Diagrammatic Hochschild cohomology via cohomology of categories, and incidence algebras

TL;DR

This paper explores the Hochschild cohomology of algebra diagrams, linking it to category cohomology, analyzing its behavior under algebra morphisms, and computing it for incidence algebra diagrams related to simplicial complexes.

Contribution

It connects Gerstenhaber-Schack complexes with Baues-Wirsching cohomology, analyzes Hochschild cohomology under homological epimorphisms, and computes cohomology for incidence algebra diagrams.

Findings

Established a connection between Gerstenhaber-Schack and Baues-Wirsching cohomology.

Analyzed Hochschild cohomology behavior under homological epimorphisms.

Computed Hochschild cohomology for diagrams of incidence algebras associated with simplicial filtrations.

Abstract

In this paper, we study the Hochschild cohomology of diagrams of algebras introduced by Gerstenhaber and Schack and provide computations for filtrations of incidence algebras. Our aims are threefold: firstly, we revisit and explore the connection between the Gerstenhaber-Schack complexes and the Baues-Wirsching cohomology of categories. Secondly, we analyse the behaviour of (diagrammatic) Hochschild cohomology in the context of homological epimorphisms of algebras. Thirdly, we study diagrams of incidence algebras. In particular, as a main application, we compute the diagrammatic Hochschild cohomology of diagrams of incidence algebras associated to finite filtrations of simplicial complexes.