On some local cohomology spectral sequences

TL;DR

This paper develops new spectral sequences related to local cohomology by formalizing derived functors over finite posets, providing conditions for their degeneration and generalizing Hochster's decomposition theorem.

Contribution

It introduces novel spectral sequences involving derived functors of limits and colimits over finite posets, extending local cohomology decomposition results.

Findings

Spectral sequences degenerate under certain conditions.

Generalization of Hochster's local cohomology decomposition.

Framework applicable to various functor types.

Abstract

We introduce a formalism to produce several families of spectral sequences involving the derived functors of the limit and colimit functors over a finite partially ordered set. The first type of spectral sequences involves the left derived functors of the colimit of the direct system that we obtain applying a family of functors to a single module. For the second type we follow a completely different strategy as we start with the inverse system that we obtain by applying a covariant functor to an inverse system. The spectral sequences involve the right derived functors of the corresponding limit. We also have a version for contravariant functors. In all the introduced spectral sequences we provide sufficient conditions to ensure their degeneration at their second page. As a consequence we obtain some decomposition theorems that greatly generalize the well-known decomposition formula for local cohomology modules given by Hochster.