Cohomologically complete intersections with vanishing of Betti numbers

TL;DR

This paper characterizes cohomologically complete intersections in Gorenstein rings through vanishing conditions of local cohomology and Betti numbers, providing new homological criteria for their identification.

Contribution

It introduces a novel characterization linking the vanishing of local cohomology modules to Betti numbers, advancing the understanding of cohomologically complete intersections.

Findings

Vanishing of $H^i_{I}(R)$ for $i eq c$ is equivalent to Betti number vanishing of $H^c_{I}(R)$.

Provides necessary and sufficient conditions for an ideal to be a cohomologically complete intersection.

Connects homological properties with cohomological vanishing to characterize the intersection property.

Abstract

Let $I$ be ideal of an $n$-dimensional local Gorenstein ring $R$. In this paper we will describe several necessary and sufficient conditions such that the ideal $I$ becomes cohomologically complete intersections. In fact, as a technical tool, it will be shown that the vanishing $H^i_{I}(R)= 0$ for all $i\neq c= \grade (I)$ is equivalent to the vanishing of the Betti numbers of $H^c_{I}(R)$. This gives a new characterization to check the cohomologically complete intersections property with the homological properties of the vanishing of Tor modules of $H^c_{I}(R)$.