Extensions of lattices overp-solvable groups

Abstract

By PETER SCHMID Suppose G is a finite p-solvable group for some prime p. Let K be a finite extension of the p- adic number field 9 and k = R/r~R its residue class field. Let U, Vbe RG-lattices ( = R-free RG-modules of finite R-rank). The objective of this note is to record the following. Theorem. Ext~o (U, V) = 0 if either (i) K|174 is irreducible belonging to a block with cyclic defect group and K is unramified over •, or (ii) k| = k| is irreducible and the ramification index of K over 9 is less than p- 1 (and hence p odd). Statement (i) generalizes a result by Berman [11 who considered the case where G has a normal cyclic Sylow p-subgroup (and K = 9 It has been already noticed by Gudivok [31 that the result does not hold, in general, if K is ramified over 9 In a certain sense, both statements rely on the Fong-Swan theorem. In the first case we use the theory of blocks with cyclic defect groups and the fact that the Brauer tree is a star if G is p-solvable. The second statement follows from the uniqueness part of the lifting theorem in [61. Thus solvability is a crucial assumption. 1. Extensions of lattices. Let G be a finite group and let U, Vbe RG-lattices. Multiplication with 7r gives the exact sequence O-*V&V-,fz~O where f'= V/IrV~k| Since U is R- projective, the induced sequence 0--Hom R (U, V)&HomR (U, V)~Hom R (U, I7") ~0 is exact as well, and Extlo (U, V) = H 1 (G, Hom R (U, V)). Lemma. Ext~o(O,V)=O if and only if the residue class map HomRc(U, V)~ Homk6 (U, V) is epimorphic. P r o o f. Application of the long exact cohomology sequence yields the exact sequence 0 ~HomRc (U, V) ~ HomRo (U, V) --*Homko (0, ~2)-~Ext~o (U, V) Z*Ext~a (U, V) .... .