Given a group algebra of a finite p-group over a field of characteristic p. Tachikawa proved that in this case $Ext^{1}(M,M) \neq 0$ for any finite dimensional non-projective module $M$. Can one give a nice explicit description of W in a possibly "canonical" exact sequence $0 \rightarrow M \rightarrow W \rightarrow M \rightarrow 0$? Maybe one can even find a description (how to build W from M with some homological tricks) which could extent the result to more general local algebras.
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