}$ is a linear algebraic group satisfying the
condition of Proposition 3.15, if $M$ is cofinite, and if $N$ is finite
dimensional, then $M\otimes N$ is cofinite.

###### Proof.

If $\\{X_{i}\\}$ is an ascending converging sequence of finite dimensional
subspaces of $\mathcal{O}(\mathbb{G})$ and if $M$ has finite coheight, then
$M=M_{X_{i}},i\gg 0$; if $M$ is cofinite, $M_{X_{i}}$ is finite dimensional.
Together, these statements imply assertion (1). Assertion (2) follows from the
left exactness of $(-)_{X}$.

Assertion (3) follows from Lemma 1.3. Any $\mathbb{G}$-module $M$ admits an
embedding $M\ \subset\ M^{tr}\otimes\mathcal{O}(\mathbb{G})^{R}$, where
$M^{tr}$ is the underlying vector space of $M$ equipped with the trivial
$\mathbb{G}$-action. Since $\mathcal{O}(\mathbb{G})^{R}$ is cofinite by
Theorem 3.4(2),(4), $M^{tr}\otimes\mathcal{O}(\mathbb{G})$ is also cofinite
provided $M$ is finite dimensional.

To prove final statement, choose $e$ such that
$N=N_{\mathcal{O}(\mathbb{G})_{\leq e}}$. If $M\otimes N$ is not cofinite,
then each $(M\otimes N)_{\mathcal{O}(\mathbb{G})_{\leq d}}$ is infinite
dimensional. On the other hand, this implies that
$M_{\mathcal{O}(\mathbb{G})_{\leq d+e}}$ must be infinite dimensional by
Proposition 3.15. Thus, $M$ is not cofinite. ∎

In the following example, we see that the tensor product of cofinite modules
need not be cofinite and that $CoFin(\mathbb{G})\ \subset\ Mod(\mathbb{G})$ is
not closed upon taking quotients.

######  Example 5.7.

Let $\mathbb{G}$ be a linear algebraic group over $k$. Then $M\otimes
k[\mathbb{G}]^{R}$ is not cofinite if $M$ is infinite dimensional (even if $M$
is cofinite). Namely, $M\otimes k[\mathbb{G}]^{R}\ \simeq\ M^{tr}\otimes
k[\mathbb{G}]^{R}$ satisfies $(M\otimes
k[\mathbb{G}]^{R})_{\leq\mathcal{O}(\mathbb{G})_{\leq
0}}=H^{0}(\mathbb{G},M\otimes k[\mathbb{G}]^{R})=M$ which is infinite
dimensional.

Let $\mathbb{G}$ be semi-simple and let $S_{\lambda},\ S_{\mu}$ be irreducible
$\mathbb{G}$ modules whose restrictions to $\mathbb{G}_{(1)}$ are irreducible
and such that

$Ext_{\mathbb{G}}^{1}(S_{\lambda},S_{\mu})\ =\
H^{1}(\mathbb{G},S_{\lambda}^{*}\otimes S_{\mu})\ =\
Ext_{\mathbb{G}}^{1}(k,S_{\lambda}^{*}\otimes S_{\mu})\ \not=\ 0.$

Choose a non-trivial extension $0\to S_{\lambda}^{*}\otimes S_{\mu}\to
E_{0}\to k\to 0$. Then $M\ \equiv\ \bigoplus_{r\geq 0}E_{0}^{(r)}$ is
cofinite, but the evident quotient $Q\equiv\bigoplus_{r\geq 0}k$ of $M$ is not
cofinite.

##  6\. Growth of cofinite $\mathbb{G}$-modules

In this section, we introduce the growth of a cofinite $\mathbb{G}$-module
$M$. This is defined in terms of an ascending converging sequence
$\\{C_{d}\\}$ of finite dimensional sub-coalgebras of
$\mathcal{O}(\mathbb{G})$. Much of our focus is on cofinite
$\mathbb{G}$-modules with polynomial growth, although we give examples in
Examples 6.11 and 6.12 of $\mathbb{G}$-modules with logarithmic growth and
with exponential growth. We give a few, straight-forward computations of
growth leading to expected answers.

