Vol. 5, No. 1, 2011

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Group algebra extensions of depth one

Robert Boltje and Burkhard Külshammer

Vol. 5 (2011), No. 1, 63–73
Abstract

A ring extension A B is said to have depth one if B is isomorphic to a direct summand of An as an (A,A)-bimodule, for some positive integer n. We prove group-theoretic characterizations of this property in the case kH kG, 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 kG with defect group P is a depth-one extension of kP.

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