A unified approach to formal local cohomology and local Tate cohomology

TL;DR

This paper develops a unified theory connecting formal local cohomology with local homology, cohomology, and Tate cohomology for complexes over Noetherian rings, providing new insights into their relationships and properties.

Contribution

It introduces a comprehensive framework for formal local cohomology of complexes and explores its connections with other local cohomological theories, including characterizations of Cohen-Macaulay complexes.

Findings

Established natural isomorphisms between formal local cohomology, local homology, and Tate cohomology.

Provided criteria for vanishing of formal local cohomology modules.

Characterized Cohen-Macaulay complexes using formal local cohomology.

Abstract

Let R be a commutative Noetherian ring. We introduce a theory of formal local cohomology for complexes of R-modules. As an application, we establish some relations between formal local cohomology, local homology, local cohomology and local Tate cohomology through some natural isomorphisms. We investigate vanishing of formal local cohomology modules. Also, we give a characterization of Cohen-Macaulay complexes.