Projectivity of modules for infinitesimal unipotent group schemes 
Christopher P. Bendel

Let $k$ be an algebraically closed field of characteristic $p > 0$ and $G$ be 
an infinitesimal group scheme over $k$, that is an affine group scheme $G$ 
over $k$ whose coordinate (Hopf) algebra $k[G]$ is a finite-dimensional local 
$k$-algebra. A rational $G$-module is equivalent to a $k[G]$-comodule and 
further equivalent to a module for the finite-dimensional cocommutative Hopf 
algebra $k[G]^* \equiv \operatorname{Hom}_k(k[G],k)$.  Since $k[G]^*$ is a 
Frobenius algebra, a rational $G$-module (even infinite-dimensional) is in 
fact projective if and only if it is injective.   Further,  for any rational 
$G$-module $M$ and any closed  subgroup scheme $H \subset G$, if $M$ is 
projective over $G$, then it remains projective  upon restriction to $H$.  
We consider the question of whether there is a ``nice'' collection of closed 
subgroups of $G$ upon which projectivity (over $G$) can be detected. 

For an example of what we mean by a ``nice'' collection, consider the 
situation of modules over a finite group.  Over a field of characteristic 
$p > 0$, a module over afinite group is projective if and only if it is 
projective upon restriction to a $p$-Sylow subgroup.   For a $p$-group (and 
hence for any finite group), L. Chouinard showed that a module is projective 
if and only if it is projective upon restriction to every elementary abelian 
subgroup.   If the module is assumed to be finite-dimensional, this result 
follows from the theory of varieties for finite groups.   Indeed, elementary 
abelian subgroups play an essential role in this theory. 

In work of A. Suslin, E. Friedlander, and the author, a theory of varieties 
for infinitesimal group schemes was developed.  In this setting, subgroups 
of the  form $\gar$ (the $r$th Frobenius kernel of the additive group scheme 
$\ga$) play the role analogous to that of elementary abelian subgroups in the 
case of finite groups.  Not surprisingly then, one concludes that a 
finite-dimensional module for an infinitesimal group scheme is projective if 
and only if it's projective upon restriction to all subgroup schemes of the 
form $\gar$. 

The goal of this paper is to show that this holds for infinite-dimensional 
modules if $G$ is assumed to be {\em unipotent}.  (An affine group scheme 
$G$ is said to  be unipotent if it  admits an embedding as a closed subgroup 
of $U_n$, the subgroup scheme in $GL_n$ of strictly upper triangular matrices, 
for some postive integer $n$.)  Specifically, we prove the following theorem 
using some of the ideas in the work with Friedlander and Suslin and an 
argument similar to that of Chouinard without appealing to the theory of 
varieties. 

\begin{main} 
Let $k$ be an algebraically closed field of characteristic $p > 0$, $r > 0$ 
be an integer,  and $G$ be an  infinitesimal unipotent group scheme over $k$ 
of height $\leq r$.  For any rational $G$-module $M$,  $M$ is projective as 
a rational $G$-module if and only if for every field extension $K/k$ and 
\rm{(}}closed{\rm{)}} $K$-subgroup scheme $H \subset G\otimes_kK$ with 
$H \simeq \gas\otimes_kK$ {\rm{(}}with $s \leq r${\rm{)}} the restriction of 
$M$ to $H$ is  projective  as a rational $H$-module. 
\end{main} 

We remind the reader that the {\em restricted} representation theory of a 
restricted Lie algebra $\mathfrak{g}$ over $k$ is equivalent to the 
representation theory of a certain (height 1) infinitesimal group scheme.  
As such, the theorem applies to {\em $p$-nilpotent}, restricted Lie algebras 
and may be stated as follows. 

\begin{thecor}  Let $k$ be an algebraically closed field of characteristic 
$p > 0$, $\mathfrak{g}$ be a $p$-nilpotent, restricted Lie algebra over $k$, 
and $M$ be a $u(\mathfrak{g})$-module, where $u(\mathfrak{g})$ denotes the 
restricted enveloping algebra of $\mathfrak{g}$.  Then $M$ is projective over 
$u(\mathfrak{g})$ if and only if for every field extension $K/k$, $M$ is 
projective upon restriction to each subalgebra $u(\langle x \rangle) \subset 
u(\mathfrak{g}\otimes_kK)$ for all $x \in \mathfrak{g}\otimes_kK$ with 
$x^{[p]} = 0$, where $\langle x \rangle \subset \mathfrak{g}\otimes_kK$ 
denotes the  one-dimensional restricted Lie subalgebra of 
$\mathfrak{g}\otimes_kK$ spanned by $x$ and $x^{[p]}$ denotes the image of 
$x$ under the restriction map on $\mathfrak{g}\otimes_kK$. 
\end{thecor}