Cousin complexes and applications

TL;DR

This thesis explores modules with finitely generated Cousin complex cohomologies, linking them to uniform local cohomological annihilators, and characterizes Cohen-Macaulay modules and loci in terms of these properties.

Contribution

It establishes the equivalence of modules with finitely generated Cousin complexes and those with uniform local cohomological annihilators over certain local rings, providing new characterizations.

Findings

Modules with finite Cousin complexes have specific properties and characterizations.

Cohen-Macaulay loci are Zariski-open in certain rings for all finitely generated modules.

Characterization of generalized Cohen-Macaulay modules via uniform annihilators.

Abstract

In this thesis, the class of modules whose Cousin complexes have finitely generated cohomologies are studied as a subclass of modules which have uniform local cohomological annihilators and it is shown that these two classes coincide over local rings with Cohen-Macaulay formal fibres. This point of view enables us to obtain some properties of modules with finite Cousin complexes and find some characterizations of them. In this connection we discuss attached prime ideals of certain local cohomology modules in terms of cohomologies of Cousin complexes. In continuation, we study the top local cohomology modules with specified set of attached primes. Our approach to study Cousin complexes leads us to characterization of generalized Cohen-Macaulay modules in terms of uniform annihilators of local cohomology. We use these results to study the Cohen-Macaulay loci of modules and find two classes of rings over which the Cohen-Macaulay locus of any finitely generated module is a Zariski--open subset of the spectrum of the ring.