Hochschild Cohomology of Reduced Incidence Algebras

TL;DR

This paper computes the Hochschild cohomology of various reduced incidence algebras using coalgebraic methods, linking it to Ext-groups of related incidence algebras, advancing algebraic understanding.

Contribution

It introduces a coalgebraic approach to compute Hochschild cohomology for reduced incidence algebras and relates it to Ext-groups of incidence algebras from quivers.

Findings

Hochschild cohomology of formal power series and related algebras computed

Coalgebraic methods effectively determine cohomology groups

Connections established between Hochschild cohomology and Ext-groups for quiver-based incidence algebras

Abstract

We compute the Hochschild cohomology of the reduced incidence algebras such as the algebra of formal power series, the algebra of exponential power series, the algebra of Eulerian power series, and the algebra of formal Dirichlet series. We achieve the result by carrying out the computation on the coalgebra ${\rm Cotor}$-groups of their pre-dual coalgebras. Using the same coalgebraic machinery, we further identify the Hochschild cohomology groups of an incidence algebra associated to a quiver with the ${\rm Ext}$-groups of the incidence algebra associated to a suspension of the quiver.