Group algebra extensions of depth one

Boltje, Robert; Külshammer, Burkhard

doi:10.2140/ant.2011.5.63

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Group algebra extensions of depth one

Robert Boltje and Burkhard Külshammer

Vol. 5 (2011), No. 1, 63–73

DOI: 10.2140/ant.2011.5.63

Abstract

A ring extension $A \subseteq B$ is said to have depth one if $B$ is isomorphic to a direct summand of $A^{n}$ as an $(A, A)$ -bimodule, for some positive integer $n$ . We prove group-theoretic characterizations of this property in the case $k H \subseteq k G$ , where $H$ is a subgroup of a finite group $G$ and $k$ is a field. We determine when the source algebra of a block of $k G$ with defect group $P$ is a depth-one extension of $k P$ .

Keywords

depth-one ring extension, centrally projective ring extension, depth-two ring extension, symmetric Frobenius extension, $p$-hypoelementary group, nilpotent block, source algebra, trivial source module

Mathematical Subject Classification 2000

Primary: 20C05

Secondary: 19A22, 16D90, 16D20, 20C20

Milestones

Received: 15 January 2010

Revised: 22 April 2010

Accepted: 6 June 2010

Published: 22 August 2011

Authors

Robert Boltje
Department of Mathematics University of California Santa Cruz, CA 95064 United States

Burkhard Külshammer
Mathematical Institute Friedrich Schiller University 07737 Jena Germany

Vol. 5, No. 1, 2011