A Block Refinement of the Green-Puig Parameterization of the Isomorphism Types of Indecomposable Modules

Let p be a prime integer, let  be a commutative complete local Noetherian ring with an algebraically closed residue field k of charateristic p and letG be a finite group. Let P be a p-subgroup of G and let X be an indecomposable P-module with vertex P. Let Λ(G, P, X) denote a set of representatives for the isomorphism classes of indecomposable G-modules with vertex-source pair (P, X) (so that Λ(G, P, X) is a finite set by the Green correspondence). As mentioned in [5, Notes on Section 26], L. Puig asserted that a defect multiplicity module determined by (P, X) can be used to obtain an extended parameterization of Λ(G, P, X). In [5, Proposition 26.3], J. Thévenaz completed this program under the hypotheses that X is -free. Here we use the methods of proof of [5, Theorem 26.3] to show that the -free hypothesis on X is superfluous. (M. Linckelmann has also proved this, cf. [3]). Let B be a block of G. Then we obtain a corresponding paramaterization of the (G)B-modules in Λ(G, P, X).