We remind the reader that the dimension of $k[x_{1},\ldots,x_{n}]_{d}$ of the
vector space of polynomials in $n$ variables of total degree $d$ equals
$n+d-1\choose d$ which is a polynomial in $d$ of degree $n-1$; the dimension
of $k[x_{1},\ldots,x_{n}]_{\leq d}$ of the vector space of polynomials in
$n$-variables of total degree less than or equal to $d$ equals $n+d\choose d$
which is a polynomial in $d$ of degree $n$. In particular

${n+d\choose d}\quad=\quad\frac{(d+1)\cdot(d+2)\cdot\cdots\cdot(d+n)}{n!}.$

######  Definition 6.1.

Let $\phi,\psi:\mathbb{N}\to\mathbb{N}$ be (not necessarily strictly)
increasing functions on non-negative integers with values in positive integers
such that $\varinjlim_{n}\phi(n)=\infty=\varinjlim_{n}\psi(n)$.

We say that $\phi,\psi$ have equal growth, writing $n\mapsto\phi(n)\ \approx\
n\mapsto\psi(n)$, provided that $\varinjlim_{n}\frac{\phi(n)}{\psi(n)}\
=\alpha\ $ for some positive number $\alpha$.

We say that such a sequence $\phi$ has polynomial growth equal to $m$ if
$n\mapsto\phi(n)\ \approx\ n\mapsto n^{m}.$

######  Remark 6.2.

We could consider a refinement of the equivalence relation $\phi\ \approx\
\psi$ of Definition 6.1 by requiring $\varinjlim_{n}\frac{\phi(n)}{\psi(n)}\
=1$. In much of what follows, it is more convenient to look at the coarser
equivalence relation of Definition 6.1.

######  Definition 6.3.

Let $\mathbb{G}$ be an affine group scheme of finite type over $k$ and let
$\\{C_{d}\\}$ be an ascending converging sequence of finite dimensional sub-
coalgebras of $\mathcal{O}(\mathbb{G})$.

We say that a cofinite $\mathbb{G}$-module $M$ has polynomial growth of degree
$m$ with respect to $\\{C_{d}\\}$ if

$d\mapsto dim(M_{C_{d}})\quad\approx\quad d\mapsto d^{m}.$

######  Example 6.4.

Take $\mathbb{G}$ to be $\mathbb{G}_{a}$ with
$\mathcal{O}(\mathbb{G}_{a})=k[t]$. Consider the action of $\mathbb{G}_{a}$ on
the affine variety $\mathbb{A}^{m}$,
$\mathbb{G}_{a}\times\mathbb{A}^{m}\to\mathbb{A}^{m}$ given functorially by
sending $(a\in\mathbb{G}_{a}(A),v_{1},\ldots v_{m}\in\mathbb{A}^{m}(A)=A^{m})$
to $(v_{1}+a,\ldots,v_{m}+a)$. This determines a $\mathbb{G}_{a}$ -module
structure on $k[A^{m}]$ given by the coaction

$\Delta_{\mathbb{A}^{m}}:k[x_{1},\ldots,x_{m}]\mapsto
k[x_{1},\ldots,x_{m}]\otimes k[t],\quad x_{i}\mapsto x_{i}\otimes 1+1\otimes
t.$

Then $k[\mathbb{A}^{m}]_{\mathcal{O}(\mathbb{G}_{a})_{\leq d}}$ has dimension
$m+d\choose d$ (spanned by monomials in the $\\{x_{i}\\}$ of total degree
$\leq d$) which is a polynomial in $d$ of degree $m$, so that
$k[\mathbb{A}^{m}]$ has polynomial growth of degree $m$ with respect to
$\\{\mathcal{O}(\mathbb{G}_{a})_{\leq d}\\}$.

######  Lemma 6.5.

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 $e$

###### Proof.

Namely, if $\varinjlim_{m}\phi(m)/m^{e}=\alpha$ and
$\varinjlim_{n}\psi(n)/n^{f}=\beta$, then

$\varinjlim_{n}(\phi\circ\psi)(n)/n^{e\cdot f}\ =\
\varinjlim_{n}\\{\phi(\psi(n))/\psi(n)^{e}\cdot(\psi(n)/n^{f})^{e}\\}\ =\
\alpha\cdot\beta^{e}.$

∎

######  Corollary 6.6.

By Proposition 3.13 and Lemma 6.5, the growth of
$d\mapsto\mathcal{O}(\mathbb{G})_{\leq d}$ is independent of a choice of
closed embedding $\mathbb{G}\to GL_{N}$.

