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

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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 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.

Original languageEnglish (US)
Pages (from-to)943-947
Number of pages5
JournalProceedings of the American Mathematical Society
Volume116
Issue number4
DOIs
StatePublished - 1992

Keywords

  • Almost split sequence
  • Finite group
  • Relatively projective module

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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