Source Algebras and Source Modules

The aim of this article is to give a self-contained approach in module theoretic terms to two fundamental results in block theory, due to Puig 6, 14.6 : first, there is an embedding of the source algebra of the Brauer correspondent of a block of some finite group into a source algebra of that block, and second, the source algebras of the Brauer correspondence can be described explicitly. Our proof of the first result Theorem 5 and its Corollary 6 below is essentially the translation to our terminology of the proof in 1, 4.10 . The second result, describing the source algebras of the Brauer correspondent, follows also from 3 , but the proofs in 3 use both the main result on the structure of nilpotent blocks in 5 as well as techniques from 6, Sects. 4, 6 , while our approach from Proposition 9 onwards requires only some classical results on the structure of blocks with a central defect group. An account of the results in 6 on source algebras of blocks with a normal defect group can also be found in 7, Sect. 45 , except for the ˆ Ž . explicit description of the central extension E see Section 10 below , as being defined in terms of the multiplicity algebra of the block, which is Ž .Ž . quoted in 7, 45.10 b without proof, referring back to 6 at the end of 7, Sect. 45 .