Restricting homology to hypersurfaces

Abstract

This paper concerns the homological properties of a module $M$ over a commutative noetherian ring $R$ relative to a presentation $R\cong P/I$, where $P$ is local ring. It is proved that the Betti sequence of $M$ with respect to $P/(f)$ for a regular element $f$ in $I$ depends only on the class of $f$ in $I/\mathfrak{n}I$, where $\mathfrak{n}$ is the maximal ideal of $P$. Applications to the theory of supports sets in local algebra and in the modular representation theory of elementary abelian groups are presented.

17 pages. This version differs from the previous one only in Section 5, where the statement of Theorem 5.1 has changed. This paper will appear in "Geometric and topological aspects of the representation theory of finite groups", to be published by Springer in the series titled "Proceedings in Mathematics"