An explanation of the independence with respect to a choice of closed
embedding $\mathbb{G}\to GL_{N}$ of the growth of
$d\mapsto\mathcal{O}(\mathbb{G})_{\leq d}$ is given by the following
proposition. In what follows, we denote by
$\mathfrak{m}_{\mathbb{G}}\subset\mathcal{O}(\mathbb{G})$ the maximal ideal at
the identity of $\mathbb{G}$. The following proposition incorporates the
computation of Proposition 3.10 asserting that

(6.6.1) $d\mapsto dim(\mathcal{O}(GL_{N})_{\leq d})\ \approx\ d^{N^{2}},\quad
d\mapsto dim(\mathcal{O}(SL_{N})_{\leq d})\ \approx\ d^{N^{2}-1}.$

######  Proposition 6.7.

Let $\mathbb{G}$ be a linear algebraic group equipped with a smooth, closed
embedding into some $GL_{N}$, and let $\mathfrak{g}$ denote the Lie algebra of
$\mathbb{G}$. Then

(6.7.1) $d\mapsto dim(\mathcal{O}(\mathbb{G})_{\leq d})\quad\approx\quad
d\mapsto d^{dim(\mathfrak{g})}.$

###### Proof.

Since $\mathcal{O}(\mathbb{M}_{N,N})\subset\mathcal{O}(GL_{N})$ sends
$\mathcal{O}(\mathbb{M}_{N,N})_{1}$ to a subspace of $\mathfrak{m}_{GL}$
inducing an isomorphism
$\mathcal{O}(\mathbb{M}_{N,N})_{1}\stackrel{{\scriptstyle\sim}}{{\to}}\mathfrak{m}_{GL}/\mathfrak{m}_{GL}^{2}$,
we conclude that $d\mapsto\mathcal{O}(M_{N,N})_{\leq d}$ has growth equal to
$d\mapsto\mathcal{O}(GL_{N})/\mathfrak{m}_{GL}^{d}\ \approx\ d^{N^{2}}$.
Because
$\mathcal{O}(GL_{N})/\mathfrak{m}_{GL}^{d}\to\mathcal{O}(\mathbb{G})/\mathfrak{m}_{\mathbb{G}}^{d}$
is surjective, we conclude that $d\mapsto A(\mathbb{G})_{\leq d}$ (where
$A(\mathbb{G})_{\leq d}$ is the image of $\mathcal{O}(\mathbb{M}_{N,N})_{\leq
d}\to\mathcal{O}(\mathbb{G})$) has growth at least
$d\mapsto\mathcal{O}(\mathbb{G})/\mathfrak{m}_{\mathbb{G}}^{d}\ \approx\
d\mapsto d^{n}$ where $n=dim(\mathfrak{g})$.

Choose $\ell$ to be the least positive integer such that the composition
$\mathcal{O}(GL_{N})_{\leq 1}\to\mathcal{O}(\mathbb{G})_{\leq
1}\to\mathcal{O}(\mathbb{G})/m_{\mathbb{G}}^{d}$ is injective. Then
$\mathcal{O}(\mathbb{G})_{\leq d}$ maps injectively to
$\mathcal{O}(\mathbb{G})/m_{\mathbb{G}}^{\ell\cdot d}$, so that Lemma 6.5
implies that the growth of
$d\mapsto\mathcal{O}(\mathbb{G})/\mathfrak{m}_{\mathbb{G}}^{d}$ is not greater
than $d\mapsto d^{n}$. ∎

As a corollary of Proposition 6.7, we obtain the following.

######  Proposition 6.8.

Let $\mathbb{G}$ be a linear algebraic group equipped with a smooth, closed
embedding into some $GL_{N}$. The $G$-modules $k[\mathbb{G}]^{R},\
k[\mathbb{G}]^{L},\ k[\mathbb{G}]^{Ad}$ each have polynomial growth of degree
equal to $dim(\mathfrak{g})$ with respect to $\\{\mathcal{O}(\mathbb{G})_{\leq
d}\\}$.

