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# Support Theory for Extended Drinfeld doubles

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

Following earlier work with Cris Negron on the cohomology of Drinfeld doubles
$D(\mathbb{G}_{(r)})$, we develop a “geometric theory” of support varieties
for “extended Drinfeld doubles” $\tilde{D}(\mathbb{G}_{(r)})$ of Frobenius
kernels $\mathbb{G}_{(r)}$ of smooth linear algebraic groups $\mathbb{G}$ over
a field $k$ of characteristic $p>0$. To a $\tilde{D}(\mathbb{G}_{(r)})$-module
$M$ we associate the space $\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}$ of
equivalence classes of “pairs of $\pi$-points” and prove most of the desired
properties of $M\mapsto\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}$. Namely, this
association satisfies the “tensor product property” and admits a natural
continuous map $\Psi_{\tilde{D}}$ to cohomological support theory. Moreover,
for $M$ finite dimensional and with suitable conditions on $\mathbb{G}_{(r)}$,
this association provides a “projectivity test”, $\Psi_{\tilde{D}}$ is a
homeomorphism, and identifies $\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}$ with the
cohomological support variety of $M$ for various classes of
$\tilde{D}(\mathbb{G}_{(r)})$-modules $M$.

###### Key words and phrases:

Drinfeld doubles, Frobenius kernels, Support varieties

###### 2010 Mathematics Subject Classification:

16G99, 16S40, 16T05

Partially supported by the Simons Foundation

##  0\. Introduction

Cohomological support varieties for $kG$-modules for a finite group $G$
constitute natural extensions of D. Quillen’s geometric description of
$\operatorname{Spec}\nolimits H^{\bullet}(G,k)$ [19], [20]. As introduced by
J. Carlson in [3], the cohomological support variety $|G|_{M}$ of a finite
dimensional $kG$-module $M$ is the closed subvariety of
$\operatorname{Spec}\nolimits H^{\bullet}(G,k)$ associated to the (radical of)
the kernel of the natural map $H^{\bullet}(G,k)\to Ext_{G}^{*}(M,M)$ of
$k$-algebras. To make this more geometric and more computable in the special
case that $G$ is an elementary abelian $p$-group, J. Carlson conjectured and
G. Avrunin and L. Scott proved (in [1]) that $|E|_{M}$ can be identified with
the “rank variety” $V(E)_{M}$ of cyclic shifted subgroups along which the
pull-backs of $M$ are not projective. This “geometric interpretation” of
cohomological support varieties for modules for an elementary abelian
$p$-group was successively extended by B. Parshall and the author to modules
for restricted Lie algebras [6], by A. Suslin, C. Bendel, and the author in
[23] to modules for infinitesimal group schemes, and to modules for arbitrary
finite group schemes by J. Pevstova and the author [8]. The interweaving of
the two approaches (one, primarily cohomological; the second, more geometric)
has led to various constructions and invariants.

In this paper, we adapt the theory of $\pi$-points for finite group schemes to
a theory of $\pi$-point pairs for a special class of finite dimensional Hopf
algebras over $k$ which are neither commutative nor cocommutative. Namely,
beginning with a smooth linear algebraic group $\mathbb{G}$ over $k$, we
consider the Frobenius kernel $\mathbb{G}_{(r)}$ (i.e., the kernel of the
$r$-th iterate of the Frobenius map, $F^{r}:\mathbb{G}\to\mathbb{G}^{(r)}$ for
some $r>0$) and take the smash product algebra $\tilde{D}(\mathbb{G}_{(r)})\
\equiv\ k[\mathbb{G}_{(r+1)}]\\#k\mathbb{G}_{(r)}$ which maps surjectively to
the Drinfeld double $D(\mathbb{G}_{(r)})$ of $\mathbb{G}_{(r)}$.

What enables our approach for $\tilde{D}(\mathbb{G}_{(r)})$-modules (and does
not work for modules for the Drinfeld double $D(\mathbb{G}_{(r)})$) is the
existence of a sub Hopf algebra
$O(\mathbb{G}_{(r)})\subset\tilde{D}(\mathbb{G}_{(r)})$ which essentially
detects the cohomology of $\tilde{D}(\mathbb{G}_{(r)})$ and which is
isomorphic as an algebra to the “group algebra” for a finite group scheme.
Indeed, almost by definition, the $\pi$-point pair space
$\Pi(\tilde{D}(\mathbb{G}_{(r)}))$ maps bijectively to the $\pi$-point pair
space $\Pi(O(\mathbb{G}_{(r)}))$ which is homeomorphic to the $\pi$-point
space of the associated finite group scheme. To establish the desired tensor
product property $\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M\otimes M^{\prime}}\ =\
\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}\cap\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{N}$
in Theorem 3.4, we utilize this tensor product property for finite group
schemes. The key step requires the verification that this property remains
valid upon changing coproduct structure on this group algebra to the coproduct
structure on $O(\mathbb{G}_{(r)})$. This invariance of coproduct of supports
of a tensor product is the content of Proposition 3.2. The tensor product
property enables a natural topology on $\Pi(\tilde{D}(\mathbb{G}_{(r)}))$.

In Section 4, we extend the arguments of C. Negron and the author in [5] for
the cohomology of Drinfeld doubles to the cohomology of the extended doubles
$\tilde{D}(\mathbb{G}_{(r)})$ which allows a comparison in Corollary 4.3 of
the cohomology of $\tilde{D}(\mathbb{G}_{(r)})$ with that of the simpler
subalgebra $O(\mathbb{G}_{(r)})$. In Proposition 4.6, we establish a
commutative square of continuous maps relating $\pi$-point pair spaces for
$O(\mathbb{G}_{(r)})$ and $\tilde{D}(\mathbb{G}_{(r)})$ with (projectivized)
prime ideal spectra of their cohomology.

In order to proceed further, we require the determination of the cohomology
algebra $H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k)$, the commutative
subalgebra of the Ext-algebra $Ext_{\tilde{D}(\mathbb{G}_{(r)})}^{*}(k,k)$ of
classes of even degree for $p>2$ (and the full Ext-algebra if $p=2$). This is
achieved in Theorem 5.4 provided that $\mathbb{G}$ admits a quasilogarithm
(see Example 5.2) and that $p^{r+1}>2dim(\mathbb{G})$, again using techniques
of [5]. This computation allows us to show that
$\Psi_{\tilde{D}}:\Pi(\tilde{D}(\mathbb{G}_{(r)}))\to\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))$
is a homeomorphism. As a consequence, we conclude in Theorem 5.6 that a finite
dimensional $\tilde{D}(\mathbb{G}_{(r)})$-module is projective if and only if
$\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}$ is empty.

The fact that $\Psi_{\tilde{D}}$ is a homeomorphism mapping
$\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}$ to
$\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))_{M}$ does not a priori
identify $\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}$ with
$\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))_{M}$. In Section 6, we
achieve this identification for various special classes of
$\tilde{D}(\mathbb{G}_{(r)})$-modules, including Carlson’s
$L_{\zeta}$-modules, modules whose action factors through the quotient
$\tilde{D}(\mathbb{G}_{(r)})\twoheadrightarrow k\mathbb{G}_{(r)}$ and modules
whose restriction to $k[\mathbb{G}_{(r+1)}]$ are projective.

In Section 7, we observe that the same techniques we use to investigate
$\tilde{D}(\mathbb{G}_{(r)})$-modules apply to modules for closely related
Hopf algebras. We also mention a couple of questions worthy of further
investigation.

We thank both Cris Negron and Julia Pevtsova for their contributions to the
foundations of this work, and the Institute for Advanced Study for its
(virtual) hospitality.

##  1\.  $D(\mathbb{G}_{(r)}),\ \tilde{D}(\mathbb{G}_{(r)}),\
O(\mathbb{G}_{(r)})$, $\mathbb{G}_{(r)}\times$
($\mathfrak{g}$${}^{(r)})_{(1)}$

Throughout this work, $\mathbb{G}$ will denote a connected linear algebraic
group smooth over $k$. If $A$ is a $k$-algebra and $K/k$ is a field extension,
then we denote by $A_{K}$ the base change $K\otimes A$; for an $A$-module $M$,
we use the notation $M_{K}$ for the $A_{K}$-module $K\otimes M$. For a finite
dimensional $k$-algebra $A$, we denote by $H^{\bullet}(A,k)\ \subset
Ext_{A}^{*}(k,k)$ the commutative subalgebra of even dimensional classes if
$p>2$ and the full algebra $Ext_{A}^{*}(k,k)$ if $p=2$.

For any $r>0$, we denote by $F^{r}:\mathbb{G}\to\mathbb{G}^{(r)}$ the $r$-th
iterate of the Frobenius map $F:\mathbb{G}\to\mathbb{G}^{(1)}$ given by the
$k$-linear map

$F^{*}:k[\mathbb{G}^{(1)}]\equiv k\otimes_{\phi}k[\mathbb{G}]\to
k[\mathbb{G}],\quad\lambda\otimes f\mapsto\lambda f^{p}$

of coordinate algebras (where $k\otimes_{\phi}-$ denotes the base change along
the $p$-th power map $\phi:k\to k$). The $r$-th Frobenius kernel of
$\mathbb{G}$ is the finite group scheme

(1.0.1) $\mathbb{G}_{(r)}\ \equiv\
ker\\{F^{r}:\mathbb{G}\to\mathbb{G}^{(r)}\\}\ \hookrightarrow\mathbb{G},$

whose coordinate algebra $k[\mathbb{G}_{(r)}]$ is the local commutative
algebra defined as the quotient of the localization $(k[\mathbb{G}])_{(e)}$ of
$k[\mathbb{G}]$ at the identity $e\in\mathbb{G}$ modulo the ideal generated by
elements $x^{p^{r}},x\in\mathfrak{m}_{e}$. For example, $k[GL_{n(r)}]\ =\
k[x_{i,j}]/(x_{i,j}^{p^{r}}-\delta_{i,j})$.

The coordinate algebra $k[\mathbb{G}]$ of $\mathbb{G}$ is a Hopf algebra with
comultiplication determined by the multiplication of $\mathbb{G}$. For a
finite group scheme $G$ such as $\mathbb{G}_{(r)}$, we consider the $k$-linear
dual $kG$ of $k[G]$ which is a finite dimensional, co-commutative Hopf algebra
(over $k$); we refer to $kG$ as the group algebra of $G$. For such a finite
group scheme $G$, a (rational) $G$-module is a $k$-vector space equipped with
a $k$-linear module structure, $kG\times M\to M$. We shall frequently refer to
this structure as the “action of $G$ on $M$”.

We recall that the Drinfeld double of a Hopf algebra $H$ is the double crossed
product of $H^{*cop}$ and $H$, where $H^{*cop}$ has the same algebra structure
as $H^{*}$ and coproduct given by the opposite of the coproduct of $H^{*}$. If
$H$ is finite dimensional and cocommutative, then $D(H)$ is isomorphic to the
smash product $H^{*op}\\#H$ (see [18, 10.3.10]).

In Hopf algebra standard notation, the coadjoint action of an element $h\in H$
on an element $f\in H^{*}$ is given by $\sum f(h_{1}\rightharpoonup
f\leftharpoonup s^{-1}(h_{2}))$ where $s$ is the antipode of $H$, $\Delta$ is
the coproduct of $H^{*}$ and $\Delta(h)$ is written as $\sum h_{1}\otimes
h_{2}$. By a result of D. Radford [21, 10.3.5], the product structure of
$D(H)$ is given for finite dimensional $H$ by

$(f\\#h)(f^{\prime}\\#h^{\prime})\ =\ \sum f(h_{1}\rightharpoonup
f^{\prime}\leftharpoonup s^{-1}(h_{2}))\\#h_{3}h^{\prime}.$

If $H$ is both finite dimensional and cocommutative, this multiplication is
that of the smash product algebra $H^{*}\\#H$:

$(f\\#h)(f^{\prime}\\#h^{\prime})\ =\ \sum fh_{1}(f)\\#h_{2}h^{\prime};$

the coproduct of $D(H)$ is the tensor product of the opposite coproduct of
$H^{*}$ and the coproduct of $H$.

######  Definition 1.1.

As discussed above, the Drinfeld double of the finite dimensional
cocommutative Hopf algebra $k\mathbb{G}_{(r)}$ has algebra structure given by
the smash product

(1.1.1) $D(\mathbb{G}_{(r)})\ \equiv\
k[\mathbb{G}_{(r)}]\\#k\mathbb{G}_{(r)}.$

with respect to the right coadjoint action of $\mathbb{G}_{(r)}$ on
$k[\mathbb{G}_{(r)}]$ and coalgebra structure given by the tensor product of
the co-opposite cooproduct on $k[\mathbb{G}_{(r)}]$ (i.e., the opposite
product structure of $k\mathbb{G}_{(r)}$) and the coproduct of
$k\mathbb{G}_{(r)}$.

The extended Drinfeld double $\tilde{D}(\mathbb{G}_{(r)})$ is the analogous
smash product with respect to the restriction to $\mathbb{G}_{(r)}$ of the
coadjoint action of $\mathbb{G}_{(r+1)}$ on $k[\mathbb{G}_{(r+1)}]$:

(1.1.2) $\tilde{D}(\mathbb{G}_{(r)})\ \equiv\
k[\mathbb{G}_{(r+1)}]\\#k\mathbb{G}_{(r)}\ \hookrightarrow\
D(\mathbb{G}_{(r+1)}).$

To verify that (1.1.2) is a well defined inclusion of Hopf algebras, we
observe that the restriction of the coadjoint action of $\mathbb{G}_{(r+1)}$
on $k[\mathbb{G}_{(r+1)}]$ along
$\mathbb{G}_{(r)}\hookrightarrow\mathbb{G}_{(r+1)}$ is the action used to
define $\tilde{D}(\mathbb{G}_{(r)})$, then check directly that the
multiplication and comultiplication of $D(\mathbb{G}_{(r+1)})$ restricts via
the evident inclusion of $k$-vector spaces $k[\mathbb{G}_{(r+1)}]\otimes
k\mathbb{G}_{(r)}\ \subset\ k[\mathbb{G}_{(r+1)}]\otimes k\mathbb{G}_{(r+1)}$
to the multiplication and comultiplication of $\tilde{D}(\mathbb{G}_{(r)})$.

We shall be interested in (left) modules for the extended Drinfeld double
$\tilde{D}(\mathbb{G}_{(r)})$ which can be viewed as $k$-vector spaces $N$
with an action $\tilde{D}(\mathbb{G}_{(r)})\times N\to N$. The condition on
such a pairing to be a left module structure can be expressed as the
restriction to elements in $\tilde{D}(\mathbb{G}_{(r)})\subset
D(\mathbb{G}_{(r+1)})$ of the condition that a pairing
$D(\mathbb{G}_{(r+1)})\times N\to N$ be a $D(\mathbb{G}_{(r+1)})$ -module
structure. This latter condition is made explicit in the following proposition
(with $H=k\mathbb{G}_{(r+1)}$).

We first observe that for any Hopf algebra $H$ there are natural algebra
embeddings of $H,H^{*}$ in $D(H)$: $h\in H\mapsto 1\\#h;\ f\in H^{*}\mapsto
f\\#1$.

######  Proposition 1.2.

[18, 10.6.16], [16] Let $H$ be a finite dimensional Hopf Algebra. The data of
a $D(H)$-module structure on a vector space $N$ consists of module structures
on $N$ for both $H^{*}$ and $H$ such that

$h\cdot(f\cdot n)\ =\ \sum(h_{1}\rightharpoonup
f_{2})\cdot((h_{2}\leftharpoonup f_{1})\cdot n)\ \in\ N.$

This is equivalent to a left/right Yetter-Drinfeld module structure on $N$
which consists of the data of a $k\mathbb{G}_{(r)}$-module structure and a
right $k\mathbb{G}_{(r)}$-comodule structure on $N$ satisfying the Yetter-
Drinfeld condition:

$\sum h_{1}\cdot n_{o}\otimes h_{2}n_{1}\ =\
\sum(h_{1}n)_{0}\otimes(h_{2}\cdot n)_{1}h_{1}\ \in\ N\otimes
k\mathbb{G}_{(r)}.$

Here, we have written the coaction $N\to N\otimes H$ as $n\mapsto\sum
n_{0}\otimes h_{1}$.

