On local-to-global spectral sequences for the cohomology of diagrams

TL;DR

This paper constructs and analyzes three potential local-to-global spectral sequences for the cohomology of diagrams of algebras with directed indexing, focusing on their $E^2$-terms as local cohomology relative to diagram components.

Contribution

It introduces three new candidate spectral sequences for computing diagram cohomology, expanding the theoretical framework for local-to-global spectral analysis.

Findings

Three candidate spectral sequences are constructed and examined.

The $E^2$-terms are interpreted as local cohomology relative to diagram maps or objects.

The paper provides foundational tools for future cohomological computations in algebra diagrams.

Abstract

The aim of this paper is to construct and examine three candidates for local-to-global spectral sequences for the cohomology of diagrams of algebras with directed indexing. In each case, the $E^2$ -terms can be viewed as a type of local cohomology relative to a map or an object in the diagram.