Restricting homology to hypersurfaces

TL;DR

This paper investigates how the homological properties of modules over certain rings are affected when restricting to hypersurfaces, revealing that Betti sequences depend only on specific classes of regular elements.

Contribution

It establishes that Betti sequences of modules over hypersurfaces depend solely on the class of the regular element in a quotient, with applications to local algebra and modular representation theory.

Findings

Betti sequences depend only on the class of the regular element in the quotient.

Results apply to support sets in local algebra.

Implications for modular representation theory of elementary abelian groups.

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.