We relate the extended Drinfeld double $\tilde{D}(\mathbb{G}_{(r)})$ to
various other structures.

######  Definition 1.3.

We define

(1.3.1) $O(\mathbb{G}_{(r)})\ \equiv\ k[(\mathbb{G}^{(r)})_{(1)}]\otimes
k\mathbb{G}_{(r)}.$

Using the natural identification of $(\mathbb{G}^{(r)})_{(1)}$ with
$\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}$, we obtain the natural embedding

(1.3.2) $i_{O}:O(\mathbb{G}_{(r)})\ \hookrightarrow\
\tilde{D}(\mathbb{G}_{(r)}).$

######  Proposition 1.4.

The embedding $\mathbb{G}_{(r)}\ \subset\mathbb{G}_{(r+1)}$ determines a
$\mathbb{G}_{(r)}$-equivariant quotient map
$k[\mathbb{G}_{(r+1)}]\twoheadrightarrow k[\mathbb{G}_{(r)}]$ of Hopf algebras
and thus a quotient map of Hopf algebras

(1.4.1) $q:\tilde{D}(\mathbb{G}_{(r)})\ \twoheadrightarrow\
D(\mathbb{G}_{(r)}).$

Similarly, the quotient map
$\mathbb{G}_{(r+1)}\twoheadrightarrow(\mathbb{G}^{(r)})_{(1)}$ determining
$i_{O}$ of (1.3.2) is an embedding of Hopf algebras

The coproduct structure inherited by $O(\mathbb{G}_{(r)})$ is the tensor
product of the co-opposite of the coproduct on $k[\mathbb{G}^{(r)})_{(1)}]$
(in other words the dual of the opposite multiplication of
$k(\mathbb{G}^{(r)})_{(1)}$) and the coproduct of $k\mathbb{G}_{(r)}$ (the
dual of the multiplication of $k[\mathbb{G}_{(r)}]$). Thus,
$O(\mathbb{G}_{(r)})$ is co-commutative if and only if
$\mathfrak{g}=Lie(\mathbb{G})$ is commutative.

###### Proof.

Since each $\mathbb{G}_{(r)}$ is normal in $\mathbb{G}$, the embedding
$\mathbb{G}_{(r)}\ \subset\mathbb{G}_{(r+1)}$ is $\mathbb{G}$-equivariant so
that $k[\mathbb{G}_{(r+1)}]\twoheadrightarrow
k[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}]$ is a $\mathbb{G}$-equivariant quotient
map of Hopf algebras; thus, this $\mathbb{G}_{(r)}$-equivariant quotient
determines the map of Hopf algebras $q:\tilde{D}(\mathbb{G}_{(r)})\
\twoheadrightarrow\ D(\mathbb{G}_{(r)})$.

Similarly, the quotient
$\mathbb{G}_{(r+1)}\twoheadrightarrow\mathbb{G}^{(r)})_{(1)}$ is also
$\mathbb{G}$-equivariant, determining the $\mathbb{G}$-equivariant embedding
$k[\mathbb{G}^{(r)})_{(1)}]\hookrightarrow k[\mathbb{G}_{(r+1)}]$ of Hopf
algebras. Observe that $\mathbb{G}_{(r)}\subset\mathbb{G}$ acts trivially on
$k[\mathbb{G}^{(r)})_{(1)}]$, so that we obtain the embedding (as algebras) of
the tensor product $k[\mathbb{G}^{(r)})_{(1)}]\otimes k\mathbb{G}_{(r)}\
\hookrightarrow\ \tilde{D}(\mathbb{G}_{(r)})$. ∎

######  Proposition 1.5.

The subalgebra $k[\mathbb{G}^{(r)})_{(1)}]\ \simeq\
k[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}]$ of the commutative algebra
$k[\mathbb{G}_{(r+1)}]$ gives $k[\mathbb{G}_{(r+1)}]$ the structure of a free
$k[\mathbb{G}^{(r)})_{(1)}]$-module with basis $\\{\tilde{f}_{i}\\}$ given by
lifting along $k[\mathbb{G}_{(r+1)}]\to k[\mathbb{G}_{(r)}]$ a $k$-basis
$\\{f_{i}\\}$ for $k[\mathbb{G}_{(r)}]$.

This structure determines a free (right) $O(\mathbb{G}_{(r)})$-module
structure on $\tilde{D}(\mathbb{G}_{(r)})$ with basis
$\\{\tilde{f}_{i}\\#1\\}$.

###### Proof.

We choose a $k$-linear section $\sigma:k[\mathbb{G}_{(r)}]\to
k[\mathbb{G}_{(r+1)}]$ of the quotient map
$k[\mathbb{G}_{(r+1)}]\twoheadrightarrow k[\mathbb{G}_{(r)}]$ and consider the
$k[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}]$-bilinear map

(1.5.1) $k[\mathbb{G}_{(r)}]\otimes k[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}]\
\to\ k[\mathbb{G}_{(r+1)}],\quad f_{i}\otimes h\mapsto\sigma(f_{i})\cdot h.$

We readily check that this induces a $k$-linear isomorphism, for example by
showing that tensors of elements of $k$-bases for $k[\mathbb{G}_{(r)}]$ and
$k[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}]$ determine a $k$-linearly independent
set in $k[\mathbb{G}_{(r+1)}]$.

Using the fact that $k[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}]\otimes 1\
\subset\tilde{D}(\mathbb{G}_{(r)})$ is central, we readily check that

$k[\mathbb{G}_{(r)}]\otimes O(\mathbb{G}_{(r)})\to\
\tilde{D}(\mathbb{G}_{(r)}),\quad f\otimes(f^{\prime}\otimes
h)\mapsto(\sigma(f)\\#1)\cdot(f^{\prime}\\#h)$

is an $O(\mathbb{G}_{(r)})$-linear isomorphism ∎

The following observation about $O(\mathbb{G}_{(r)})$ will prove central in
our investigation of $\tilde{D}(\mathbb{G}_{(r)})$-modules. The reader should
not confuse the finite group scheme $(\underline{\mathfrak{g}}^{(r)})_{(1)}$
occurring in the proposition below with $(\mathbb{G}^{(r)})_{(1)}$.

######  Proposition 1.6.

Let $\underline{\mathfrak{g}}$ denote the vector group scheme
$\operatorname{Spec}\nolimits S^{\bullet}(\mathfrak{g}^{*})$ (ignoring the Lie
algebra structure on $\mathfrak{g}$). Then the group algebra of the finite
group scheme $(\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)}$ is
isomorphic as an algebra to $\mathcal{O}(\mathbb{G}_{(r)})$.

On the other hand, the dual of the Hopf algebra $O(\mathbb{G}_{(r)})$ is not
commutative (unlike the coordinate algebra of
$(\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)}$).

###### Proof.

Essentially by definition, elements $X^{(r)}\in Lie(\mathbb{G}^{(r)})$ are
distributions at the identity of $\mathbb{G}^{(r)}$ of $k[\mathbb{G}^{(r)}]$
of order 1 (see [13, I.7.10]), This identification determines an embedding of
$\mathfrak{g}^{(r)*}$ into the maximal ideal of
$k[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}]$ which necessarily sends each
$(X^{(r)})^{p}\in S^{p}(\mathfrak{g}^{(r)*})$ to 0. The resulting map

$\nu:S^{\bullet}(\mathfrak{g}^{(r)*})/((X^{(r)})^{p},X\in\mathfrak{g}^{*})\
\to\ k[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}]$

is injective, and thus surjective by dimension reasons. The proof is completed
by observing that the group algebra of
$(\underline{\mathfrak{g}}^{(r)})_{(1)}$ equals
$S^{\bullet}(\mathfrak{g}^{(r)*})/((X^{(r)})^{p},X\in\mathfrak{g}^{*})$. ∎

We conclude this section with some remarks about $\pi$-point spaces
$\Pi(G_{1}\times G_{2})$ of a product of finite group schemes which we shall
use for $G_{1}\times
G_{2}=(\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)}$. Recall
that a $\pi$-point $\alpha_{K}:K[t]/t^{p}\to kG$ of a finite group scheme $G$
over $k$ is a flat map of algebras which factors through the group algebra of
some unipotent abelian subgroup scheme $C_{K}\subset G_{K}$, where $K$ is some
field extension of $k$. Thus, a $\pi$-point of $G_{1}\times G_{2}$ is a pair
of $\pi$-points $\alpha_{K,1}:K[t]/t^{p}\to KG_{1},\
\alpha_{K,2}:K[t]/t^{p}\to KG_{2}$ corresponding to a flat map

$\alpha_{K,1}+\alpha_{2}:K[t]/t^{p}\to KC_{1}\otimes KC_{2},\quad
t\mapsto\alpha_{K,1}(t)\otimes 1+1\otimes\alpha_{K,1}(t)$

with $C_{K,i}\subset G_{K,i}$ unipotent abelian subgroup schemes for $i=1,2$.

It is important here for $\alpha_{K,1}(t),\ \alpha_{K,1}(t)$ to commute in
order that $\alpha_{K,1}+\alpha_{K,2}$ is well defined.

By [8, Thm 3.6], two $\pi$-points $\alpha_{K}:K[t]/t^{p}\to
KG,\beta_{L}:L[t]/t^{p}\to LG$ are equivalent if and only if the two ideals of
$H^{\bullet}(G,k)$

$ker\\{\alpha_{K}^{*}:H^{\bullet}(G,K)\to H^{*}(K[t]/t^{p},K)\\}\ \cap\
H^{\bullet}(G,k),$ $ker\\{\beta_{L}^{*}:H^{\bullet}(G,L)\to
H^{*}(L[t]/t^{p},L)\\}\ \cap\ H^{\bullet}(G,k)$

are equal. In other words, this gives a natural bijection of sets (which is
sharpened to an isomorphism of schemes in [8])

(1.6.1) $\Pi_{G}:\Pi(G)\ \stackrel{{\scriptstyle\sim}}{{\to}}\
\mathbb{P}H^{\bullet}(G,k).$

Observe that the flat map $(\alpha_{K,1},\alpha_{K,2}):K[t]/t^{p}\to
KG_{1}\times KG_{2}$ induces the map

$(\alpha_{K,1}^{*}\otimes
1)+(1\otimes\alpha_{K,2})^{*}:H^{*}(C_{1},K)\otimes_{K}H^{\bullet}(C_{2},K)\to
H^{*}(K[t]/t^{p},K).$

We denote the composition of this map with the restriction map

$(\alpha_{K,1}+\alpha_{K,2})^{*}:H^{*}(G_{1}\times G_{2},K)\ \to\
H^{*}(K[t]/t^{p},K).$

Thus, we have proved the following proposition.

######  Proposition 1.7.

Two $\pi$-points $(\alpha_{K,1},\alpha_{K,2})$ and $(\beta_{L,1},\beta_{L,2})$
of the finite group scheme $G_{1}\times G_{2}$ are equivalent if and only if
the two ideals of $H^{\bullet}(G_{1}\times G_{2},k)$

$ker\\{(\alpha_{K,1}+\alpha_{K,2})^{*}:H^{\bullet}(G_{1}\times G_{2},K)\to
H^{*}(K[t]/t^{p},K)\\}\ \cap\ H^{\bullet}(G_{1}\times G_{2},k),$
$ker\\{(\beta_{L,1}+\beta_{L,2})^{*}:H^{\bullet}(G_{1}\times G_{2},L)\to
H^{*}(L[t]/t^{p},L)\\}\ \cap\ H^{\bullet}(G_{1}\times G_{2},k)$

are equal.

In particular, if $\alpha_{K_{i}}$ is equivalent to $\beta_{L,i}$ for $i=1,2$,
then $(\alpha_{K,1},\alpha_{K,2})$ and $(\beta_{L,1},\beta_{L,2})$ are
equivalent.

##  2\. The space $\Pi(\tilde{D}(\mathbb{G}_{(r)}))$ of $\pi$-point pairs

In this section we introduce an extension of “rank varieties” which applies to
$\tilde{D}(\mathbb{G}_{(r)})$-modules. Our construction is a modification of
the “$\pi$-point construction” of [8]. The strategy we follow is relatively
straight-forward. In view of the homological computations of [5], it seems
natural to use pairs of $\pi$-points $(\alpha_{K},\beta_{K})$ with
$\alpha_{K}:K[t]/t^{p}\to K[\mathbb{G}_{r+1)}]$ and $\beta_{K}:K[T]/t^{p}\to
K\mathbb{G}_{(r)}$. Yet the most natural way to do this (which would also
apply to $D(\mathbb{G}_{(r)})$) appears to fail because
$\alpha_{K}(t),\beta_{K}(t)$ need not commute. We work around this difficulty
by restricting $\tilde{D}(\mathbb{G}_{(r)})$-modules to $O(\mathbb{G}_{(r)})$.

######  Definition 2.1.

A $\pi$-point pair $(\alpha_{K},\beta_{K})$ of $\tilde{D}(\mathbb{G}_{(r)})$
is a pair of maps of $K$-algebras for some field extension $K/k$,

$\alpha_{K}:K[t]/t^{p}\to
K[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}]\hookrightarrow
K[\mathbb{G}_{(r+1)}],\quad\beta_{K}:K[t]/t^{p}\to K\mathbb{G}_{(r)},$

such that $\beta_{K}$ either sends $t$ to 0 or is a $\pi$-point of
$\mathbb{G}_{(r)}$, $\alpha_{K}$ either sends $t$ to 0 or is flat, and at
least one of $\alpha_{K},\beta_{K}$ is flat.

The following lemma provides an essential property of a $\pi$-point pair
$(\alpha_{K},\beta_{K})$.

######  Lemma 2.2.

A $\pi$-point pair $(\alpha_{K},\beta_{K})$ of $\tilde{D}(\mathbb{G}_{(r)})$
determines a flat map

$\alpha_{K}+\beta_{K}:K[t]/t^{p}\ \to\ O(\mathbb{G}_{(r)})_{K},\quad
t\mapsto\alpha_{K}(t)\\#1+1\\#\beta_{K}(t)$

whose composition with $i_{O}:O(\mathbb{G}_{(r),K})\ \hookrightarrow\
\tilde{D}(\mathbb{G}_{(r),K})$ is also flat.

###### Proof.

We consider the composition

$K[t]\ \stackrel{{\scriptstyle\Delta}}{{\to}}\ K[t]/p^{\otimes 2}\
\stackrel{{\scriptstyle\alpha_{K}\boxtimes\beta_{K}}}{{\to}}\
K[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}]\otimes
K\mathbb{G}_{(r)}=O(\mathbb{G}_{(r)})_{K},\quad
t\mapsto\alpha_{K}(t)\otimes\beta_{K}(t).$

Since $\alpha_{K}(t),\beta_{K}(t)$ commute in $O(\mathbb{G}_{(r)})_{K}$, this
composition factors through $\alpha_{K}+\beta_{K}:K[t]/t^{p}\ \to\
O(\mathbb{G}_{(r)})_{K}.$ Since $\Delta:K[t]/t^{p}\
\stackrel{{\scriptstyle\Delta}}{{\to}}\ K[t]/p\otimes K[t]/t^{p}$ is flat,
since both $\alpha_{K}$ and $\beta_{K}$ are either flat or “trivial”, since
$O(\mathbb{G}_{(r)})$ is flat over each of its tensor factors, and since
$\alpha_{K}\boxtimes\beta_{K}$ is flat if
$\alpha_{K}(t)\not=0\not=\beta_{K}(t)$, we conclude that
$\alpha_{K}+\beta_{K}:K[t]/t^{p}\ \to\ O(\mathbb{G}_{(r)})$ is flat. By
Proposition 1.5, this implies the composition
$i_{O}\circ(\alpha_{K}+\beta_{K}):K[t]/t^{p}\ \to\
\tilde{D}(\mathbb{G}_{(r)})$ is also flat. ∎

The following definition of equivalence for $\pi$-point pairs
$(\alpha_{K},\beta_{K}),(\alpha^{\prime}_{L},\beta^{\prime}_{L})$ is motivated
by Proposition 1.7.

