G-algebras, Jacobson radical and almost split sequences
Let k be an algebraically closed field of characteristic \(p\neq 0\) and G a finite group; let H be a subgroup of G. Let M be an indecomposable kG- module and X the almost split exact sequence terminating in M. The behaviour of X under restriction to H is looked at; in particular, as to when the restriction is an SAS-sequence (a direct sum of split and almost split exact sequences). Induction up to G of an almost split exact sequence of kH-modules (terminating in an indecomposable kH-module) is also looked at. For restriction, J. A. Green obtained a criterion for the restriction to be an SAS-sequence in terms of various relative Jacobson radicals ideals lying in \(End_ kM\). One generalization is to the corresponding condition for G-algebra A (instead of \(End_ kM).\) However here a very local criterion for restriction to behave is obtained. This is in terms of the multiplicity module \(V_{p_{\gamma}}\) corresponding to a vertex P and source of M; \(V_{p_{\gamma}}\) is an indecomposable projective module over a twisted group algebra on the group \(\overline{N_ G(P_{\gamma})}=N_ G(P_{\gamma})/P\) and the condition is that Res\(\begin{matrix} \overline{N_ G(P_{\gamma})}\\ \overline{N_ H(P)}\end{matrix} (Soc(V_{P_{\gamma}}))\) be semisimple. A similar analysis is given for the behaviour of almost split exact sequences under induction. Indeed the whole behaviour and result of induction and restriction on almost split exact sequences are given. The analysis and terminology follow that introduced by \textit{L. Puig} [Math. Z. 176, 265-292 (1981; Zbl 0464.20007)].
scientific article; zbMATH DE number 4093736