Projective resolutions and Poincaré duality complexes

Abstract

Let k be a field lof characteristic p > 0 p > 0 and let G be a finite group. We investigate the structure of the cohomology ring H ∗ ( G , k ) {H^\ast }(G,k) in relation to certain spectral sequences determined by systems of homogeneous parameters for the cohomology ring. Each system of homogeneous parameters is associated to a complex of projective kG-modules which is homotopically equivalent to a Poincaré duality complex. The initial differentials in the hypercohomology spectral sequence of the complex are multiplications by the parameters, while the higher differentials are matric Massey products. If the cohomology ring is Cohen-Macaulay, then the duality of the complex assures that the Poincaré series for the cohomology satisfies a certain functional equation. The structure of the complex also implies the existence of cohomology classes which are in relatively large degrees but are not in the ideal generated by the parameters. We consider several other questions concerned with the minimal projective resolutions and the convergence of the spectral sequence.