Hochschild cohomology of some finite category algebras as simplicial cohomology

TL;DR

This paper explores the relationship between Hochschild cohomology of finite category algebras and simplicial cohomology, extending known results and providing new isomorphisms under certain conditions.

Contribution

It constructs a new category and establishes a graded isomorphism between Hochschild cohomology and simplicial cohomology for certain categories.

Findings

Hochschild cohomology ring is isomorphic to simplicial cohomology ring for poset categories

A new category $\\mathcal{D}$ is constructed under specific assumptions

Degree one cohomology relates derivations to characters on the constructed category

Abstract

By a result of Gerstenhaber and Schack the simplicial cohomology ring $H^*(\mathcal{C};k)$ of a poset $\mathcal{C}$ is isomorphic to the Hochschild cohomology ring $HH^*(k\mathcal{C})$ of the category algebra $k\mathcal{C}$, where the poset is viewed as a category and $k$ is a field. Extending results of Mishchenko [6], under certain assumptions on a category $\mathcal{C}$, we construct a category $\mathcal{D}$ and a graded $k$-linear isomorphism $HH^*(k\mathcal{C})\cong H^*(\mathcal{D}; k)$. Interpreting the degree one cohomology, we also show how the $k$-space of derivations on $k\mathcal{C}$, graded by some semigroup, corresponds to the $k$-space of characters on $\mathcal{D}$.