Projective resolutions of simple modules and Hochschild cohomology for incidence algebras

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