D. J. Benson
The nucleus, and extensions between modules for a 
finite group

Submitted to the proceedings of ICRA9, Beijing 2000.

Abstract:
Let $G$ be a finite group and $k$ be a field of characteristic
$p$ dividing $|G|$. If we are given two finitely generated
$kG$-modules $M$ and $N$, how do we tell whether there are
extensions (i.e., whether $\text{\rm Ext}^i_{kG}(M,N) \ne 0$
for some $i>0$)? Certainly, if the varieties of $M$ and $N$ do
not overlap, then there are no extensions. The converse is
false. For modules in the principal block, if there are no
extensions then the overlap between the varieties has to lie
in a certain subvariety of the cohomology variety, called the
nucleus. This variety can be described explicitly in terms of
the subgroup structure; it is the union of the images of the
varieties of subgroups whose centralizers are not $p$-nilpotent.

A number of attempts have been made to understand the
corresponding problem for nonprincipal blocks. The situation is
certainly harder than in the case of the principal block. An
analysis of some examples leads to some very explicit questions
about the finitely generated modules for some finite dimensional
algebras.

A typical example which we do not properly understand leads
to the following question. Let $A$ be the $k$-algebra generated
by elements $x$ and $y$, with relations $x^p=0$, $y^p=0$
($p=\text{\rm char}(k)$ an odd prime), $xy+yx=0$. If $M$ and $N$ are finitely
generated indecomposable $A$-modules whose restriction to the subalgebra
generated by $x$ is not free, is it necessarily true that
$\text{\rm Ext}^i_A(M,N)\ne 0$ for some $i>0$?