Decent intersection and Tor-rigidity for modules over local hypersurfaces

TL;DR

This paper explores the properties of decency and rigidity for modules over local hypersurfaces, establishing new links between Hochster's function and these properties using algebraic and intersection theory techniques.

Contribution

It demonstrates that the vanishing of Hochster's function implies both decency and rigidity, providing new insights into module behavior over hypersurfaces.

Findings

Vanishing of Hochster's function implies rigidity.

New results on decency and rigidity over hypersurfaces.

Application of algebraic and intersection theory methods.

Abstract

We study two properties of modules over a local hypersurface $R$: decency and rigidity. We show that the vanishing of Hochster's function $\theta^R(M,N)$, known to imply decent intersection, also implies rigidity. We investigate the vanishing of $\theta^R(M,N)$ to obtain new results about decency and rigidity over hypersurfaces. We employ a mixture of techniques from Commutative Algebra and Intersection Theory of algebraic cycles.