TY - JOUR

T1 - On the almost split sequences for relatively projective modules over a finite group

AU - Kleiner, Mark

N1 - Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 1992/12

Y1 - 1992/12

N2 - Let G be a finite group with a subgroup H. Given a field k of characteristic p dividing the order of G, denote by mod kG the category of finite-dimensional over k left G-modules, and let be the full subcategory of mod kG determined by the relatively projective modules. Let 0⟶L⟶M⟶N⟶0 be an exact sequence in mod kG with L, M, N ∈. It is proved that the sequence is an almost split sequence in if and only if it is an almost split sequence in mod kG. This implies, together with a recent result of Carlson and Happel, that 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.

AB - Let G be a finite group with a subgroup H. Given a field k of characteristic p dividing the order of G, denote by mod kG the category of finite-dimensional over k left G-modules, and let be the full subcategory of mod kG determined by the relatively projective modules. Let 0⟶L⟶M⟶N⟶0 be an exact sequence in mod kG with L, M, N ∈. It is proved that the sequence is an almost split sequence in if and only if it is an almost split sequence in mod kG. This implies, together with a recent result of Carlson and Happel, that 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.

KW - Almost split sequence

KW - Finite group

KW - Relatively projective module

UR - http://www.scopus.com/inward/record.url?scp=84966206546&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84966206546&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-1992-1100656-X

DO - 10.1090/S0002-9939-1992-1100656-X

M3 - Article

AN - SCOPUS:84966206546

VL - 116

SP - 943

EP - 947

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 4

ER -