######  Definition 2.3.

Two $\pi$-point pairs
$(\alpha_{K},\beta_{K}),(\alpha^{\prime}_{L},\beta^{\prime}_{L})$ of
$\tilde{D}(\mathbb{G}_{(r)})$ are said to be equivalent (respectively,
$O(\mathbb{G}_{(r)})$-equivalent) if there exists some common field extension
$\Omega$ of both $K$ and $L$ such that for all finite dimensional
$\tilde{D}(\mathbb{G}_{(r)})$-modules $M$ (resp., all finite dimensional
$O(\mathbb{G}_{(r)})$-modules $M$)
$(\alpha_{\Omega}+\beta_{\Omega})^{*}(M_{\Omega})$ is projective as an
$\Omega[t]/t^{p}$-module if and only if
$(\alpha_{\Omega}^{\prime}+\beta_{\Omega}^{\prime})^{*}(M_{\Omega})$ is
projective.

We denote by $\Pi(\tilde{D}(\mathbb{G}_{(r)}))$ the set of equivalence classes
of $\pi$-point pairs of $\tilde{D}(\mathbb{G}_{(r)})$, and we denote by
$\Pi(O(\mathbb{G}_{(r)}))$ the set of $O(\mathbb{G}_{(r)})$-equivalence
classes of $\pi$-point pairs of $\tilde{D}(\mathbb{G}_{(r)})$. For any
$\tilde{D}(\mathbb{G}_{(r)})$-module $M$, we define the $\Pi$-pair set

$\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}\quad\subset\quad\Pi(\tilde{D}(\mathbb{G}_{(r)}))$

to be the subset of equivalence classes of $\pi$-point pairs of
$\tilde{D}(\mathbb{G}_{(r)})$ such that $(\alpha_{K}+\beta_{K})^{*}(M)$ is not
projective. Similarly, for any $O(\mathbb{G}_{(r)})$-module $M^{\prime}$, we
define
$\Pi(O(\mathbb{G}_{(r)}))_{M^{\prime}}\subset\Pi(O(\mathbb{G}_{(r)}))$ to be
the subset of equivalence classes of $\pi$-point pairs of
$\tilde{D}(\mathbb{G}_{(r)})$ such that
$(\alpha_{K}+\beta_{K})^{*}(M^{\prime})$ is not projective.

######  Proposition 2.4.

The natural surjection $\Pi_{\tilde{D}}:\Pi(O(\mathbb{G}_{(r)}))\
\twoheadrightarrow\ \Pi(\tilde{D}(\mathbb{G}_{(r)}))$ is a bijection.

Moreover, for any finite dimensional $\tilde{D}(\mathbb{G}_{(r)})$-module $M$,
$\Pi_{\tilde{D}}$ restricts to a bijection

$\Pi_{\tilde{D},M}:\Pi(O(\mathbb{G}_{(r)}))_{M}\
\stackrel{{\scriptstyle\sim}}{{\to}}\ \Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}.$

###### Proof.

If $(\alpha_{K},\beta_{K}),(\alpha^{\prime}_{L},\beta^{\prime}_{L})$ are
$O(\mathbb{G}_{(r)})$-equivalent, then they are necessarily equivalent since
equivalence involves the same condition as $O(\mathbb{G}_{(r)})$-equivalence
except that it only requires consideration of finite dimensional
$\tilde{D}(\mathbb{G}_{(r)})$-modules. Thus, we have well defined, surjective
map $\Pi(O(\mathbb{G}_{(r)}))\ \twoheadrightarrow\
\Pi(\tilde{D}(\mathbb{G}_{(r)}))$.

To prove injectivity, let
$(\alpha_{K},\beta_{K}),(\alpha^{\prime}_{L},\beta^{\prime}_{L})$ represent
distinct points in $\Pi(\tilde{D}(\mathbb{G}_{(r)}))$ and choose some finite
dimensional $O(\mathbb{G}_{(r)})$-module $N$ such that
$(\alpha_{\Omega}+\beta_{\Omega})^{*}(N_{\Omega})$ is projective whereas
$(\alpha_{\Omega}^{\prime}+\beta_{\Omega}^{\prime})^{*}(N_{\Omega})$ is not
projective for some some common field extension $\Omega$ of both $K$ and $L$.
Consider the $\tilde{D}(\mathbb{G}_{(r)})$-module $M\ \equiv\
\tilde{D}(\mathbb{G}_{(r)})\otimes_{O(\mathbb{G}_{(r)})}N$ and set
$N^{\prime}\equiv i_{O}^{*}(M)$. By Proposition 1.5, $N^{\prime}$ is
isomorphic to a direct sum of copies of $N$ (indexed by a $k$-basis of
$k[\mathbb{G}_{(r)}]$). Thus,
$(\alpha_{\Omega}+\beta_{\Omega})^{*}(N^{\prime}_{\Omega})$ is projective
whereas
$(\alpha_{\Omega}^{\prime}+\beta_{\Omega}^{\prime})^{*}(N^{\prime}_{\Omega})$
is not projective. Since $N^{\prime}$ is the restriction of the
$D(\mathbb{G}_{(r)})$-module $M$, we conclude that the images of
$(\alpha_{K},\beta_{K}),(\alpha^{\prime}_{L},\beta^{\prime}_{L})$ in
$\Pi(\tilde{D}(\mathbb{G}_{(r)}))$ are distinct.

Finally, the bijectivity of $\Pi_{\tilde{D},M}$ follows immediately from the
definitions and the bijectivity of $\Pi_{\tilde{D}}$. ∎

We next observe that the condition of equivalence of $\pi$-point pairs is
equivalent to the seemingly stronger equivalence relation obtained by dropping
the finite dimensionality condition on $M$ in Definition 2.3.

######  Proposition 2.5.

Two $\pi$-point pairs
$(\alpha_{K},\beta_{K}),(\alpha^{\prime}_{L},\beta^{\prime}_{L})$ of
$\tilde{D}(\mathbb{G}_{(r)})$ (respectively $O(\mathbb{G}_{(r)})$) are
equivalent if and only if there exists some common field extension $\Omega$ of
both $K$ and $L$ such that for all $\tilde{D}(\mathbb{G}_{(r)})$-modules $M$
(resp., all $O(\mathbb{G}_{(r)})$-modules $M$)
$(\alpha_{\Omega}+\beta_{\Omega})^{*}(M_{\Omega})$ is projective as an
$\Omega[t]/t^{p}$-module if and only if
$(\alpha_{\Omega}^{\prime}+\beta_{\Omega}^{\prime})^{*}(M_{\Omega})$ is
projective.

###### Proof.

Denote by $\tilde{\Pi}(\tilde{D}(\mathbb{G}_{(r)}))$ (respectively,
$\tilde{\Pi}(O(\mathbb{G}_{(r)}))$) the set of equivalence classes of
$\pi$-point pairs of $\tilde{D}(\mathbb{G}_{(r)})$ (resp.,
$O(\mathbb{G}_{(r)})$) using the equivalence relations of the statement of
this proposition. We readily verify that we have a commutative square of sets

(2.5.1)
$\textstyle{\tilde{\Pi}(O(\mathbb{G}_{(r)}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Pi_{\tilde{D}}}$$\scriptstyle{\tilde{\Pi}_{O}}$$\textstyle{\tilde{\Pi}(\tilde{D}(\mathbb{G}_{(r)}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\Pi}_{D}}$$\textstyle{\Pi(O(\mathbb{G}_{(r)}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Pi_{D}}$$\textstyle{\Pi(\tilde{D}(\mathbb{G}_{(r)})).}$

The map $\Pi_{D}$ is a bijection as seen in Proposition 2.4; the proof of that
proposition also proves that $\Pi_{\tilde{D}}$ is a bijection. The fact that
$\tilde{\Pi}_{O}$ is a bijection follows from [8, Thm 4.6] applied to the
finite group scheme
$(\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)}$ (see
Proposition 2.7 below). Thus, the fact that $\tilde{\Pi}_{D}$ is a bijection
follows from the surjectivity of $\Pi_{\tilde{D}}$ and a simple diagram chase.
∎

The following properties of $M\ \mapsto\ \Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}$
are each verified by restricting along flat maps
$(\alpha_{\Omega}+\beta_{\Omega})^{*}$ associated to $\pi$-point pairs
$(\alpha_{K},\beta_{K})$, thereby reducing each assertion to the corresponding
assertion about $K[t]/t^{p}$-modules.

######  Proposition 2.6.

Let $M_{1},\ M_{2},\ M_{3}$ be $\tilde{D}(\mathbb{G}_{(r)})$-modules.

  1. (1)

$\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{k}\ =\ \Pi(\tilde{D}(\mathbb{G}_{(r)}))$.

  2. (2)

If $M_{1}$ is a projective $\tilde{D}(\mathbb{G}_{(r)})$-module, then
$\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M_{1}}\ =\ \emptyset$.

  3. (3)

If $0\to M_{1}\to M\to M_{2}\to$ is exact and if $\sigma$ is a permutation of
$\\{1,2,3\\}$, then

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

  4. (4)

$\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M_{1}\oplus M_{3}}\ =\
\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M_{1}}\cup\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M_{3}}.$

We proceed to show that $\Pi(O(\mathbb{G}_{(r)}))$ can be “identified” with
the set of Zariski points of the $\pi$-point scheme of the finite group scheme
$(\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)}$ introduced
below.

######  Proposition 2.7.

Let $\underline{\mathfrak{g}}\ \simeq\ (\mathbb{G}_{a})^{\times
dim(\mathfrak{g})}$ denote the vector group scheme associated to the
underlying vector space of $\mathfrak{g}=Lie(\mathbb{G})$. As an algebra
$O(\mathbb{G}_{(r)})$ is isomorphic to the group algebra of the infinitesimal
group scheme $(\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)}$.

Choose an isomorphism of $k$-algebras
$O(\mathbb{G}_{(r)})\stackrel{{\scriptstyle\sim}}{{\to}}k((\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)})$.
Then sending a $\pi$-point pair $(\alpha_{K},\beta_{K})$ of
$\tilde{D}(\mathbb{G}_{(r)})$ to

$\alpha_{K}+\beta_{K}:K[t]/t^{p}\ \to\ O(\mathbb{G}_{(r),K})\
\stackrel{{\scriptstyle\sim}}{{\to}}\
K((\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)})$

determines a bijection

(2.7.1) $\Pi_{O}:\Pi(O(\mathbb{G}_{(r)}))\stackrel{{\scriptstyle\sim}}{{\to}}\
\Pi((\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)}),$

where we abuse notation by interpreting the right hand side as the set of
(Zariski) points of the indicated $\Pi$-point scheme.

Moreover, for every $O(\mathbb{G}_{(r)})$-module $M$, $\Pi_{O}$ restricts to a
bijection

(2.7.2) $\Pi_{O,M}:\Pi(O(\mathbb{G}_{(r)}))_{M}\ \simeq\
\Pi((\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)})_{M}.$

###### Proof.

Since $(\underline{\mathfrak{g}}^{(r)})_{(1)}$ is an abelian unipotent finite
group scheme, since $\beta_{K}$ is a $\pi$-point of $\mathbb{G}_{(r)}$ and
thus factors through an abelian unipotent subgroup scheme, and since
$\alpha_{K}+\beta_{K}:K[t]/t^{p}\to KG$ is flat by Lemma 2.2, we conclude that
$\alpha_{K}+\beta_{K}$ is a $\pi$-point of the finite group scheme
$(\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)}$. Moreover, the
discussion preceding shows that every point of
$\Pi((\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)})$ (i.e.,
every equivalence class of $\pi$-points of
$\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)}$) is represented
by such a flat map $\alpha_{K}+\beta_{K}$ associated to a $\pi$-point pair of
$\tilde{D}(\mathbb{G}_{(r)})$.

Thus, to prove that $\Pi_{O}$ of (2.7.1) is well defined and injective (as
well as surjective), it suffices to show that two $\pi$-point pairs
$(\alpha_{K},\beta_{K}),\ (\alpha_{K}^{\prime},\beta_{K}^{\prime})$ of
$\tilde{D}(\mathbb{G}_{(r)})$ are $O(\mathbb{G}_{(r)})$-equivalent if and only
if $\alpha_{K}+\beta_{K},\ \alpha_{K}^{\prime}+\beta_{K}^{\prime}$ are
equivalent $\pi$-points of
$(\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)}$. This follows
immediately from Definition 2.3, Proposition 1.7, and the bijection
$\Psi_{\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)}}$ of
(1.6.1).

The fact that (2.7.1) restricts to a bijection $\Pi(O(\mathbb{G}_{(r)}))_{M}\
\stackrel{{\scriptstyle\sim}}{{\to}}\ \
\Pi((\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)})_{M}$ follows
immediately from the definitions (i.e., from Definition 2.3 plus Proposition
2.4 and [7, Defn 3.2]). ∎

##  3\. The tensor product property for $\tilde{D}(\mathbb{G}_{(r)})$-modules

Propositions 2.4 and 2.7 identify
$M\mapsto\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}$ with
$M\mapsto\Pi((\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)})_{M}$.
This immediately tells us that $M\mapsto\Pi(\tilde{D}(\mathbb{G}_{(r)})_{M}$
satisfies most of the properties required for a good theory of support
varieties of $\tilde{D}(\mathbb{G}_{(r)})$-modules. However, this
identification does not imply that
$M\mapsto\Pi(\tilde{D}(\mathbb{G}_{(r)})_{M}$ satisfies the “tensor product
property” (namely, that $\Pi(\tilde{D}(\mathbb{G}_{(r)})_{M\otimes
M^{\prime}}\ =\
\Pi(\tilde{D}(\mathbb{G}_{(r)})_{M}\cap\Pi(\tilde{D}(\mathbb{G}_{(r)})_{M^{\prime}}$)
because $O(\mathbb{G}_{(r),K})\ \stackrel{{\scriptstyle\sim}}{{\to}}\
K((\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)})$ is not a map
of Hopf algebras. In this section, we prove this tensor product property.

The following proposition of [7] is the key to our proof of Theorem 3.4
establishing the tensor product property for
$M\mapsto\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}$.

######  Proposition 3.1.

[7, Prop 2.2] Let $V$ be a $k$-vector space and $\alpha,\beta,\gamma$ be
pairwise commuting $V$-endomorphisms. Assume further $\alpha,\beta$ are
$p$-nilpotent and $\gamma$ is $p^{r}$-nilpotent in $End_{k}(V)$ for some $\geq
1$. Then $V$ is projective as a $k[u]/u^{p}$-module where the action of $u$ is
given by $\alpha$ if and only if it is projective as a $k[v]/v^{p}$module
where the action of $v$ is given by $\alpha+\beta\gamma$.

We apply Proposition 3.1 to prove the following comparison of pull-backs along
a given $\pi$-point of tensor products of $A$-modules with two different
$A$-module structures determined by different coproducts on $A$.

######  Proposition 3.2.

Let $A$ be a finite dimensional, local, commutative $k$-algebra with maximal
ideal $\mathfrak{m}$ satisfying the property that $a^{p}=0$ for all
$a\in\mathfrak{m}$. Assume that $A$ is equipped with two $A$-linear coproducts
$\Delta,\ \Delta^{\prime}:A\to A\otimes A$ with the property that

(3.2.1) $(\Delta(a)-a\otimes 1-1\otimes a),\ (\Delta^{\prime}(a)-a\otimes
1-1\otimes a)\ \in\ \mathfrak{m}\otimes\mathfrak{m},\quad\forall
a\in\mathfrak{m}.$

Let $M,\ M^{\prime}$ be $A$-modules and let $M\boxtimes M^{\prime}$ be the
$A\otimes A$ module given as the external tensor product. Then for any flat
map $\alpha:k[t]/t^{p}\to A$, the pull-back of $M\boxtimes M^{\prime}$ via
$\Delta\circ\alpha$ is a projective $k[t]/t^{p}$-module if and only if the
pull-back of $M\boxtimes M^{\prime}$ via $\Delta^{\prime}\circ\alpha$ is a
projective $k[t]/t^{p}$-module.

