On the almost split sequences for relatively projective modules over a finite group

Abstract:

Let $G$ be a finite group with a subgroup $H$. Given a field $k$ of characteristic $p$ dividing the order of $G$, denote by $\bmod kG$ the category of finite-dimensional over $k$ left $G$-modules, and let $\mathcal {C}$ be the full subcategory of $\bmod kG$ determined by the relatively projective modules. Let $0 \to L \to M \to N \to 0$ be an exact sequence in $\bmod kG$ with $L,M,N \in \mathcal {C}$. It is proved that the sequence is an almost split sequence in $\mathcal {C}$ if and only if it is an almost split sequence in $\bmod kG$. This implies, together with a recent result of Carlson and Happel, that $\mathcal {C}$ has almost split sequences if and only if it is closed under extensions, i.e., if and only if $p$ is coprime to either the order of $H$ or the index of $H$ in $G$.