Hochschild Cohomology of Some Finite Category Algebras as Simplicial Cohomology

a result of Gerstenhaber and Schack, the simplicial cohomology ring H-center dot(C , k) of a poset C is isomorphic to the Hochschild cohomology ring HH center dot(kC) of the category algebra kC, where the poset is viewed as a category and k is a field. Extending results of Mishchenko, under certain assumptions on a category C, we construct a category D and a graded k-linear isomorphism HH center dot(kC) expressionpproximexpressiontely equexpressionl to H-center dot(D , k). Interpreting the degree one cohomology, we also show how the k-space of derivations on kC graded by some semigroup corresponds to the k-space of characters on D.