###### Proof.

Since the $\mathcal{O}(\mathbb{G})$-module $k[\mathbb{G}]^{R}$ equals
$\mathcal{O}(\mathbb{G})$ with comodule structure determined by the coaction
given by the coproduct, the growth of $k[\mathbb{G}]^{R}$ with respect to
$\\{\mathcal{O}(\mathbb{G})_{\leq d}\\}$ equals $dim(\mathfrak{g})$ by
Proposition 6.7.

Consider the anitipode $\sigma:\mathcal{O}(GL_{N})\to\mathcal{O}(GL_{N})$, the
map on coordinate algebras induced by $(-)^{-1}:GL_{N}\to GL_{N}$. For any
invertible $N\times N$-matrix $A$, Kramer’s rule tell us that $A$ has inverse
$B^{-1}=(b_{i,j})$ where $b_{i,j}=(-1)^{i+j}det(A_{i,j})\cdot det(A)^{-1}$
where $A_{i,j}$ is the minor of $A$ at $(i,j)$. Thus, $\sigma(x_{i,j})$ is the
product of an $N-1$ degree polynomial $det(\\{x_{s,t},s\not=i,y\not=j\\})$ by
$det(\\{x_{i,j})^{-1}$ (which is given filtration degree $N$). Consequently,
the algebra homomorphism $\sigma:\mathcal{O}(GL_{N})\to\mathcal{O}(GL_{N})$
multiplies filtration degree (with respect to
$\\{\mathcal{O}(\operatorname{GL}\nolimits_{N})_{\leq d}\\}$) by $2N-1$. The
coaction for $k[GL_{N}]^{L}$ is the composition

$\tau\circ(\sigma_{GL_{N}}\otimes
id_{k[GL_{N}]^{L}})\circ\Delta_{GL_{N}}:k[GL_{N}]^{L}\ \to\
\mathcal{O}(GL_{N})\otimes k[GL_{N}]^{L}\to\ $ $\mathcal{O}(GL_{N})\otimes
k[GL_{N}]^{L}\ \to\ k[GL_{N}]^{L}\otimes\mathcal{O}(GL_{N}).$

Thus, $dim((k[GL_{N}]^{L})_{\mathcal{O}(GL_{N})_{\leq(2N-1)d}})\ =\
dim(\mathcal{O}(GL_{N})_{\leq d})$. We conclude that
$d\mapsto(k[GL_{N}]^{L})_{\mathcal{O}(GL_{N})_{\leq d}}\ \approx\ d\mapsto
d^{N^{2}}$ has growth $N^{2}$ by applying Lemma 6.5.

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

$\mu_{1,3}\circ(\sigma_{GL_{N}}\otimes 1\otimes
1)\circ(1\circ\Delta_{GL_{N}})\circ\Delta_{GL_{N}}:k[\operatorname{GL}\nolimits_{N}]^{Ad}\
\to\ \mathcal{O}(GL_{N})\otimes k[GL_{N}]^{Ad}\ \to\ $
$\mathcal{O}(GL_{N})\otimes k[GL_{N}]^{Ad}\otimes\mathcal{O}(GL_{N})\ \to\
\mathcal{O}(GL_{N})\otimes k[GL_{N}]^{Ad}\otimes\mathcal{O}(GL_{N})\ \to\
k[GL_{N}]^{Ad}\otimes\mathcal{O}(GL_{N}).$

If we write this coaction as $\Delta_{Ad}(x_{i,j})=\sum_{s,t}x_{s,t}\otimes
f_{s,t}^{i,j}$, then we see 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 $\leq 2N$. As for $k[GL_{N}]^{L}$, we
conclude that $d\mapsto(k[GL_{N}]^{Ad})_{\mathcal{O}(GL_{N})_{\leq d}}$ has
growth $N^{2}$ by applying Lemma 6.5.