More generally, let $H$ be finite dimensional, local, commutative Hopf algebra
with maximal ideal $\mathfrak{m}_{H}$ such that $c^{p}=0$ for all
$c\in\mathfrak{m}_{H}$ and equip $A\otimes H$ with the coproducts
$\Delta\otimes\Delta_{H},\ \Delta^{\prime}\otimes\Delta_{H}$. Consider
$(A\otimes H)$-modules $M,\ M^{\prime}$, and let $M\boxtimes M^{\prime}$
denote the $(A\otimes H)\otimes(A\otimes H)$ module given as the external
tensor product. Then for any flat map $\alpha:k[t]/t^{\mathfrak{p}}\to
A\otimes H$, the pull-back of $M\boxtimes M^{\prime}$ via
$(\Delta\otimes\Delta_{H})\circ\alpha$ is a projective $k[t]/t^{p}$ module if
and only if the pull-back of $M\boxtimes M^{\prime}$ via
$(\Delta^{\prime}\otimes\Delta_{H})\circ\alpha$ is a projective
$k[t]/t^{p}$-module.

###### Proof.

We apply Proposition 3.1 to the $k$-vector space $V=M\otimes M^{\prime}$ and
consider triples of $p$-nilpotent elements in the image of $A\otimes A\to
End_{k}(V)$ given by viewing $V$ as the external tensor product $M\boxtimes
M^{\prime}$ (and thus an $A\otimes A$ module for the commutative $k$-algebra
$A\otimes A$).

Consider an element $a\in\mathfrak{m}\subset A$ and write
$\Delta(a)-\Delta^{\prime}(a)\ =\ \sum_{i=1}^{t}\ b_{i}\otimes
b_{i}^{\prime}\in\mathfrak{m}\otimes\mathfrak{m}.$ Observe that
$\Delta(a)-\Delta^{\prime}(a)\in A\otimes A$ acts on $V=M\boxtimes M^{\prime}$
as the sum of the actions commuting elements $(b_{i}\otimes 1)\cdot(1\otimes
b_{i}^{\prime}),1\leq i\leq,t$ each of which is $p$-nilpotent.. We apply
Proposition 3.1 with $V=M\boxtimes M^{\prime}$ successively to the triples

$\alpha=((\Delta^{\prime}(a)+\sum_{s=1}^{i}b_{s}\otimes
b_{s}^{\prime}),\quad\beta=(b_{i+1}\otimes 1),\quad\gamma=(1\otimes
b_{i+1}^{\prime})$

with $i+1\leq t$ to conclude that $V$ is projective for $k[u]/u^{p}$ with $u$
acting as $\Delta^{\prime}(a)+\sum_{s=1}^{i}b_{s}\otimes b_{s}^{\prime}$ if
and only if $V$ is projective for $k[v]/t^{p}$ with $v$ acting as
$\Delta^{\prime}(a)+\sum_{s=1}^{i+1}b_{s}\otimes b_{s}^{\prime}$. This
establishes the first assertion.

We extend this argument to $A\otimes H$, a commutative algebra with coproducts
$\Delta\otimes\Delta_{H},\ \Delta^{\prime}\otimes\Delta_{H}$. We denote by
$\mathfrak{n}\subset A\otimes H$ the maximal ideal of $A\otimes H$ generated
by $\mathfrak{m}\otimes 1,\ 1\otimes\mathfrak{m}_{H}$, and observe that every
element in $\mathfrak{n}$ has $p$-th power 0. Moreover,
$(\Delta\otimes\Delta_{H})-(\Delta^{\prime}\otimes\Delta_{H})\ =\
(\Delta-\Delta^{\prime})\otimes\Delta_{H}$ applied to $a\otimes
1\in\mathfrak{n}$ lies in $\mathfrak{n}\otimes\mathfrak{n}$ whereas
$(\Delta-\Delta^{\prime})\otimes\Delta_{H}$ applied to $1\otimes h\in
1\otimes\mathfrak{m}_{H}$ is 0. Consequently,

$((\Delta\otimes\Delta_{H})-(\Delta^{\prime}\otimes\Delta_{H}))(b)\ \in\
\mathfrak{n}\otimes\mathfrak{n},\quad\forall b\in\mathfrak{n}.$

Thus we may repeat the preceding argument with $(A,\mathfrak{m})$ replaced by
$(A\otimes H,\mathfrak{n})$ to complete the proof. ∎

######  Definition 3.3.

Consider $O(\mathbb{G}_{(r)})$-modules $M,\ M^{\prime}$.

We denote by $M\otimes M^{\prime}$ the $O(\mathbb{G}_{(r)})$-module determined
by the coproduct of $O(\mathbb{G}_{(r)})$ defined as by restriction of the
coproduct of $\tilde{D}(\mathbb{G}_{(r)})$ (see Proposition 1.4).

We denote by $M\otimes_{G}M^{\prime}$ the $k$-vector space $M\otimes
M^{\prime}$ equipped with the $O(\mathbb{G}_{(r)})\ \simeq\
(k((\underline{\mathfrak{g}}^{(r)})_{(1)})\times\mathbb{G}_{(r)})$-module
structure determined by the coproduct for the group algebra
$k((\underline{\mathfrak{g}}^{(r)})_{(1)})\times\mathbb{G}_{(r)})$ of the
finite group scheme
$(\underline{\mathfrak{g}}^{(r)})_{(1)})\times\mathbb{G}_{(r)}$.

In the following theorem, we prove the “tensor product property” for
$\tilde{D}(\mathbb{G}_{(r)})$-modules; namely, $M\ \mapsto\
\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}$ sends tensor product of modules to
intersection of the supports of its tensor factors.

######  Theorem 3.4.

Let $(\alpha_{K},\beta_{K})$ be a $\pi$-point pair of
$\tilde{D}(\mathbb{G}_{(r)})$ as in Definition 2.1 and let $M,\ M^{\prime}$ be
$O(\mathbb{G}_{(r)})$-modules. Then $(\alpha_{K}+\beta_{K})^{*}(M_{K}\otimes
M^{\prime}_{K})$ is a free $K[t]/t^{p}$-module if and only if
$(\alpha_{K}+\beta_{K})^{*}(M_{K}\otimes_{G}M^{\prime}_{K})$ is a free
$K[t]/t^{p}$-module. Thus,

(3.4.1) $\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M\otimes M^{\prime}}\ =\
\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}\cap\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M^{\prime}}.$

###### Proof.

We recall that any $\beta_{K}:K[t]/t^{p}\to K\mathbb{G}_{(r)}$ is equivalent
to a map which factors through a map of group algebras induced by a map of
finite group schemes $\mathbb{G}_{a(r),L}\to\mathbb{G}_{(r),L}$ [7, Prop 4.2].
By replacing $K$ if necessary by a field extension $K^{\prime}/K$, we may
assume $K=L$. Thus when comparing restrictions along a given $\pi$-point pair,
we may restrict to $K[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}]\otimes
K\mathbb{G}_{a(r)}\hookrightarrow O(\mathbb{G}_{(r)})\otimes K$ and replace
$M,\ M^{\prime}$ by their restrictions to
$K[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}]\otimes K\mathbb{G}_{a(r)}$.

We apply Proposition 3.2 with $A$ equal to
$K[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}]$ and $H=K\mathbb{G}_{a(r)}$, and equip
$A\otimes K\mathbb{G}_{a(r)}$ with the two coproduct structures obtained by
restriction of the two coproduct structures on $O(\mathbb{G}_{(r)})$ discussed
in Definition 3.3. As shown in [13, I.2.4], $\Delta,\ \Delta^{\prime}$ satisfy
condition (3.2.1) of Proposition 3.2. In our application of Proposition 3.2,
we take $\alpha:K[t]/t^{p}\ \to\ K[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}]\otimes
K\mathbb{G}_{a(r)}$ to be the composition of the coproduct $K[t]/t^{p}\to
K[t]/t^{p}\otimes K[t]/t^{p},\ t\mapsto t\otimes 1+1\otimes t$ and the
external tensor product $\alpha_{K}\otimes\beta_{K}:K[t]/t^{p}\to
K[t]/t^{p}\otimes K[t]/t^{p}\to K[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}]\otimes
K\mathbb{G}_{a(r)}$.

By applying Proposition 3.2 with these values for $A,H,\alpha$, we conclude
that it suffices to prove the equality (3.4.1) after replacing the tensor
product for
$K[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}]\otimes K\mathbb{G}_{a(r)}$-modules
(given by the restriction of the coproduct of $O(\mathbb{G}_{(r)})$) by the
tensor product given by the restriction of the tensor product of
$(\mathfrak{g}^{(r)})_{(1)}\times\mathbb{G}_{(r)}$-modules. To conclude the
proof of equality (3.4.1), we apply the tensor product property for modules
for the (infinitesimal) group scheme $G\ =\
(\mathfrak{g}^{(r)})_{(1)}\times\mathbb{G}_{(r)}$ [8, Prop 3.2]; in other
words, for $M\otimes_{G}M^{\prime}$ in the notation of Definition 3.3. ∎

Theorem 3.4 enables topological structures on the sets
$\Pi(\tilde{D}(\mathbb{G}_{(r)}))$ and $\Pi(O(\mathbb{G}_{(r)}))$.

######  Corollary 3.5.

Defining a subset of $\Pi(\tilde{D}(\mathbb{G}_{(r)}))$ to be closed if and
only if it is of the form
$\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}\subset\Pi(\tilde{D}(\mathbb{G}_{(r)}))$
for some finite dimensional $\tilde{D}(\mathbb{G}_{(r)})$-module $M$ defines a
topology on $\Pi(\tilde{D}(\mathbb{G}_{(r)}))$. Similarly, defining a subset
of $\Pi(O(\mathbb{G}_{(r)}))$ to be closed if and only if it is of the form
$\Pi(O(\mathbb{G}_{(r)}))_{M}\subset\Pi(O(\mathbb{G}_{(r)}))$ for some finite
dimensional $O(\mathbb{G}_{(r)})$-module $M$ defines a topology on
$\Pi(O(\mathbb{G}_{(r)}))$.

With these topologies, the bijection
$\Pi_{\tilde{D}}:\Pi(\tilde{D}(\mathbb{G}_{(r)}))\ \twoheadrightarrow\
\Pi(O(\mathbb{G}_{(r)}))$ of Proposition 2.4 is a homeomorphism.

###### Proof.

The asserted formulation of a topology on $\Pi(\tilde{D}(\mathbb{G}_{(r)}))$
is justified by Proposition 2.6(4) and Theorem 3.4. This applies as well to
justifiy the topology on $\Pi(O(\mathbb{G}_{(r)}))$, since
$i_{O}:O(\mathbb{G}_{(r)})\to\tilde{D}(\mathbb{G}_{(r)})$ is a map of Hopf
algebras. ∎

##  4\.
$\Psi_{\tilde{D}}:\Pi(\tilde{D}(\mathbb{G}_{(r)}))\to\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))$

Cohomological support varieties can be readily defined for any finite
dimension Hopf algebra $H$ whose cohomology is finitely generated (i.e., such
that $H^{*}(H,k)$ is a finitely generated $k$-algebra and that $H^{*}(H,M)$ is
a finite $H^{*}(H,k)$-module for any finite dimensional $H$-module). In this
section, we exhibit in Proposition 4.6 a continuous map $\Psi_{\tilde{D}}$
from the $\pi$-point pair space $\Pi(\tilde{D}(\mathbb{G}_{(r)}))$ introduced
in Section 2 to the (projectivized) cohomology support variety
$\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))$. To do this, we
utilize the subalgebra $O(\mathbb{G}_{(r)})\subset\tilde{D}(\mathbb{G}_{(r)})$
and the related finite group scheme
$\mathbb{G}_{(r)}\times(\underline{\mathfrak{g}}^{(r)})_{(1)}$, relying upon
the isomorphism
$\Pi(\mathbb{G}_{(r)}\times(\underline{\mathfrak{g}}^{(r)})_{(1)})\
\stackrel{{\scriptstyle\sim}}{{\to}}\
\mathbb{P}(H^{\bullet}(\mathbb{G}_{(r)}\times(\underline{\mathfrak{g}}^{(r)})_{(1)},k)$.

Associated to a finite dimensional $\tilde{D}(\mathbb{G}_{(r)})$-module $M$,
we consider the annihilator ideal $I_{M}\subset
H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k)$ of the
$H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k)$-module
$Ext_{\tilde{D}(\mathbb{G}_{(r)})}^{*}(M,M)$ which equals the annihilator of
$1\in Ext_{\tilde{D}(\mathbb{G}_{(r)})}^{*}(M,M)$ which in turn equals the
kernel of the graded map of $k$-algebras
$H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k)\to
Ext_{\tilde{D}(\mathbb{G}_{(r)})}^{*}(M,M)$. We shall consider the projective
scheme $\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))$ whose
(Zariski) points are (non-trivial) homogenous prime ideals of
$H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k)$ and the reduced projective scheme
$\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))_{M}$ whose points are
(non-trivial) homogenous prime ideals of
$H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k)$ containing $I_{M}$.

Consider the quotient map $\mathbb{G}_{(r+2)}\ \twoheadrightarrow\
(\mathbb{G}^{(r+1)})_{(1)}$, the image of
$F^{r+1}:\mathbb{G}_{(r+2)}\to(\mathbb{G}^{(r+1)})_{(r+2)}$ whose kernel is
$\mathbb{G}_{(r+1)}$. This quotient map determines the flat map of
$k$-algebras

$k[(\mathbb{G}^{(r+1)})_{(1)}]\ \simeq\
k[\mathbb{G}_{(r+2})/\mathbb{G}_{(r+1)}]\ \to\ k[\mathbb{G}_{(r+2)}]$

with the property that $k[\mathbb{G}_{(r+1)}]\ \simeq\
k[\mathbb{G}_{(r+2)}]\otimes_{k[\mathbb{G}_{(r+2})/\mathbb{G}_{(r+1)}]}k$.

The following proposition is a summary of a central technique of [5].

######  Proposition 4.1.

The deformation $k[\mathbb{G}_{(r+2})/\mathbb{G}_{(r+1)}]\ \to\
k[\mathbb{G}_{(r+2)}]$ of $k[\mathbb{G}_{(r+1)}]$ (parametrized by
$k[\mathbb{G}_{(r+2})/\mathbb{G}_{(r+1)}]\simeq
k[(\mathbb{G}^{(r+1)})_{(1)}]$) determines a $\mathbb{G}$-equivariant map

(4.1.1) $\mathfrak{g}^{(r+1)}\ \to\ H^{2}(k[\mathbb{G}_{(r+1)}],k).$

The deformation $k[\mathbb{G}_{(r+2})/\mathbb{G}_{(r+1)}]\\#k\mathbb{G}_{(r)}\
\to\ k[\mathbb{G}_{(r+2)}]\\#k\mathbb{G}_{(r)}$ (also parametrized by
$k[\mathbb{G}_{(r+2})/\mathbb{G}_{(r+1)}]$) provides a
$\mathbb{G}_{(r)}$-equivariant lifting of (4.1.1)

(4.1.2) $\sigma_{\tilde{D}}:\mathfrak{g}^{(r+1)}\ \to\
H^{2}(\tilde{D}(\mathbb{G}_{(r)}),k).$

.

Consequently, we obtain a map of $k$-algebras

(4.1.3) $\tilde{\theta}_{r}:H^{*}(\mathbb{G}_{(r)},k)\otimes
S^{\bullet}(\mathfrak{g}^{(r+1)}[2])\ \to\
H^{*}(\tilde{D}(\mathbb{G}_{(r)}),k)$

defined as the product of the inflation map on $H^{*}(\mathbb{G}_{(r)},k)$ and
the algebra extension of $\sigma_{\tilde{D}}$ on
$S^{\bullet}(\mathfrak{g}^{(r+1)}[2])$.

