A Lifting Theorem with Applications to Blocks and Source Algebras

Abstract

The main purpose of this paper is to present a lifting theorem generalizing the well-known Wedderburn᎐Malcev theorem. We then show how this result can be applied to various questions concerning the structure of blocks and source algebras. In particular, we indicate how it can be used to give an alternative proof of Puig's main theorem on nilpotent blocks.In the following, we fix a complete discrete valuation ring R with Ž . algebraically closed residue field F s RrJ R of characteristic p ) 0 and field of fractions K of characteristic 0. For an element ␣ g R, we denote by ␣ its image in F. All our R-algebras will be associative with identity element and finitely generated R-modules. We do not require our R-algebra homomorphisms to be unitary.Let us fix an R-algebra A. Every A-module will be considered as an R-module in the usual way. We will tacitly assume that all our A-bimodules M satisfy rm s mr for r g R, m g M. Then we can regard

Blocks with cyclic defect groups and Clifford extensions

“…Let b * be the unique block of OK * covering b, let K = K * /Z p , and letb be the image of b * under the epimorphism OK * → O K. Then by [7,Theorem 7], there is a natural isomorphism OZ p ⊗ Ob O K b * OK * . Note that the categories (b * OK * | V p )-mod and (b * OK * | V )-mod of module lying over V p and V , respectively, coincide, and by Clifford theory, the categories bO α K-mod and (b * OK * | V )-mod are equivalent.…”

Modular Representation Theory of Blocks with Trivial Intersection Defect Groups

“…This is studied in [10] when A has cyclic defect groups, and in [25] when has Abelian defect groups. Strong results concerning the role of fusion are obtained in [22], and it is these that we use, in particular to consider the case where A has plocal rank one and generalised quaternion defect groups. We first need a definition from [22].…”

“…Strong results concerning the role of fusion are obtained in [22], and it is these that we use, in particular to consider the case where A has plocal rank one and generalised quaternion defect groups. We first need a definition from [22]. Note that although here we use B and G to represent blocks and groups, we are not constraining ourselves to Hypotheses 4.…”

“…By part (a) and Theorem 7 of [22] (N G (D), B, λ, [r]). But applying Proposition 6.1 to B and its Brauer correspondent in N G (D) gives…”

Induction for Interior Algebras

“…Similarly, if N ′ is a C − B−bimodule, we denote by Hom B 0 (M, N ′ ) the R−module consisting of all homomorphisms from M to N ′ viewed as right B−modules; again, this has a canonical structure of C − A−bimodules given by (c.ψ.a)(m) = cψ(am) for any a ∈ A, c ∈ C and ψ ∈ Hom B 0 (M, N ′ ). Following the terminology of [1] (which extends that of [2, 3.1]), if A is an R−algebra, an interior A−algebra is an R−algebra B endowed with a unitary algebra homomorphism σ : A → B. Note that in particular B becomes an A − A−bimodule through σ.…”