Projective resolutions of simple modules and Hochschild cohomology for incidence algebras

TL;DR

This paper presents an algorithmic approach to compute minimal projective resolutions of simple modules in incidence algebras, enabling calculations of Ext groups, Hochschild cohomology, and topological invariants.

Contribution

It introduces a practical method for calculating projective resolutions in incidence algebras, facilitating various cohomological computations.

Findings

Efficient algorithm for minimal projective resolutions

Calculation of Ext groups between simple modules

Determination of Hochschild cohomology groups

Abstract

We give a practical, algorithmic method to calculate minimal projective resolutions of simple modules for a finite dimensional incidence $k$-algebra $\Lambda$, where $k$ is a field. We apply the method to the calculation of Ext groups between simple $\Lambda$-modules, Hochschild cohomology groups $\HH^i(\Lambda, \Lambda)$, and singular cohomology groups of finite $T_0$ topological spaces with coefficients in $k$.