Generalized local cohomology modules and homological Gorenstein dimensions

TL;DR

This paper explores bounds and properties of generalized local cohomology modules using Gorenstein homological dimensions, establishing new inequalities and vanishing theorems in Cohen-Macaulay rings.

Contribution

It introduces new upper bounds for generalized local cohomology modules and extends classical vanishing theorems to this broader context.

Findings

Derived upper bounds for _{a}(M,N) using Gorenstein homological dimensions.

Proved an equality for _{m}(M,N) in Cohen-Macaulay local rings under certain finiteness conditions.

Established an analogue of the Hartshorne-Lichtenbaum Vanishing Theorem for generalized local cohomology modules.

Abstract

Let \fa be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let \cd_{\fa}(M,N) denote the supremum of the i's such that H^i_{\fa}(M,N)\neq 0. First, by using the theory of Gorenstein homological dimensions, we obtain several upper bounds for \cd_{\fa}(M,N). Next, over a Cohen-Macaulay local ring (R,\fm), we show that \cd_{\fm}(M,N)=\dim R-\grade(\Ann_RN,M), provided that either projective dimension of M or injective dimension of N is finite. Finally, over such rings, we establish an analogue of the Hartshorne-Lichtenbaum Vanishing Theorem in the context of generalized local cohomology modules.