Since the restriction $i^{*}:\mathcal{O}(GL_{N})\to\mathcal{O}(\mathbb{G})$ is
a surjective map of Hopf algebras and since $\mathcal{O}(\mathbb{G})_{\leq d}$
is defined to be $i^{*}(\mathcal{O}(\mathbb{G})_{\leq d})$, we may apply
$i^{*}$ to the above arguments for $GL_{N}$ to conclude the corresponding
statements for $\mathbb{G}$. ∎

######  Example 6.9.

Let $P$ be a polynomial representation of $GL_{N}$ of dimension $q$ 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 $M_{\mathcal{O}(GL_{N})_{n\cdot
s}}\ =\ M_{\mathcal{O}(M_{N,N})_{n\cdot s}}\ =\ S^{n}(P)$. Thus,

$d\mapsto dim(M_{\mathcal{O}(GL_{N})_{\leq d\cdot s}})\quad\approx\quad
d\mapsto dim(S^{\leq d}(P))\quad\approx\quad d\mapsto d^{q}.$

Thus, by Lemma 6.5, $d\mapsto dim(M_{\mathcal{O}(GL_{N})_{\leq
d}})\quad\approx\quad d\mapsto d^{q},$ so that $S^{*}(P)$ has polynomial
growth equal to $q$ with respect to $\\{\mathcal{O}(GL_{N})_{\leq d}\\}$.

As another class of examples, we consider homogeneous varieties (see, for
example, [2, II.6]). The projection map
$\pi:\mathbb{G}\to\mathbb{G}/\mathbb{H}$ is $\mathbb{G}$-equivariant, so that
$\mathcal{O}(\mathbb{G}/\mathbb{H})\subset k[\mathbb{G}]^{R}$ is a map of
$\mathbb{G}$-modules. Recall that $\pi$ is an open mapping provided
$\mathbb{H}$ is closed in $\mathbb{G}$. Since we are interested in the growth
of affine varieties, we shall require that $\mathbb{G}/\mathbb{H}$ is affine.
If both $\mathbb{G}$ and $\mathbb{H}$ are reductive, then
$\mathbb{G}/\mathbb{H}$ is affine. Moreover, if the projection
$\pi:\mathbb{G}\to\mathbb{G}/\mathbb{H}$ is locally in the étale topology a
product projection, then $\pi$ is a smooth map.

As discussed in [22, Thm3.14], we can choose arbitrarily “small” étale
neighorhoods of the closed point $[\mathbb{H}]\in\mathbb{G}/\mathbb{H}$ to be
“standard étale maps” $j:V\to\mathbb{G}/\mathbb{H}$. Such a map has the
following form: $U\equiv j(V)\subset\mathbb{G}/\mathbb{H}$ is an affine open
neighborhood of $[H]\in\mathbb{G}/\mathbb{H}$; $\tilde{U}\to U$ is given by
defining $\mathcal{O}(\tilde{U})=\mathcal{O}(U)[x]/h(x)$ for some monic
polynomial $h(x)\in\mathcal{O}(U)[x]$; and $V\subset\tilde{U}$ is an affine
open subset of $\tilde{U}$ such that $h(x),h^{\prime}(x)$ have no common zero.

######  Proposition 6.10.

Let $f:\mathbb{H}\hookrightarrow\mathbb{G}$ be a closed embedding of smooth
linear algebraic groups with the properties that the projection
$\pi:G\to\mathbb{G}/\mathbb{H}$ is locally in the étale topology a product
projection and that $\mathbb{G}/\mathbb{H}$ affine. These conditions are
satisfied if either both $\mathbb{H}$ and $\mathbb{G}$ are reductive (see [5,
Lem4.1]) or if $\mathbb{G}$ is unipotent (see [5, Prop1.15]). Then

$d\mapsto
dim(\mathcal{O}(\mathbb{G}/\mathbb{H})_{\mathcal{O}(\mathbb{G})_{\leq
d}})\quad\approx\quad d\mapsto d^{n},\quad n\equiv
dim(\mathfrak{g}/\mathfrak{h}).$

