Stable cohomology over local rings

Abstract

The focus of this paper is on a poorly understood invariant of a commutative noetherian local ring $R$ with residue field $k$: the stable cohomology modules $\hat{Ext}^{n}_R(k,k)$, defined for each $n\in\mathbb{Z}$ by Benson and Carlson, Mislin, and Vogel; it coincides with Tate cohomology when $R$ is Gorenstein. It is proved that important properties of $R$, such as being regular, complete intersection, or Gorenstein, are detected by the $k$-rank of $\hat{Ext}^{n}_R(k,k)$ for an arbitrary $n\in\mathbb{Z}$. Such numerical characterizations are made possible by results on the structure of $\mathbb{Z}$-graded $k$-algebra carried by $\hat{Ext}^{n}_R(k,k)$. It is proved that in many cases this algebra is determined by the absolute cohomology algebra through a canonical homomorphism ${Ext}^{n}_R(k,k)\to\hat{Ext}^{n}_R(k,k)$.