###### Proof.

Each element of the tangent space at the identity of
$(\mathbb{G}^{(r+1)})_{(1)}$ (naturally identified with
$\mathfrak{g}^{(r+1)}$) corresponds to a map $k[(\mathbb{G}^{(r+1)})_{(1)}]\to
k[\epsilon]/\epsilon^{2}$ of $k$-algebras. Extending the deformation
$k[\mathbb{G}_{(r+2})/\mathbb{G}_{(r+1)}]\ \to\ k[\mathbb{G}_{(r+2)}]$ along
such a map determines a deformation of $k[\mathbb{G}_{(r+1)}]$ parametrized by
$k[\epsilon]/\epsilon^{2}$. Thus, the map (4.1.1) is a consequence of the
naturality of Gerstenhaber’s bijection [12] between deformations of
$k[\mathbb{G}_{(r+1)}]$ parametrized by $k[\epsilon]/\epsilon^{2}$ and the
second Hochschild cohomology group $HH^{2}(k[\mathbb{G}_{(r+1)}])$ of
$k[\mathbb{G}_{(r+1)}]$ together with the natural map
$HH^{2}(k[\mathbb{G}_{(r+1)}])\to H^{2}(k[\mathbb{G}_{(r+1)}],k))$ (see [5,
Prop 3.4]). The map $\sigma_{\tilde{D}}$ of (4.1.2) follows exactly the same
way. Moreover, the naturality of Gerstenhaber’s bijection implies that
$\sigma_{\tilde{D}}$ lifts (4.1.1).

The construction of $\tilde{\theta}_{r}:H^{*}(\mathbb{G}_{(r)},k)\otimes
S^{\bullet}(\mathfrak{g}^{(r+1)}[2])\ \to\
H^{*}(\tilde{D}(\mathbb{G}_{(r)}),k)$ is implicit in the last statement of
this proposition. ∎

Finite generation of the cohomology algebra $H^{*}(D(\mathbb{G}_{(r)}),k)$ and
the finiteness of $H^{*}(D(\mathbb{G}_{(r)}),M)$ as an
$H^{*}(D(\mathbb{G}_{(r)}),k)$-module (for a finite dimensional
$D(\mathbb{G}_{(r)})$-module $M$) was established in [5, Thm 5.3]. Using
Proposition 4.1, we extend this finite generation property to
$\tilde{D}(\mathbb{G}_{(r)})$ and $O(\mathbb{G}_{(r)})$ as stated below; the
proof is a straight-forward adaptation of that of [5, Thm 5.3] which in turn
is an adaptation of [11, Thm 1.1].

######  Theorem 4.2.

Consider the map $\tilde{\theta}_{r}:H^{*}(\mathbb{G}_{(r)},k)\otimes
S^{\bullet}(\mathfrak{g}^{(r+1)}[2])\ \to\
H^{*}(\tilde{D}(\mathbb{G}_{(r)}),k)$ of (4.1.3).

  1. (1)

The composition

(4.2.1) $i_{O}^{*}\circ\tilde{\theta}_{r}:H^{*}(\mathbb{G}_{(r)},k)\otimes
S^{\bullet}(\mathfrak{g}^{(r+1)}[2])\ \to\
H^{*}(\tilde{D}(\mathbb{G}_{(r)}),k)$ $\to\ H^{*}(O(\mathbb{G}_{(r)}),k)\
\simeq\ H^{\bullet}(\mathbb{G}_{(r)},k)\otimes
H^{*}(k[(\mathbb{G}^{(r)})_{(1)}],k)$

is a split injection with $H^{*}(\mathbb{G}_{(r)},k)\otimes
S^{\bullet}(\mathfrak{g}^{(r+1)}[2])$-module complement

$H^{*}(\mathbb{G}_{(r)},k)\otimes\Lambda^{>0}(\mathfrak{g}[1])\otimes\
S^{\bullet}(\mathfrak{g}^{(r+1)}[2])$

which consists entirely of nilpotent elements

  2. (2)

For any finite dimensional $\tilde{D}(\mathbb{G}_{(r)})$-module $M$,
$H^{*}(\tilde{D}(\mathbb{G}_{(r)}),M)$ is a finitely generated
$H^{\bullet}(\mathbb{G}_{(r)},k)\otimes
S^{\bullet}(\mathfrak{g}^{(r+1)}[2])$-module (with action given by the
restriction along $\tilde{\theta}_{r}$ of the natural action of
.$H^{*}(\tilde{D}(\mathbb{G}_{(r)}),k)$ on
$H^{*}(\tilde{D}(\mathbb{G}_{(r)}),M)$).

  3. (3)

For any finite dimensional $O(\mathbb{G}_{(r)})$-module $M$,
$H^{*}(O(\mathbb{G}_{(r)}),M)$ is also a finitely generated
$H^{\bullet}(\mathbb{G}_{(r)},k)\otimes
S^{\bullet}(\mathfrak{g}^{(r+1)}[2])$-module.

###### Proof.

We utilize the spectral sequence

(4.2.2) $E_{2}^{s,t}=H^{s}(\mathbb{G}_{(r)},H^{t}(k[G_{(r+1)}],k))\
\Rightarrow\ H^{s+t}(\tilde{D}(\mathbb{G}_{(r)}),k),$

which can be derived as a special case of Grothendieck’s spectral sequence for
the derived functors of a composition of left exact functors exactly as in [5,
Prop 5.2].

The composition $i_{O}^{*}\circ\tilde{\theta}_{r}$ restricts to the identity
on the tensor factor $H^{*}(\mathbb{G}_{(r)},k)$ by the definition of
$\tilde{\theta}_{r}$ and the fact that the composition $k\mathbb{G}_{(r)}\to
O(\mathbb{G}_{(r)})\to\tilde{D}(\mathbb{G}_{(r)})$ is the evident inclusion.
The deformation construction leading to the definition of $\tilde{\theta}_{r}$
is designed to insure that the composition
$S^{\bullet}(\mathfrak{g}^{(r+1)}[2])\to
H^{*}(\tilde{D}(\mathbb{G}_{(r)}),k)\to
H^{*}(k[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}],k)$ sends elements of
$\mathfrak{g}^{(r+1)}[2]$ to elements of
$H^{2}(k[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}],k)$ which generate
$H^{*}(k[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}],k)$ modulo nilpotent elements.
The identification of the restriction of the map
$i_{O}^{*}\circ\tilde{\theta}_{r}$ to $1\otimes
S^{\bullet}(\mathfrak{g}^{(r+1)}[2])$ and the computation of
$H^{*}(k[(\mathbb{G}^{(r)})_{(1)}],k)$ are given in [5, §4], thereby
establishing the asserted split surjection.

The fact that $H^{*}(\tilde{D}(\mathbb{G}_{(r)}),M)$ is a finitely generated
$H^{\bullet}(\mathbb{G}_{(r)},k)\otimes
S^{\bullet}(\mathfrak{g}^{(r+1)}[2])$-module for any finite dimensional
$\tilde{D}(\mathbb{G}_{(r)})$-module $M$ arises from the natural pairing of
(4.2.2) with the corresponding spectral sequence

(4.2.3) $E_{2}^{s,t}(M)=H^{s}(\mathbb{G}_{(r)},H^{t}(k[G_{(r+1)}],M))\
\Rightarrow\ H^{s+t}(\tilde{D}(\mathbb{G}_{(r)}),M)$

with $M$ coefficients (as in the proof of [5, Prop 5.2]). The same argument
with $\tilde{D}(\mathbb{G}_{(r)})$ replaced by $O(\mathbb{G}_{(r)})$ implies
that $H^{*}(O(\mathbb{G}_{(r)}),M)$ is a finitely generated
$H^{\bullet}(\mathbb{G}_{(r)},k)\otimes
S^{\bullet}(\mathfrak{g}^{(r+1)}[2])$-module. ∎

######  Corollary 4.3.

The map of commutative $k$-algebras $\tilde{\theta}_{r}$ induces a finite,
surjective map

$\tilde{\Theta}_{r}:\operatorname{Spec}\nolimits
H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k)\to\operatorname{Spec}\nolimits
H^{\bullet}(\mathbb{G}_{(r)},k)\times\underline{\mathfrak{g}^{*}}^{(r+1)}.$

Consequently,

$dim(\operatorname{Spec}\nolimits H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))\
=\ dim(\operatorname{Spec}\nolimits
H^{\bullet}(\mathbb{G}_{(r)},k)+dim(\mathfrak{g}).$

###### Proof.

By Theorem 4.2, the composition $i^{*}\circ\tilde{\theta}_{r}$ is injective,
so that $\tilde{\theta}_{r}$ is injective; this theorem also tells us that
$\tilde{\theta}_{r}$ is a finite map. Consequently, $\tilde{\theta}_{r}$ is an
integral extension which implies that $\Theta_{r}$ is finite and surjective
(see [17, Thm 9.3]).

The determination of $dim(\operatorname{Spec}\nolimits
H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))$ follows from the observation that
$dim(\operatorname{Spec}\nolimits
H^{\bullet}(\mathbb{G}_{(r)},k)\times\underline{\mathfrak{g}^{*}}^{(r+1)})\ =\
dim(\operatorname{Spec}\nolimits
H^{\bullet}(\mathbb{G}_{(r)},k))+dim(\mathfrak{g}).$ ∎

We establish the following notation.

######  Notation 4.4.

We denote by $\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))$
(respectively, $\mathbb{P}(H^{\bullet}(O(\mathbb{G}_{(r)}),k))$; respectively,
$\mathbb{P}(H^{\bullet}(\mathbb{G}_{(r)},k))$) the projective scheme
associated to the commutative graded $k$-algebra
$H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k)$ (resp.,
$H^{\bullet}(O(\mathbb{G}_{(r)}),k)$; resp.,
$H^{\bullet}(\mathbb{G}_{(r)},k))$). Thus, the points of
$\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))$ (resp.,
$\mathbb{P}(H^{\bullet}(O(\mathbb{G}_{(r)}),k)$; resp.,
$\mathbb{P}(H^{\bullet}(\mathbb{G}_{(r)},k))$) are the non-trivial homogeneous
prime ideals of $H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k)$ (resp.,
$H^{\bullet}(O(\mathbb{G}_{(r)}),k)$; resp.,
$H^{\bullet}(\mathbb{G}_{(r)},k))$). For any finite dimensional
$\tilde{D}(\mathbb{G}_{(r)})$-module $M$, we denote by

$\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))_{M}\ \subset\
\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k)),$
$\mathbb{P}(H^{\bullet}(O(\mathbb{G}_{(r)})),k))_{M}\ \subset\
\mathbb{P}(H^{\bullet}(O(\mathbb{G}_{(r)})),k)),$
$\mathbb{P}(H^{\bullet}(\mathbb{G}_{(r)},k))_{M}\ \subset\
\mathbb{P}(H^{\bullet}(\mathbb{G}_{(r)},k))$

the reduced closed subscheme given by the radicals of the ideals

$ker\\{H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k)\to
Ext^{*}_{\tilde{D}(\mathbb{G}_{(r)})}(M,M)\\},$
$ker\\{H^{\bullet}(O(\mathbb{G}_{(r)}),k)\to
Ext^{*}_{O(\mathbb{G}_{(r)}))}(M,M)\\},$
$ker\\{H^{\bullet}(\mathbb{G}_{(r)},k)\to Ext^{*}_{\mathbb{G}_{(r)}}(M,M)\\}$

respectively.

The equivalence relation on $\pi$-point pairs $(\alpha_{K},\beta_{K})$ of
$\tilde{D}(\mathbb{G}_{(r)})$ (and thus on $O(\mathbb{G}_{(r)})$) has been
designed to enable the validity of the following proposition.

######  Proposition 4.5.

Define
$\Psi_{O}:\Pi(O(\mathbb{G}_{(r)}))\to\mathbb{P}(H^{*}(O(\mathbb{G}_{(r)}),k))$
to be the map of sets sending the equivalence class of
$(\alpha_{K},\beta_{K})\in\Pi(O(\mathbb{G}_{(r)}))$ to

$ker\\{(\alpha_{K}+\beta_{K})^{*}:H^{\bullet}(O(\mathbb{G}_{(r)})_{K},K)\to
H^{\bullet}(K[t]/t^{p},K)\\}\ \cap\ H^{*}(O(\mathbb{G}_{(r)}),k)$

in $\mathbb{P}(H^{*}(O(\mathbb{G}_{(r)}),k))$. A choice of isomorphism of $k$
algebras $j:O(\mathbb{G}_{(r)})\ \stackrel{{\scriptstyle\sim}}{{\to}}\
k(\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)})$ as in
Proposition 1.6 determines the vertical maps in the following commutative
square of homeomorphisms

(4.5.1)
$\textstyle{\Pi(O(\mathbb{G}_{(r)}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{O}}$$\scriptstyle{\Pi_{O}}$$\textstyle{\mathbb{P}(H^{\bullet}(O(\mathbb{G}_{(r)}),k))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{P}_{O}}$$\textstyle{\Pi(\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{G}}$$\textstyle{\mathbb{P}(H^{\bullet}(\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)},k))}$

whose lower horizontal map is the natural homeomorphism of C for $G$ the
finite group scheme
$(\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)}$.

Moreover, for every finite dimensional $O(\mathbb{G}_{(r)})$-module $M$,
$\Psi_{O}$ restricts to a bijection $\Psi_{O,M}:\Pi(O(\mathbb{G}_{(r)}))_{M}\
\stackrel{{\scriptstyle\sim}}{{\to}}\
\mathbb{P}(H^{\bullet}(O(\mathbb{G}_{(r)}),k))_{M}$.

###### Proof.

The definition of $\Psi_{O}$ is compatible with the definition of
$\Psi_{G}:\Pi(\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)})\
\to\
\mathbb{P}(H^{\bullet}(\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)},k))$
(see $(\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)}$ and [7,
Prop 3.5]), so that (4.5.1) commutes. The fact that $\Psi_{O}$ is well defined
follows from the fact that $\Psi_{G}$ is well defined and that the maps
$\Pi_{O},\ \mathbb{P}_{O}$ are bijections. The topology on
$\Pi(\tilde{D}(\mathbb{G}_{(r)}))\ =\ \Pi(O(\mathbb{G}_{(r)}))$ introduced in
Theorem 3.4 is equal to that induced by the bijection $\Pi_{O}$, so that this
bijection is a homeomorphism. The bijection $\mathbb{P}_{O}$ is a
homeomorphism since the isomorphism $j$ induces an isomorphism between the
category of finite dimensional $O(\mathbb{G}_{(r)})$-modules and the category
of finite dimensional
$k(\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)})$-modules.

The fact that $\Psi_{O,M}$ is a bijection for any finite dimensional
$O(\mathbb{G}_{(r)})$-module follows from the fact that $\Psi_{O}$ restricts
to a bijection
$\Psi_{O,M}:\Pi(\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)})\
\stackrel{{\scriptstyle\sim}}{{\to}}\
\mathbb{P}(H^{\bullet}(\underline{\mathfrak{g}}^{(r)})_{(1)}\times\mathbb{G}_{(r)},k))_{M}$.
∎

The following proposition complements Proposition 4.5 while introducing
$\Psi_{\tilde{D}}$.

######  Proposition 4.6.

The composition
$\Psi_{\tilde{D}}\equiv\mathbb{P}_{\tilde{D}}\circ\Psi_{O}\circ\Pi_{\tilde{D}}^{-1}$:

$\Pi(\tilde{D}(\mathbb{G}_{(r)}))\stackrel{{\scriptstyle\sim}}{{\to}}\Pi(O(\mathbb{G}_{(r)}))\
\stackrel{{\scriptstyle\sim}}{{\to}}\
\mathbb{P}(H^{\bullet}(O(\mathbb{G}_{(r)}),k))\ \to\
\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))$

is continuous, fitting in the commutative square

(4.6.1)
$\textstyle{\Pi(\tilde{D}(\mathbb{G}_{(r)}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{\tilde{D}}}$$\textstyle{\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))}$$\textstyle{\Pi(O(\mathbb{G}_{(r)}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Pi_{\tilde{D}}}$$\scriptstyle{\simeq}$$\scriptstyle{\Psi_{O}}$$\scriptstyle{\sim}$$\textstyle{\mathbb{P}(H^{\bullet}(O(\mathbb{G}_{(r)}),k))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{P}_{\tilde{D}}}$

with the properties that $\Pi_{\tilde{D}}$ and $\Psi_{O}$ are homeomorphisms,
and that the map $\mathbb{P}_{\tilde{D}}$ (induced by
$i_{O}:O(\mathbb{G}_{(r)})\to\tilde{D}(\mathbb{G}_{(r)})$) is a closed
immersion.

