Duality Complexes

Abstract

At the end of the last chapter, we related complexity to multiple complexes. We were concerned with constructions of periodic complexes based on choices of a system of homogeneous parameters for the cohomology ring H* (G, k). Spectral sequences arise from truncated versions of the periodic complexes. In this chapter, we use the constructions to show that for any kG-module M, the cohomology Hⁿ(G, M) can not equal zero for too many consecutive values of n without necessitating that it vanish for all values of n. We introduce the hypercohomology spectral sequence (HSS) to prove this and subsequent results. We also show that the complexes satisfy a Poincaré duality. In some special cases, such as when the cohomology ring is Cohen-Macaulay, the Poincaré duality implies some special properties of the cohomology ring. This includes a functional equation for the Poincaré series or Hilbert series of the cohomology ring. Finally, we characterize the Poincaré duality in terms of a cap product with a cohomology class in negative degree and investigate the relationship to maximal elementary abelian subgroups.