###### Proof.

We choose a standard étale neighborhood $j:V\to\mathbb{G}/\mathbb{H}$ of
$[\mathbb{H}]\in\mathbb{G}/\mathbb{H}$ so that
$\pi:\mathbb{G}\times_{\mathbb{G}/\mathbb{H}}V\to V\ \simeq\ \mathbb{H}\times
V$. Let $U=j(V)\subset\mathbb{G}/\mathbb{H}$ be an affine open subset which we
may assume to be a principal affine open, with
$\mathcal{O}(U)=\mathcal{O}(\mathbb{G}/\mathbb{H})[f^{-1}]$ for some
$f\in\mathcal{O}(\mathbb{G}/\mathbb{H})$ and let $e$ be minimal such that
$f\in\mathcal{O}(\mathbb{G}/\mathbb{H})_{\leq e}$. We define
$\mathcal{O}(U)_{\leq d}$ to be the subspace of fractions $h/f^{j}$ such that
$h\in\mathcal{O}(\mathbb{G}/\mathbb{H})_{\leq d-j\cdot e}$. As in the proof of
Theorem 3.6(4), we conclude that $dim(\mathcal{O}(U)_{\leq d})\leq
dim(\mathcal{O}(\mathbb{G}/\mathbb{H})_{\leq
d})+dim(\mathcal{O}(\mathbb{G}/\mathbb{H})_{\leq d-e})$, so that $d\mapsto
dim(\mathcal{O}(U)_{\leq d})\ \approx\ d\mapsto
dim(\mathcal{O}(\mathbb{G}/\mathbb{H})_{\leq d})$.

Let $h(x)\in\mathcal{O}(U)[x]$ be a monic polynomial of degree $b$ and define
$\tilde{U}\ =\ $
$\operatorname{Spec}\nolimits\mathcal{O}(U)[x]/h(x)$. We define
$(\tilde{U})_{\leq d}$ to consist of elements represented by polynomials
$a_{0}+a_{1}x+\cdots+a_{b-1}x^{b-1}\in\mathcal{O}(U)[x]$ with the property
that there exist non-negative numbers $d_{i}$ with $\sum d_{i}\leq d$ and such
that each $a_{i}\in(\mathcal{O}(U))_{\leq d_{i}}$. Finally, we give a
filtration on $\mathcal{O}(V)$ for $V$ a principal affine open subset of
$\tilde{U}$ as argued for $U\subset\mathbb{G}/\mathbb{H}$.

Thus, with these definitions, we conclude that

$d\mapsto dim(\mathcal{O}(\mathbb{G}/\mathbb{H})_{\leq d})\quad\approx\quad
d\mapsto dim(\mathcal{O}(U)_{\leq d})\quad\approx\quad d\mapsto
dim(\mathcal{O}(\tilde{U})_{\leq d})$ $\quad\approx\quad d\mapsto
dim(\mathcal{O}(V)_{\leq d}).$

A completely parallel argument shows that

$d\mapsto d^{dim(\mathfrak{g})}\quad\approx\quad d\mapsto
dim(\mathcal{O}(\mathbb{G})_{\leq d})\quad\approx\quad d\mapsto
dim(\mathcal{O}(\pi^{-1}(U))_{\leq d})$ $\quad\approx\quad d\mapsto
dim(\mathcal{O}(\pi^{-1}(\tilde{U}))_{\leq d})\quad\approx\quad d\mapsto
dim(\mathcal{O}(\pi^{-1}(V))_{\leq d}),$

where $\pi^{-1}(\tilde{U})\ \equiv\
\mathbb{G}\times_{\mathbb{G}/\mathbb{H}}\tilde{U}$ and $\pi^{-1}(V)\ \equiv\
\mathbb{G}\times_{\mathbb{G}/\mathbb{H}}V$.

We observe that
$\mathcal{O}(\pi^{-1}(V))\simeq\mathcal{O}(\mathbb{H})\otimes\mathcal{O}(V)$
and that

$\mathcal{O}(\mathbb{H})_{\leq\mathcal{O}(\mathbb{G})_{\leq
d}}\otimes\mathcal{O}(V)_{\leq d}\ \hookrightarrow\
\mathcal{O}(\pi^{-1}(V))_{\leq
2d}\hookrightarrow\mathcal{O}(\mathbb{H})_{\mathcal{O}(\mathbb{G})_{\leq
2d}}\otimes\mathcal{O}(V)_{\leq 2d}.$

Moreover, by Proposition 3.13,

$d\mapsto d^{dim(\mathfrak{h})}\quad\approx\quad
d\mapsto\mathcal{O}(\mathbb{H})_{\leq d}\quad\approx\quad d\mapsto
dim(\mathcal{O}(\mathbb{H})_{\leq\mathcal{O}(\mathbb{G})_{\leq d}}),$

so that

$d\mapsto d^{dim(\mathfrak{g})}\quad\approx\quad d\mapsto
dim(\mathcal{O}(\pi^{-1}(V))_{\leq d}$ $\quad\approx\quad d\mapsto
dim(\mathcal{O}(\mathbb{H})_{\leq d})\cdot dim(\mathcal{O}(V)_{\leq
d}\quad\approx\quad d\mapsto
d^{dim(\mathfrak{h})}d^{dim(\mathbb{G}/\mathbb{H})}.$

∎

In the following simple example, the growth $d\mapsto
dim(M_{\mathcal{O}(\mathbb{G}_{a})_{\leq d}})$ is logarithmic.