For any finite dimensional $\tilde{D}(\mathbb{G}_{(r)})$-module $M$,
$\Psi_{\tilde{D}}$ restricts to

(4.6.2) $\Psi_{\tilde{D},M}:\Pi(\tilde{D})_{M}\ \hookrightarrow\
\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))_{M}.$

###### Proof.

By Proposition 2.4, $\Pi_{\tilde{D}}$ is a homeomorphism, whereas $\Psi_{O}$
is a homeomorphism by Proposition 4.5. Theorem 4.2(1) implies that $i_{O}^{*}$
is a finite map and is surjective modulo nilpotents which implies that
$\mathbb{P}_{O}$ is a closed embedding. Thus, $\Psi_{\tilde{D}}$ is also a
closed embedding and thus is continuous.

To prove that $\Psi_{\tilde{D}}(\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M})$ is
contained in $\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))_{M}$, we
consider some $(\alpha_{K},\beta_{K})$ representing a point of
$\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}$; in other words, assume that
$(\alpha_{K}+\beta_{K})^{*}(M_{K})$ is not projective. Consider the following
commutative square

(4.6.3)
$\textstyle{H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\alpha_{K}+\beta_{K})^{*}}$$\textstyle{H^{\bullet}(K[t]/t^{p},K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Ext^{*}_{\tilde{D}(\mathbb{G}_{(r)}))}(M,M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Ext^{*}_{K[t]/t^{p}}((\alpha_{K}+\beta_{K})^{*}(M_{K}),(\alpha_{K}+\beta_{K})^{*}(M_{K}))}$

whose horizontal maps are induced by restriction along $\alpha_{K}+\beta_{K}$.
Since $(\alpha_{K}+\beta_{K})^{*}(M_{K})$ is not projective, the right
vertical map is injective. A simple diagram chase then implies that
$ker\\{(\alpha_{K}+\beta_{K})^{*}:H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)})),k)\to
H^{\bullet}(K[t]/t^{p},K)\\}$ contains
$ker\\{H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)})),k)\to
Ext^{*}_{K[t]/t^{p}}((\alpha_{K},\beta_{K})^{*}(M_{K}),(\alpha_{K},\beta_{K})^{*}(M_{K}))\\}$,
so that $\Psi_{\tilde{D}}$ applied to the class of $(\alpha_{K},\beta_{K})$
lies in $\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))_{M}$. ∎

##  5\. $\Pi(\tilde{D}(\mathbb{G}_{(r)}))$ for $\mathbb{G}$ admitting a
quasilogarithm

One missing aspect of our understanding of $\Psi_{\tilde{D}}$ in (4.6.1) is
whether or not it is a homeomorphism. By Proposition 4.6, this is equivalent
to whether or not $\mathbb{P}_{\tilde{D}}$ is surjective. With the added
hypothesis that $\mathbb{G}$ admits a quasilogarithm and that
$p^{r+1}>2dim(\mathbb{G})$, we prove this surjectivity of
$\mathbb{P}_{\tilde{D}}$ in Theorem 5.4. This enables us to conclude in
Theorem 5.6 that $M\mapsto\Pi(\tilde{D}(\mathbb{G}_{(r)})_{M}$ detects
projectivity: a finite dimensional $\tilde{D}(\mathbb{G}_{(r)})$-module $M$ is
projective if and only if $\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}$ is empty.

We first recall the definition of a quasilogarithm for $\mathbb{G}$.

######  Definition 5.1.

[14], [5, Defn 6.2] Let $\mathbb{G}$ be a linear algebraic group over $k$. A
quasilogarithm for $\mathbb{G}$ is a $\mathbb{G}$-equivariant map of
$k$-schemes

(5.1.1) $L:\mathbb{G}\ \to\ \operatorname{Spec}\nolimits
S^{\bullet}(\mathfrak{g}^{*})\equiv\underline{\mathfrak{g}}$

such that $L(e_{\mathbb{G}})=0$ and such that the differential
$d_{e_{\mathbb{G}}}L:T_{e_{\mathbb{G}}}\mathbb{G}\to
T_{0}\underline{\mathfrak{g}}=\mathfrak{g}$ is the identity.

We denote by $I_{r}\ \subset\ S^{\bullet}(\mathfrak{g}^{*})$ the
$\mathbb{G}$-stable ideal generated by $\\{f^{p^{r}},\
f\in\mathfrak{g}^{*}\\}$.

######  Example 5.2.

[2, Lem C3], [5, 6.4] If $\mathbb{G}$ is a simple algebraic group for which
$p$ is very good, then $\mathbb{G}$ admits a quasilogarithm. Furthermore any
Borel subgroup of $\mathbb{G}$ also admits a quasilogarithm. If $\mathbb{U}$
is a unipotent subgroup of a semi-simple algebraic group $\mathbb{G}$ which is
normalized by a maximal torus and if $p$ is greater than the nilpotent class
of $\mathbb{U}$, then $\mathbb{U}$ admits a quasilogarithm.

For us, a critical consequence of the condition that $\mathbb{G}$ admits a
quasilogarithm $L:\mathbb{G}\to\underline{\mathfrak{g}}$ is the following
result, a summary of the discussion prior to and including Lemma 6.6 of [5].

######  Proposition 5.3.

Assume that $\mathbb{G}$ is a linear algebraic group smooth over $k$ which
admits a quasilogarithm. The $\mathbb{G}$-equivariant map on coordinate
algebras associated to the composition $L\circ
i:\mathbb{G}_{(r)}\subset\mathbb{G}\to\underline{\mathfrak{g}}$ has the form

$S^{\bullet}(\mathfrak{g}^{*})\ \to\ S^{\bullet}(\mathfrak{g}^{*})/I_{r}\
\stackrel{{\scriptstyle\sim}}{{\to}}\ k[\mathbb{G}_{(r)}].$

Moreover, the isomorphism $S^{\bullet}(\mathfrak{g}^{*})/I_{r}\
\stackrel{{\scriptstyle\sim}}{{\to}}\ k[\mathbb{G}_{(r)}]$ induces an
isomorphism of $k$-algebras

$S^{\bullet}(\mathfrak{g}^{*})/I_{r}\\#k\mathbb{G}_{(r)}\
\stackrel{{\scriptstyle\sim}}{{\to}}\ k[\mathbb{G}_{(r)}]\\#k\mathbb{G}_{(r)}\
\simeq D(\mathbb{G}_{(r)}).$

Similarly, $L$ determines the isomorphism of $k$-algebras

$S^{\bullet}(\mathfrak{g}^{*})/I_{r+1}\\#k\mathbb{G}_{(r)}\
\stackrel{{\scriptstyle\sim}}{{\to}}\
k[\mathbb{G}_{(r+1)}]\\#k\mathbb{G}_{(r)}\ \simeq\tilde{D}(\mathbb{G}_{(r)}).$

These smash products algebras
$S^{\bullet}(\mathfrak{g}^{*})/I_{r}\\#k\mathbb{G}_{(r)},\
S^{\bullet}(\mathfrak{g}^{*})/I_{r+1}\\#k\mathbb{G}_{(r)}$ are associated to
the coadjoint action of $\mathbb{G}_{(r)}$ on $\mathfrak{g}^{*}$. They both
admit a $\mathbb{G}_{(r)}$-equivariant grading, with $\mathfrak{g}^{*}$
homogenous of degree 1 and $k\mathbb{G}_{(r)}$ homogeneous of degree 0.

Using the structure of a quasilogarithm on $\mathbb{G}$, we obtain the
following strengthening of Corollary 4.3. The proof of this theorem is
essentially that of [5, Thm 6.9] which gives the analogous result for
$D(\mathbb{G}_{(r)})$.

######  Theorem 5.4.

Assume that $\mathbb{G}$ is a linear algebraic group smooth over $k$ which
admits a quasilogarithm. Further assume that $p^{r+1}>2dim(\mathbb{G})$. Then
the map

$\tilde{\theta}_{r}:H^{*}(\mathbb{G}_{(r)},k)\otimes
S^{\bullet}(\mathfrak{g}^{(r+1)}[2])\ \to
H^{*}(\tilde{D}(\mathbb{G}_{(r)}),k)$

of Theorem 4.2 is a split injection with $H^{*}(\mathbb{G}_{(r)},k)\otimes
S^{\bullet}(\mathfrak{g}^{(r+1)}[2])$-module complement consisting entirely of
nilpotent elements.

Similarly, the map $i_{O}^{*}:H^{*}(\tilde{D}(\mathbb{G}_{(r)}),k)\ \to\
H^{*}(O(\mathbb{G}_{(r)}),k)$ has nilpotent kernel in
$H^{*}(\tilde{D}(\mathbb{G}_{(r)}),k)$; and the image in
$H^{*}(O(\mathbb{G}_{(r)}),k)$ has $H^{*}(\mathbb{G}_{(r)},k)\otimes
S^{\bullet}(\mathfrak{g}^{(r+1)}[2])$-module complement consisting entirely of
nilpotent elements.

Thus,
$\mathbb{P}_{O}:\mathbb{P}(H^{*}(O(\mathbb{G}_{(r)}),k))\to\mathbb{P}(H^{*}(\tilde{D}(\mathbb{G}_{(r)}),k))$
is a homeomorphism. Consequently,
$\Psi_{\tilde{D}}:\Pi(\tilde{D}(\mathbb{G}_{(r)}))\ \to\
\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))$ is also a
homeomorphism

###### Proof.

The statement that $\tilde{\theta}_{r}$ is a split injection is that of
Theorem 4.2(1).

The gradings on $S^{\bullet}(\mathfrak{g}^{*})/I_{r}\\#k\mathbb{G}_{(r)},\
S^{\bullet}(\mathfrak{g}^{*})/I_{r+1}\\#k\mathbb{G}_{(r)}$ (and thus also on
$O(\mathbb{G}_{(r)})$) determine internal gradings on the spectral sequences
of Theorem 4.2; we view these gradings as $\mathbb{Z}/p^{r+1}$-gradings. As in
[5, Lem 6.7], the internal grading on $\mathfrak{g}^{(r+1)}[2]\subset
H^{2}(k[\mathbb{G}_{(r+1)}],k)$ is $p^{r+1}$, hence $\ \equiv 0\ mod\ p^{r+1}$
and the internal grading of $\mathfrak{g}[1]\subset
H^{1}(k[\mathbb{G}_{(r+1)}],k)$ is 1.

We conclude that the map $\tilde{\theta}_{r}$ of Theorem 4.2 is an isomorphism
onto the degree $\equiv 0$ portion of the cohomoloy of
$\tilde{D}(\mathbb{G}_{(r)})$. As argued in the proof of [5, Thm 6.9], the
hypothesis that $p^{r+1}>2dim(\mathbb{G})$ implies that this image has an
$H^{*}(\mathbb{G}_{(r)},k)\otimes S^{\bullet}(\mathfrak{g}^{(r+1)}[2])$-module
complement in $H^{*}(\tilde{D}(\mathbb{G}_{(r)}),k)$ which is contained in the
nilradical.

We may repeat this argument with the spectral sequence (4.2.2) for
$\tilde{D}(\mathbb{G}_{(r)})$ mapping to the spectral sequence

(5.4.1)
$E_{2}^{s,t}=H^{s}(\mathbb{G}_{(r)},H^{t}(k[G_{(r+1)}/\mathbb{G}_{(r)}],k))\
\Rightarrow\ H^{s+t}(O(\mathbb{G}_{(r)}),k).$

Thus, the composition $H^{*}(\mathbb{G}_{(r)},k)\otimes
S^{\bullet}(\mathfrak{g}^{(r+1)}[2])\ \to\
H*(\tilde{D}(\mathbb{G}_{(r)}),k)\to\ H^{*}(O(\mathbb{G}_{(r)}),k)$ is also a
split injection with $H^{*}(\mathbb{G}_{(r)},k)\otimes
S^{\bullet}(\mathfrak{g}^{(r+1)}[2])$-module complement consisting entirely of
nilpotent elements. This readily implies the second statement concerning
$i_{O}^{*}:H^{*}(\tilde{D}(\mathbb{G}_{(r)}),k)\ \to\
H^{*}(O(\mathbb{G}_{(r)}),k)$. Consequently, $\mathbb{P}_{O}$ is a bijection
as well as a closed embedding, and thus is a homeomorphism. This, together
with Proposition 4.6 implies that $\Psi_{\tilde{D}}$ is also a homeomorphism.
∎

We “recall” the following projectivity criterion for a finite dimensional
$\tilde{D}(\mathbb{G}_{(r)})$-module in terms of cohomological support
varieties.

######  Proposition 5.5.

(see [6, Prop 1.5]) Let $M$ be a finitely generated
$\tilde{D}(\mathbb{G}_{(r)})$-module. Then $M$ is projective if and only if
$\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))_{M}\ =\ \emptyset.$

Similarly, if $M$ is a finitely generated $O(\mathbb{G}_{(r)})$-module, them
$M$ is projective if and only if
$\mathbb{P}(H^{\bullet}(O(\mathbb{G}_{(r)}),k))_{M}\ =\ \emptyset.$

###### Proof.

As shown in [15] for any finite dimensional Hopf algebra with finitely
generated cohomology, $\tilde{D}(\mathbb{G}_{(r)})$ and $O(\mathbb{G}_{(r)})$
are Frobenius algebras. This enables the proof of [6, Prop 1.5] to apply, even
though that result was stated only for Hopf algebras which are restricted
enveloping algebras. ∎

Combining Theorem 5.4, with Proposition 5.5, we conclude that the support
theory $M\mapsto\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}$ provides a test for
projectivity.

######  Theorem 5.6.

Assume the hypotheses on $\mathbb{G}$ of Theorem 5.4 and consider a finite
dimensional $\tilde{D}(\mathbb{G}_{(r)})$-module $M$. Then $M$ is projective
as a $\tilde{D}(\mathbb{G}_{(r)})$-module if and only if
$\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}\ =\ \emptyset.$

###### Proof.

Assume that $M$ is projective as a $\tilde{D}(\mathbb{G}_{(r)})$-module. Then
the pull-back along $(\alpha_{K}+\beta_{K})^{*}$ of $M$ is projective (and
thus free) as a $K[t]/t^{p}$-module for any $\pi$-point pair
$(\alpha_{K},\beta_{K})$, so that $\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}\ =\
\emptyset$.

Conversely, assume that $\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}\ =\ \emptyset$.
Since all the maps of the commutative square (4.6.1) are bijective (in fact,
homeomorphisms), we conclude that
$\mathbb{P}H^{\bullet}((\tilde{D}(\mathbb{G}_{(r)}),k)_{M}\ =\ \emptyset.$
Thus, by Proposition 5.5, $M$ is a projective
$\tilde{D}(\mathbb{G}_{(r)})$-module. ∎

The arguments of Sections 4 and 5 apply to $\tilde{D}(G^{\prime})\equiv
k[\mathbb{G}_{(r+1)}]\\#G^{\prime}\ \subset\ \tilde{D}(\mathbb{G}_{(r)})$ with
merely a change of notation, as we state explicitly in the following corollary
(and use in the next section).

######  Corollary 5.7.

Let $G^{\prime}\ \hookrightarrow\ \mathbb{G}_{(r)}$ be a subgroup scheme of
$\mathbb{G}_{(r)}$. Theorems 4.2 and 5.4 remain valid upon replacing
$i_{O}:O(\mathbb{G}_{(r)})\to\tilde{D}(\mathbb{G}_{(r)})$ by

$i_{O^{\prime}}:k[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}]\\#kG^{\prime}\
\hookrightarrow k[\mathbb{G}_{(r+1)}]\\#kG^{\prime}.$

Consequently, as in Proposition 5.5 and Theorem 5.6, a finite dimensional
$k[\mathbb{G}_{(r+1)}]\\#kG^{\prime}$-module $M$ is projective if and only if
its restriction along $i_{O^{\prime}}$ is a projective
$k[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}]\\#kG^{\prime}$-module.

