Varieties and transfers

Abstract

be a minimal KG-projective resolution of the trivial kG-module k. The restriction of (P, E) to H is a kH-projective resolution. Any <e H;(k) is represented by a kHcocycle f E Horn&P,, k). The transfer to G of i is then Trz([)=cIsCfl) where f’(m) = xi= 1 f(xip ‘m) for x t, . . . ,x, a complete set of representatives of the left cosets of H in G and m E P, . The transfer map is only a k-module homomorphism. Its image in Hz(k) is an ideal. The transfer has been studied for many years. However when p divides IG: HI, its utility is somewhat limited because it is often zero. We show here that, collectively, the images of the transfers are relatively large provided the Sylow p-subgroup of G is not commutative. Specifically we prove the following.