Abstract
By results of Boltje and Kulshammer, if a source algebra A of a principal p-block of a nite group with a defect group P with inertial quotient E is a depth two extension of the group algebra of P, then A is isomorphic to a twisted group algebra of the group P o E. We show in this note that this is true for arbitrary blocks. We observe further that the results of Boltje and Kulshammer imply that A is a depth two extension of its hyperfocal subalgebra, with a criterion for when this is a depth one extension. By a result of Watanabe, this criterion is satised if the defect groups are abelian.
Highlights
By results of Boltje and Kulshammer, if a source algebra A of a principal p-block of a finite group with a defect group P with inertial quotient E is a depth two extension of the group algebra of P, A is isomorphic to a twisted group algebra of the group P E
Following terminology in [4], a ring extension B → A is called of depth one if A is isomorphic, as a B-B-bimodule, to a direct summand of Bn for some positive integer n, and a ring extension B → A is called of of depth two if A ⊗B A is isomorphic, as an A-B-bimodule, to a direct summand of An, for some positive integer n
Boltje and Kulshammer showed in [2, 2.4] that if A is isomorphic to a twisted group algebra of the form Oα(P E) for some p -subgroup E of Aut(P ) and some α ∈ H2(E; k×), inflated trivially to P E, the canonical map OP → A is an extension of depth two
Summary
By results of Boltje and Kulshammer, if a source algebra A of a principal p-block of a finite group with a defect group P with inertial quotient E is a depth two extension of the group algebra of P , A is isomorphic to a twisted group algebra of the group P E. Let A be a source algebra of a block algebra over O of a finite group, with a defect group P .
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