##  6\. Relating $\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}$ to
$\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))_{M}$

In this section, we verify for certain finite dimensional
$\tilde{D}(\mathbb{G}_{(r)})$-modules $M$ that the natural inclusion
$\Psi_{\tilde{D},M}:\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}\ \hookrightarrow\
\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))_{M}$ is a bijection and
thus a homeomorphism.

We begin with an important class of examples (due to J. Carlson; see [3]) of
modules constructed to have given cohomological support. Let $P_{*}\to k$ be a
minimal resolution of $k$ as a $\tilde{D}(\mathbb{G}_{(r)})$-module and define
$\Omega_{\tilde{D}(\mathbb{G}_{(r)})}^{n}(k)$ for $n\geq 1$ to be the kernel
of $P_{n-1}\to P_{n-2}$ (where $P_{-1}$ is set equal to $k$). Recall that
$\zeta\in H^{n}(\tilde{D}(\mathbb{G}_{(r)}),k)$ naturally corresponds to a map
$\tilde{\zeta}:\Omega_{\tilde{D}(\mathbb{G}_{(r)})}^{n}(k)\to k$ in the stable
module category $stmod(\tilde{D}(\mathbb{G}_{(r)}))$. The
$\tilde{D}(\mathbb{G}_{(r)})$-module $L_{\zeta}$ is defined by the short exact
sequence of $\tilde{D}(\mathbb{G}_{(r)})$-modules

(6.0.1) $0\to L_{\zeta}\ \to\ \Omega_{\tilde{D}(\mathbb{G}_{(r)})}^{n}(k)\
\stackrel{{\scriptstyle\tilde{\zeta}}}{{\to}}\ k\to 0.$

For $\xi\in H^{n}(O(\mathbb{G}_{(r)}),k)$ with corresponding map
$\tilde{\zeta}:\Omega_{O(\mathbb{G}_{(r)})}^{n}k)\to k$ in the stable module
category $stmod(O(\mathbb{G}_{(r)}))$, we similarly define the
$O(\mathbb{G}_{(r)})$-module $L_{\xi}$ by the short exact sequence of
$O(\mathbb{G}_{(r)})$-modules

$0\to L_{\xi}\ \to\ \Omega_{O\mathbb{G}_{(r)})}^{n}(k)\
\stackrel{{\scriptstyle\tilde{\xi}}}{{\to}}\ k\to 0.$

We restrict attention to $n=2d>0$. For such $n=2d$,
$\Omega_{k[t]/t^{p}}^{2d}(k)$ is stably isomorphic to $k$.

######  Proposition 6.1.

Consider the $\tilde{D}(\mathbb{G}_{(r)})$-module $L_{\zeta}$ associated to a
cohomology class $\zeta\in H^{2d}(\tilde{D}(\mathbb{G}_{(r)}),k)$. Then a
$\pi$-point pair $(\alpha_{K},\beta_{K})$ of $\tilde{D}(\mathbb{G}_{(r)})$
satisfies the condition $(\alpha_{K}+\beta_{K})^{*}(L_{\zeta})$ is projective
as a $K[t]/t^{p}$-module if and only if
$(\alpha_{K}+\beta_{K})^{*}(\zeta)\not=0\in H^{2d}(K[t]/t^{p},K)$.

Consequently, $\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{L_{\zeta}}$ consists of
equivalence classes of $\pi$-point pairs $(\alpha_{K},\beta_{K})$ such that
$(\alpha_{K}+\beta_{K})^{*}(\zeta)=0\in H^{2d}(K[t]/t^{p},K)$.

###### Proof.

Carlson’s identification of the support variety of $L_{\zeta}$ in the context
of a group algebra of a finite group applies to $\tilde{D}(\mathbb{G}_{(r)})$,
for his argument merely requires that the algebra
$\tilde{D}(\mathbb{G}_{(r)})$ be a Frobenius algebra and that
$\alpha_{K}+\beta_{K}:K[t]/t^{p}\to\tilde{D}(\mathbb{G}_{(r)})$ be flat. The
key observation of Carlson’s proof is that the pullback along
$\alpha_{K}+\beta_{K}$ of (6.0.1) is split if and only if
$(\alpha_{K}+\beta_{K})^{*}(\zeta)\not=0\in H^{2d}(K[t]/t^{p},K)$. ∎

Using Proposition 4.6 and Theorem 5.4, we verify that
$\Psi_{\tilde{D},L_{\zeta}}$ is a homeomorphism.

######  Proposition 6.2.

Assume the hypotheses on $\mathbb{G}$ of Theorem 5.4. Consider $\zeta\in
H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k)$ and denote by $\zeta_{O}$ the
restriction of $\zeta$ to $H^{\bullet}(O(\mathbb{G}_{(r)}),k)$. Then

  1. (1)

The restriction of the $\tilde{D}(\mathbb{G}_{(r)})$-module $L_{\zeta}$ to
$O(\mathbb{G}_{(r)})$ is stably equivalent to $L_{\zeta_{O}}$.

  2. (2)

$\mathbb{P}i_{O}^{*}$ restricts to a bijection

$\mathbb{P}_{O}:\mathbb{P}(H^{\bullet}(O(\mathbb{G}_{(r)}),k))_{L_{\zeta_{O}}}\
\stackrel{{\scriptstyle\sim}}{{\to}}\
\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))_{L_{\zeta}}.$

  3. (3)

$\Psi_{\tilde{D}}$ restricts to a homeomorphism

$\Psi_{\tilde{D},L_{\zeta}}:\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{L_{\zeta}}\
\stackrel{{\scriptstyle\sim}}{{\to}}\
\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))_{L_{\zeta}}.$

###### Proof.

To prove the first assertion, observe that the $O(\mathbb{G}_{(r)})$-module
$\Omega_{O(\mathbb{G}_{(r)}}^{2d}(k)$ is stably equivalent to the restriction
of $\Omega_{\tilde{D}(\mathbb{G}_{(r)})}^{2d}(k)$ because the restriction of
$\tilde{D}(\mathbb{G}_{(r)})$-projectives to $O(\mathbb{G}_{(r)})$ are
projective $O(\mathbb{G}_{(r)})$-modules. Thus, restricting the short exact
sequence (6.0.1) of $\tilde{D}(\mathbb{G}_{(r)})$-modules along
$i_{O}:O(\mathbb{G}_{(r)})\to\tilde{D}(\mathbb{G}_{(r)})$ takes the form of a
distinguished triangle

$L_{\zeta_{O}}\ \to\ \Omega_{O(\mathbb{G}_{(r)}}^{2d}(k)\
\stackrel{{\scriptstyle\zeta_{O}}}{{\to}}\ k$

in the stable module category $stmod(O(\mathbb{G}_{(r)})$.

Using Proposition 6.1, we see that
$\mathbb{P}(H^{\bullet}(O(\mathbb{G}_{(r)}),k))_{L_{\zeta_{O}}}\ \subset\
\mathbb{P}(H^{\bullet}(O(\mathbb{G}_{(r)}),k))$ is the zero locus of the
homogeneous “function” $\zeta_{O}$ on
$\mathbb{P}(H^{\bullet}(O(\mathbb{G}_{(r)}),k))$, and similarly that
$\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))_{L_{\zeta}}\ \subset\
\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))$ is the zero locus of
the homogeneous “function” $\zeta$ on
$\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))$. The second assertion
thus follows from the fact that $\zeta_{O}\ =\ i_{O}^{*}(\zeta)$ and the fact
that $\mathbb{P}_{O}$ is a homeomorphism by Theorem 5.4.

The last assertion now follows from the commutative square (4.6.1) of
Proposition 4.6 and the fact that $\Psi_{O}$ restricts to a bijection
$\Psi_{O,L_{\zeta_{O}}}:\Psi(\tilde{O}(\mathbb{G}_{(r)}))_{L_{\zeta_{O}}}\
\stackrel{{\scriptstyle\sim}}{{\to}}\
\mathbb{P}(H^{\bullet}(\tilde{O}(\mathbb{G}_{(r)}),k))_{L_{\zeta_{O}}}$ by
Proposition 4.5. ∎

The following proposition gives some insight into the topology on
$\Pi(\tilde{D}(\mathbb{G}_{(r)}))$ beyond knowing that it is the topology
inherited from $\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))$.

######  Proposition 6.3.

Assume the hypotheses on $\mathbb{G}$ of Theorem 5.4 so that the maps of the
commutative square (4.6.1) are homeomorphisms. Then

$\\{\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{L_{\zeta}},\ \zeta\in
H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k)\ {\text{h}omogeneous}\\}$

is a closed base for the topological space $\Pi(\tilde{D}(\mathbb{G}_{(r)}))$.

###### Proof.

Observe that

$\\{\mathbb{P}(H^{\bullet}(O(\mathbb{G}_{(r)}),k))_{L_{\xi}}\ \xi\in
H^{\bullet}(O(\mathbb{G}_{(r)}),k)\ {\text{h}omogeneous}\\}$

is a closed base for the topology of
$\mathbb{P}(H^{\bullet}(O(\mathbb{G}_{(r)}),k))$. Thus, the fact that
$\mathbb{P}_{O}:\mathbb{P}(H^{\bullet}(O(\mathbb{G}_{(r)}),k))\ \to\
\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))$ is a homeomorphism (by
Theorem 5.4) together with Proposition 6.2(2) implies that

$\\{\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))_{L_{\zeta}},\
\zeta\in H^{\bullet}(D(\mathbb{G}_{(r)}),k)\ {\text{h}omogeneous}\\}$

is a closed base for the topology of
$\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))$. Since
$\Psi_{\tilde{D}}$ is a homeomorphism, Proposition 6.2(3) implies that

$\\{\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{L_{\zeta}},\ \zeta\in
H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k)\ {\text{h}omogeneous}\\}$

is a closed base for the topological space $\Pi(\tilde{D}(\mathbb{G}_{(r)}))$.
∎

We next consider the class of $\tilde{D}(\mathbb{G}_{(r)})$-modules given as
the inflation along $\tilde{D}(\mathbb{G}_{(r)})\to\mathbb{G}_{(r)}$ of finite
dimensional $\mathbb{G}_{(r)}$-modules (equivalently, as finite dimensional
$\tilde{D}(\mathbb{G}_{(r)})$-module whose restriction to
$k[\mathbb{G}_{(r+1)}]\subset\tilde{D}(\mathbb{G}_{(r)})$ is trivial).

######  Proposition 6.4.

Assume the hypotheses on $\mathbb{G}$ of Theorem 5.4. Let $M$ be a finite
dimensional $\tilde{D}(\mathbb{G}_{(r)})$-module whose restriction to
$k[\mathbb{G}_{(r+1)}]\subset\tilde{D}(\mathbb{G}_{(r)})$ is trivial. Then

  1. (1)

$\mathbb{P}_{O,M}:\mathbb{P}(H^{\bullet}(O(\mathbb{G}_{(r)}),k))_{M}\ \to\
\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))_{M}$ is a
homeomorphism.

  2. (2)

$\mathbb{P}(H^{\bullet}(O(\mathbb{G}_{(r)}),k))_{M}$ can be identified as the
“geometric join” of
$\operatorname{Proj}\nolimits(S^{\bullet}(\mathfrak{g}^{(r+1)}))$ and
$\mathbb{P}(H^{\bullet}(\mathbb{G}_{(r)},k))_{M}$.

  3. (3)

$\Psi_{\tilde{D},M}:\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}\ \to\
\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))_{M}$ is also a
homeomorphism.

###### Proof.

We consider the commutative square

(6.4.1)
$\textstyle{H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{O}^{*}}$$\textstyle{H^{\bullet}(O(\mathbb{G}_{(r)}),k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),M^{*}\otimes
M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{O}^{*}}$$\textstyle{H^{\bullet}(O(\mathbb{G}_{(r)}),M^{*}\otimes
M).}$

We observe that there are multiplicative spectral sequences for $M^{*}\otimes
M$ replacing $k$ corresponding to the spectral sequences (4.2.2) and (5.4.1).
Equipping $M^{*}\otimes M$ with internal grading degree 0, (6.4.1) determines
a commutative square of multiplicative spectral sequences with internal
grading whose $E_{2}$-page has the form

(6.4.2)
$\textstyle{H^{s}(\mathbb{G}_{(r)},H^{t}(k[\mathbb{G}_{(r+1)}],k))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{s}(\mathbb{G}_{(r)},H^{t}(k[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}],k))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{s}(\mathbb{G}_{(r)},H^{t}(k[\mathbb{G}_{(r+1)}],k)\otimes
M^{*}\otimes
M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{s}(\mathbb{G}_{(r)},H^{t}(k[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}],k)\otimes
M^{*}\otimes M).}$

The restriction map $H^{*}(k[\mathbb{G}_{(r+1)}],k)\to
H^{*}(k[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}],k)$ has nilpotent kernel and
cokernel which are nilpotent, $\mathbb{G}_{(r)}$-summands of
$H^{*}(k[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}],k)$. Thus, arguing as in the
proof of Theorem 5.4, classes at the $E_{2}$-level with internal grading not
congruent to 0 modulo $p^{r+1}$ are torsion; moreover, the lower horizontal
arrow of (6.4.2) restricted to internal degrees congruent to 0 modulo
$p^{r+1}$ has nilpotent kernel and cokernel which are nilpotent,
$\mathbb{G}_{(r)}$-summands.

Since $\mathbb{P}_{O}$ is a homeomorphism, to prove (1) it suffices to verify
that if $\mathfrak{p}\subset H^{\bullet}(O(\mathbb{G}_{(r)}),k)$ is a
homogeneous prime ideal whose inverse image $\mathfrak{q}\equiv
i_{O}^{-1}(\mathfrak{p})$ contains the kernel of the left vertical map of
(6.4.1), then $\mathfrak{p}$ contains the kernel of the right vertical map.
The validity of this statement is equivalent to the same statement of the
commutative square obtained from (6.4.1) by replacing the lower horizontal
arrow by its associated map on reduced algebras (i.e., by dividing out be the
nilradicals of the domain and range of this map). By the preceding discussion,
this latter map is an isomorphism. Thus, having made this replacement, the
required statement about $i_{O}^{-1}(\mathfrak{p})$ is immediate.

Assertion (2) follows from the familiar interpretation of the projectivization
of the product of two affine varieties.

Assertion (3) follows from the commutativity of (4.6.1) and assertion (1). ∎

We consider a third class of finite dimensional
$\tilde{D}(\mathbb{G}_{(r)})$-modules, those with the property that their
restrictions to $k[\mathbb{G}_{(r+1)}]$ are projective. We first recall the
following detection theorem of A. Suslin whose formulation we quote from [7,
Thm 4.10].

######  Theorem 6.5.

[22] Let $G$ be a finite group scheme, $\Lambda$ a unital associative
$G$-algebra, and $\zeta\in H^{\bullet}(G,k)$ be a homogeneous cohomology class
of even degree. Then $\zeta$ is nilpotent if and only if $\zeta_{K}$ restricts
to a nilpotent class in the cohomology of every quasi-elementary subgroup
scheme $\mathcal{E}_{K}\subset G_{K}$ for any field extension $K/k$.

Let $M$ be a finite dimensional $\tilde{D}(\mathbb{G}_{(r)})$-module with the
property that the restriction of $M$ to $k[\mathbb{G}_{(r+1)}]$ is projective.
If $M$ satisfies this property, then the spectral sequences (4.2.2) and (4.2.3
degenerate so that the edge homomorphisms provide the isomorphisms of algebras

(6.5.1) $H^{*}(\mathbb{G}_{(r)},H^{0}(k[\mathbb{G}_{(r+1)}],M^{*}\otimes M))\
\stackrel{{\scriptstyle\sim}}{{\to}}\
H^{*}(\tilde{D}(\mathbb{G}_{(r)}),M^{*}\otimes M),$
$H^{*}(\mathbb{G}_{(r)},H^{0}(k[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}],M^{*}\otimes
M))\ \stackrel{{\scriptstyle\sim}}{{\to}}\
H^{*}(O(\mathbb{G}_{(r)}),M^{*}\otimes M).$

We use (6.5.1) in conjunction with Theorem 6.5 to prove that

$\Psi_{\tilde{D}_{M}}:\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}\ \to\
\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))_{M}$

is a homeomorphism for these modules.