######  Example 6.11.

Denote by $M$ the $\mathbb{G}_{a}$-submodule of $k[\mathbb{G}_{a}]^{R}=k[t]$
spanned as a $k$-vector space by $\\{t^{p^{i}},i\geq 0\\}$; in other words,
$M$ is the subspace of $\mathcal{O}(\mathbb{G}_{a})$ of primitive elements.
Thus, for $p^{r-1}\leq d\leq p^{r}-1$, the dimension of
$M_{\mathcal{O}(\mathbb{G}_{a})_{\leq d}}$ equals $r=log_{p}(p^{r})$. Since
$\varinjlim_{r}(log_{p}(p^{r-1}))/(log_{p}(p^{r}-1))=1$, we conclude that
$\varinjlim_{d}dim(M_{\mathcal{O}(\mathbb{G}_{a})_{\leq d}})/log_{p}(d)=1$.

This example should be compared/contrasted with the example of the mock
injective module $(k[\mathbb{G}_{a}]^{R})^{\mathbb{Z}/p}$ given prior to
Theorem 1.13.

We conclude our examples with an example of a cofinite $\mathbb{G}$-module
with exponential growth.

######  Example 6.12.

Let $V$ be the natural representation of $GL_{N}$ of dimension $N$, a
homogeneous polynomial representation of degree $1$ (see Example 6.9) and
define $M$ to be $M\ \equiv\ \bigoplus_{r\geq 0}(V^{(r)})^{\oplus p^{r^{e}}}.$
Then

$p^{r}\mapsto dim(M_{\mathcal{O}(GL_{N})_{\leq\cdot p^{r}}})\quad\approx\quad
p^{r}\mapsto N\cdot p^{r^{e}}\quad\approx\quad p^{r}\mapsto(p^{r})^{e}.$

Thus, $d\ \mapsto\ dim(M_{\mathcal{O}(GL_{N})_{\leq d}})$ has exponential
growth equal to $e$.

##  7\. General questions

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

######  Question 7.1.

To what extent does our discussion of $\mathbb{G}$-modules for an affine group
scheme of finite type over a field $k$ of positive characteristic extend to
group schemes of finite type over a (commutative) Noetherian $k$-algebra?

######  Question 7.2.

Much of the discussion of this paper applies to Hopf algebras $A$ of countable
dimension over $k$ which are more general than those of the form
$\mathcal{O}(\mathbb{G})$ (see [23]), although a braiding is required at some
points. Does a similar consideration of ascending converging sequences of
subspaces lead to interesting modules for various Hopf algebras which are not
co-commutative?

######  Question 7.3.

Let $mod(\mathbb{G},X)\ \subset\ mod(\mathbb{G})$ denote the abelian category
of finite dimensional $X$-comodules. Can one give a useful description of the
Balmer spectrum of the triangulated category $D^{b}(mod(\mathbb{G},X))$ for
various subspaces $X\subset\mathcal{O}(\mathbb{G})$?

######  Question 7.4.

Let $\mathbb{G}$ be a reductive group. How are the filtrations
$\\{\mathcal{O}_{\pi_{n}}(\mathbb{G})\\}$ of Theorem 3.2 and
$\\{\mathcal{O}(\mathbb{G})_{\leq d}\\}$ of Theorem 3.8 related?

######  Question 7.5.

For $\mathbb{U}$ a unipotent linear algebraic group, can we utilize the
ascending converging sequences of sub-coalgebras of $\mathcal{O}(\mathbb{U})$
to enable computations of the Hochschild cohomology of $\mathbb{U}$? (See
[13].)

######  Question 7.6.

Are there constraints on what positive numbers $d$ can be the degree of
polynomial growth of an indecomposable, cofinite $GL_{N}$-module with respect
to $\\{\mathcal{O}(GL_{N})_{\leq d}\\}$?

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