Cohomology of diagrams of algebras

TL;DR

This paper develops a cohomology theory for diagrams of algebras using Beck's comonad approach and introduces a spectral sequence linking diagram cohomology to algebra cohomology.

Contribution

It extends cohomology theory to diagrams of algebras and applies it to $\\Psi$-rings, establishing a spectral sequence connecting diagram and algebra cohomology.

Findings

Established a spectral sequence relating diagram cohomology to algebra cohomology.

Applied the theory specifically to the case of $\Psi$-rings.

Provided a framework for computing cohomology of algebra diagrams.

Abstract

We consider cohomology of diagrams of algebras by Beck's approach, using comonads. We then apply this theory to computing the cohomology of $\Psi$-rings. Our main result is that there is a spectral sequence connecting the cohomology of the diagram of an algebra to the cohomology of the underlying algebra.