######  Proposition 6.6.

Assume the hypotheses on $\mathbb{G}$ of Theorem 5.4. Let $M$ be a finite
dimensional $\tilde{D}(\mathbb{G}_{(r)})$-module with the property that the
restriction of $M$ to $k[\mathbb{G}_{(r+1)}]$ is projective. Then the
restriction of $\Psi_{\tilde{D}}$ is a homeomorphism

$\Psi_{\tilde{D},M}:\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}\
\stackrel{{\scriptstyle\sim}}{{\to}}\
\mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k))_{M}.$

.

###### Proof.

Let $\Lambda(M)$ denote unital associative $\mathbb{G}_{(r)}$-algebra
$H^{0}(k[\mathbb{G}_{(r+1)}],M^{*}\otimes M)$ and let $\Lambda_{O}(M)$ denote
$H^{0}(k[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}],M^{*}\otimes M)$. Using the
isomorphisms (6.5.1), we rewrite (6.4.1) as

(6.6.1)
$\textstyle{H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{O}^{*}}$$\textstyle{H^{\bullet}(O(\mathbb{G}_{(r)}),k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{\bullet}(\mathbb{G}_{(r)},\Lambda(M))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{O}^{*}}$$\textstyle{H^{\bullet}(\mathbb{G}_{(r)},\Lambda_{O}(M)).}$

We set $I_{M}$ to be the kernel of the left vertical map and $J_{M}$ the
kernel of the right vertical map of (6.6.1).

Since we know that $\Psi_{\tilde{D},M}$ is well defined and injective and that
$\Psi_{\tilde{D}}$ is bijective, to prove the proposition we must show for any
homogeneous prime ideal $\mathfrak{q}\subset
H^{\bullet}(O(\mathbb{G}_{(r)}),k)$ with inverse image $\mathfrak{p}\subset
H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k)$ that $J_{M}\subset\mathfrak{q}$
(i.e., $\mathfrak{p}$ is a point of
$\mathbb{P}(H^{\bullet}(O(\mathbb{G}_{(r)}),k))_{M}$) whenever
$I_{M}\subset\mathfrak{p}$. Assume that $\zeta\in\mathfrak{p}\subset
H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k)$ is a homogeneous element which is
not in $I_{M}$, so that its image $\overline{\zeta}\in
H^{\bullet}(\mathbb{G}_{(r)},\Lambda(M))$ is not nilpotent. Using Theorem 6.5,
let $\gamma:\mathbb{G}_{a(r),K}\to\mathbb{G}_{(r),K}$ be a 1-parameter
subgroup with the property that $\gamma^{*}(\overline{\zeta})\in
H^{\bullet}(\mathbb{G}_{a(r),K},\gamma^{*}(\Lambda(M_{K})))$ is not nilpotent.

In other words, $\gamma^{*}(\zeta)\in
H^{\bullet}(K[\mathbb{G}_{(r+1)}]\\#\mathbb{G}_{a(r)},k)$ has image
$\overline{\gamma^{*}(\zeta)}\in
Ext_{K[\mathbb{G}_{(r+1)}]\\#\mathbb{G}_{a(r),K}]}^{*}(\gamma^{*}(M_{K}),\gamma^{*}(M_{K}))$
which is not nilpotent. By Corollary 5.7, the image of
$\overline{\gamma^{*}(\zeta)}$ in
$Ext_{K[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}]\\#\mathbb{G}_{a(r),K}]}^{*}(\gamma^{*}(M_{K}),\gamma^{*}(M_{K}))$
is not nilpotent as well. Consequently, $i_{O}^{*}(\zeta)$ does not lie in
$J_{M}$. Thus, we conclude if $\mathfrak{p}$ contains $I_{M}$ then
$\mathfrak{q}$ must contain $J_{M}$.

∎

##  7\. Further remarks

We begin this final section with a natural question.

######  Question 7.1.

Assume that $\mathbb{G}$ satisfies the conditions of Theorem 5.4. Can one
extend the results of Section 6 to prove that

$\Psi_{\tilde{D}(\mathbb{G}_{(r)}),M}:\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}\
\to\ \mathbb{P}(H^{\bullet}(\tilde{D}(\mathbb{G}_{(r)}),k)_{M}$

is a homeomorphism for all finite dimensional
$\tilde{D}(\mathbb{G}_{(r)})$-modules?

We proceed to briefly introduce Hopf subalgebras
$\tilde{D}^{(s)}(\mathbb{G}_{(r)})\ \subset\tilde{D}(\mathbb{G}_{(r)})$
(generalizing $O(\mathbb{G}_{(r)})\subset\tilde{D}(\mathbb{G}_{(r)})$) and
quotient maps of Hopf algebras $k[\mathbb{G}_{r+s}]\ \twoheadrightarrow\
D(\mathbb{G}_{(r)})$ (generalizing
$\tilde{D}(\mathbb{G}_{(r)})\twoheadrightarrow D(\mathbb{G}_{(r)}))$. These
generalizations behave much as the algebras we have consider in previous
sections.

######  Proposition 7.2.

The $\mathbb{G}$-equivariant quotient map $\mathbb{G}_{(r+1)}\
\twoheadrightarrow\
\mathbb{G}_{(r+1)}/\mathbb{G}_{(1)}\simeq(\mathbb{G}^{(1)})_{(r)}$ determines
a Hopf algebra embedding of smash products

(7.2.1) $i_{D}:\tilde{D}^{(1)}(\mathbb{G}_{(r)})\ \equiv
k[(\mathbb{G}^{(1)})_{(r)}]\\#k\mathbb{G}_{(r)}\ \hookrightarrow\
k[\mathbb{G}_{(r+1}]\\#k\mathbb{G}_{(r)}\equiv\tilde{D}(\mathbb{G}_{(r)}).$

This is a map of free (right) $O(\mathbb{G}_{(r)})$-modules

The composition

$\tilde{D}^{(1)}(\mathbb{G}_{(r)})\ \hookrightarrow\
\tilde{D}(\mathbb{G}_{(r)})\ \ \twoheadrightarrow\ D(\mathbb{G}_{(r)})$

is the map $F\\#id:k[(\mathbb{G}^{(1)})_{(r)}]\\#k\mathbb{G}_{(r)}\ \to\
k[\mathbb{G}_{(r)}]\\#k\mathbb{G}_{(r)}$, where $F$ is the Frobenius map.

###### Proof.

As in the justification of (1.1.2), the map of (7.2.1) is an embedding of Hopf
algebras. We choose a $k$-linear section
$\sigma:k[\mathbb{G}_{(r)}/\mathbb{G}_{(1)}]\to
k[\mathbb{G}_{(r+1)}/\mathbb{G}_{(1)}]$ of the quotient map induced by the
embedding
$\mathbb{G}_{(r)}/\mathbb{G}_{(1)}\subset\mathbb{G}_{(r+1)}/\mathbb{G}_{(1)}$
and consider the

$k[(\mathbb{G}_{(r+1)}/\mathbb{G}_{(1)})/(\mathbb{G}_{(r)}/\mathbb{G}_{(1)})]\
\simeq\ k[\mathbb{G}_{(r+1)}/\mathbb{G}_{(r)}]\ \simeq\
k[(\mathbb{G}^{(r)})_{(1)}]$

bilinear map

$k[(\mathbb{G}^{(1)})_{(r-1)}]\otimes k[(\mathbb{G}^{(r)})_{(1)}]\ \to\
k[(\mathbb{G}^{(1)})_{(r)}]$

analogous to (1.5.1). Since $k[(\mathbb{G}^{(1)})_{(r-1)})]\otimes 1$ is
central in $\tilde{D}^{(1)}(\mathbb{G}_{(r)})$, we conclude as in the proof of
Proposition 1.5 that this provides $\tilde{D}^{(1)}(\mathbb{G}_{(r)})$ with
the structure of a free (right) $O(\mathbb{G}_{(r)})$-module.

To verify that the asserted composition equals $F\\#id$, we use the
identification
$\mathbb{G}_{(r+1)}/\mathbb{G}_{(1)}\simeq(\mathbb{G}_{(r)})^{(1)}$. ∎

We can view $\tilde{D}^{(1)}(\mathbb{G}_{(r)})$ as a twisted form of
$D(\mathbb{G}_{(r)})$ for which the action of $\mathbb{G}_{(r)}$ on
$k[(\mathbb{G}^{(1)})_{(r)}]$ equals the Frobenius twist of the coadjoint
action of $\mathbb{G}_{(r)}$ on $k[\mathbb{G}_{(r)}]$.

The proof of the following theorem is a basically identical to the proofs
given in previous sections of the corresponding properties for
$\tilde{D}(\mathbb{G}_{(r)})$-modules.

######  Proposition 7.3.

For any $\tilde{D}^{(1)}(\mathbb{G}_{(r)})$-module $M$, we can define
$M\mapsto\Pi(\tilde{D}^{(1)}(\mathbb{G}_{(r)}))_{M}$ in strict analogy with
$M\mapsto\Pi(\tilde{D}(\mathbb{G}_{(r)}))_{M}$. So defined, $\pi$-point pairs
of $\tilde{D}^{(1)}(\mathbb{G}_{(r)})$ are $\pi$-point pairs of
$O(\mathbb{G}_{(r)})$ with the equivalence relation determined by
consideration of finite dimensional
$\tilde{D}^{(1)}(\mathbb{G}_{(r)})$-modules. For any finite dimensional
$\tilde{D}^{(1)}(\mathbb{G}_{(r)})$-module $M$, we define
$\Pi(\tilde{D}^{(1)}(\mathbb{G}_{(r)}))_{M}\subset\Pi(\tilde{D}^{(1)}(\mathbb{G}_{(r)}))$
to consist of those equivalence classes of $\pi$-point pairs
$(\alpha_{K},\eta_{K})$ with the property that
$(\alpha_{K}+\beta_{K})^{*}(M_{K})$ is not projective.

There is a natural continuous map
$\Psi_{\tilde{D}^{(1)}(\mathbb{G}_{(r)}}:\Pi(\tilde{D}^{(1)}(\mathbb{G}_{(r)}))\to\mathbb{P}(H^{\bullet}(\tilde{D}^{(1)}(\mathbb{G}_{(r)}),k)$.
Assuming that $\mathbb{G}$ admits a quasilogarithm and
$p^{r+1}>2dim(\mathbb{G})$, this map is a homeomorphism and restricts to a
homeomorphism

$\Psi_{\tilde{D}^{(1)}(\mathbb{G}_{(r)},M}:\Pi(\tilde{D}^{(1)}(\mathbb{G}_{(r)}))_{M}\
\stackrel{{\scriptstyle\sim}}{{\to}}\
\mathbb{P}(H^{\bullet}(\tilde{D}^{(1)}(\mathbb{G}_{(r)}),k)_{M}$

provided that $M$ is in one of the classes analogous to those classes of
$\tilde{D}(\mathbb{G}_{(r)})$-modules considered in Section 6.

######  Remark 7.4.

The triple of Hopf algebras over $k$

$O(\mathbb{G}_{(r)})\ \hookrightarrow\ \tilde{D}^{(1)}(\mathbb{G}_{(r)})\
\hookrightarrow\ \tilde{D}(\mathbb{G}_{(r)})$

can be refined to a sequence of embeddings of Hopf algebras which are free
over $O(\mathbb{G}_{(r)})$

$O(\mathbb{G}_{(r)})\equiv D^{(r)}(\mathbb{G}_{(r)})\ \hookrightarrow\
\cdots\hookrightarrow\tilde{D}^{(1)}(\mathbb{G}_{(r)})\hookrightarrow\tilde{D}^{(0)}(\mathbb{G}_{(r)})\equiv\tilde{D}(\mathbb{G}_{(r)}),$

where

$\tilde{D}^{(s)}(\mathbb{G}_{(r)})\ \equiv\
k[(\mathbb{G}^{(s)}_{(r+1-s)}]\\#k\mathbb{G}_{(r)},\quad 0\leq s\leq r.$

One readily checks that replacing $\tilde{D}^{(1)}(\mathbb{G}_{(r)})$ in
Proposition 7.3 by $\tilde{D}^{(s)}(\mathbb{G}_{(r)})$ gives analogous
properties for $\tilde{D}^{(s)}(\mathbb{G}_{(r)})$-modules.

######  Remark 7.5.

For any $s\geq 1$, we define the Hopf subalgebra

$k[\mathbb{G}_{(r+s)}]\\#k\mathbb{G}_{(r)}\ \subset\
k[\mathbb{G}_{(r+s)}]\\#k\mathbb{G}_{(r+s)}\equiv D(\mathbb{G}_{(r+s)}.$

This Hopf algebra admits a quotient map of Hopf algebras

$k[\mathbb{G}_{(r+s)}]\\#k\mathbb{G}_{(r)}\ \twoheadrightarrow\
D(\mathbb{G}_{(r))}$

determined by the quotient map $k[\mathbb{G}_{(r+s)}]\twoheadrightarrow
k[\mathbb{G}_{(r+s)}]\to k[\mathbb{G}_{(r)}]$ as in (1.4.1). We obtain a
sequence of quotient Hopf algebras

$\cdots\twoheadrightarrow
k[\mathbb{G}_{(r+s)}]\\#k\mathbb{G}_{(r)}\twoheadrightarrow
k[\mathbb{G}_{(r+s-1)}]\\#k\mathbb{G}_{(r)}\twoheadrightarrow\cdots\to\tilde{D}(\mathbb{G}_{(r)})\twoheadrightarrow
D(\mathbb{G}_{(r)}).$

We leave to the reader to check that our results for
$\tilde{D}(\mathbb{G}_{(r)})$-modules extend to
$k[\mathbb{G}_{(r+s)}]\\#k\mathbb{G}_{(r)}$ for $s\geq 1$.

######  Remark 7.6.

We recall that a theorem of Pevtsova, Suslin, and the author [10, Thm 4.10]
asserts that if $M$ is a finite dimensional $G$-module for a finite group
scheme $G$ over $k$ and if $\alpha_{K}:K[t]/t^{p}\to G_{K}$ is a $\pi$-point
of $G$ at which the Jordan type $\alpha_{K}^{*}(M_{K})$ is maximal among all
Jordan types $\beta_{L}^{*}(M_{L})$ as $\beta_{L}:L[t]/^{p}\to G_{L}$ varies
among all $\pi$-points of $G$, then the Jordan type of
$\alpha_{K}^{\prime*}(M_{K})$ equals the Jordan type of
$\alpha_{K}^{*}(M_{K})$ whenever the $\pi$-point $\alpha_{K}^{\prime}$ is
equivalent to $\alpha_{K}$.

This immediately applies to $O(\mathbb{G}_{(r)})$-modules, since
$O(\mathbb{G}_{(r)})$ is isomorphic as a $k$-algebra to the group algebra of a
finite group scheme as seen in Proposition 1.6. By the definition of the
equivalence relation on $\pi$-point pairs of $\tilde{D}(\mathbb{G}_{(r)})$,
this remains valid for a $\tilde{D}(\mathbb{G}_{(r)})$-module $M$ whose
maximal Jordan type is that of a projective module.

######  Question 7.7.

Does this “independence of representative of equivalence class” remain valid
for any finite dimensional $\tilde{D}(\mathbb{G}_{(r)})$-module $M$? Namely,
if the equivalence class of a $\pi$-point pair of
$\tilde{D}(\mathbb{G}_{(r)})$ contains some some representative
$(\alpha_{K},\beta_{K})$ at which $(\alpha_{K}+\beta_{K})^{*}(M_{K})$ has
maximal Jordan type, then does $(\alpha_{L}^{\prime}+\beta_{L})^{*}(M_{L})$
have this same Jordan type for any $(\alpha_{K}^{\prime}+\beta_{K}^{\prime})\
\sim\ (\alpha_{K}+\beta_{K})$?

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