On cohomologically complete intersections

TL;DR

This paper investigates cohomologically complete intersections in local Gorenstein rings, revealing that their defining vanishing conditions are fully characterized by the homological properties, especially Bass numbers, of their top local cohomology modules.

Contribution

It establishes that the vanishing of local cohomology outside the expected degree is entirely determined by the homological features of the top local cohomology module.

Findings

Vanishing of $H^i_I(R)$ for $i eq c$ is characterized by Bass numbers of $H^c_I(R)$.

Cohomologically complete intersections are linked to specific homological properties of their local cohomology.

Main result connects vanishing conditions to Bass numbers, providing a homological criterion.

Abstract

An ideal $I$ of a local Gorenstein ring $(R, \mathfrak m)$ is called cohomologically complete intersection whenever $H^i_I(R) = 0$ for all $i \not= \height I.$ Here $H^i_I(R), i \in \mathbb Z,$ denotes the local cohomology of $R$ with respect to $I.$ For instance, a set-theoretic complete intersection is a cohomologically complete intersection. Here we study cohomologically complete intersections from various homological points of view, in particular in terms of their Bass numbers of $H^c_I(R), c = \height I.$ As a main result it is shown that the vanishing $H^i_I(R) = 0$ for all $i \not= c$ is completely encoded in homological properties of $H^c_I(R),$ in particular in its Bass numbers.