Cohomology of incidence algebras and simplicial complexes (Q1824672)

Let Q be a finite oriented graph without oriented cycles such that for each arrow \(s\bullet\) \(\to^{\alpha}\bullet t\) there is no oriented path other than \(\alpha\) joining s to t. Q is called an ordered quiver. There is a well-known bijection between finite posets and ordered quivers. Let R be a commutative ring and let I be the two-sided ideal of the path algebra RQ generated by all \(\gamma\)-\(\delta\), where \(\gamma\) and \(\delta\) are paths with the same source and end vertices. The algebra RQ/I is the incidence algebra of the finite poset associated to Q. This paper contains 3 main results. First, a new proof is given of a result of Gerstenhaber and Schack showing that the Hochschild cohomology of RQ/I is isomorphic to the cohomology with coefficients in R of the simplicial complex \(\Sigma_ Q\) whose i-simplices are chains of length i in Q. The second result computes the reduced cohomology of \(\Sigma_ Q\) in terms of Ext(A,B) for certain modules A,B over an enlarged quiver \(\bar Q.\) The third computes the cohomology groups of a finite simplicial complex in terms of the ideals of an associated incidence algebra.