A BLOCK REFINEMENT OF THE GREEN-PUIG PARAMETERIZATION OF THE ISOMORPHISM TYPES OF INDECOMPOSABLE MODULES

摘要

Let p be a prime integer, let O 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 OP-module with vertex P. Let Lambda(G, P, X) denote a set of representatives for the isomorphism classes of indecomposable OG-modules with vertex-source pair (P, X) (so that Lambda( 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 Lambda(G, P, X). In [5, Proposition 26.3], J. Thevenaz completed this program under the hypotheses that X is O-free. Here we use the methods of proof of [5, Theorem 26.3] to show that the O-free hypothesis on X is superfluous. (M. Linckelmann has also proved this, cf. [3]). Let B be a block of OG. Then we obtain a corresponding paramaterization of the (OG)B-modules in Lambda(G, P, X).