Automorphisms of finitary incidence rings

Abstract

Let \(P\) be a quasiordered set, \(R\) an associative unital ring,  \(\mathcal{C}(P,R)\) a partially ordered category associated with the pair \((P,R)\)[6], \(FI(P,R)\) a finitary incidence ring of \(\mathcal{C}(P,R)\)[6]. We prove that the group \({\rm Out}FI\) of outer automorphisms of \(FI(P,R)\) is isomorphic to the group \({\rm Out}\mathcal{C}\) of outer automorphisms of \(\mathcal{C}(P,R)\) under the assumption that \(R\) is indecomposable. In particular, if \(R\) is local, the equivalence classes of \(P\) are finite and \(P=\bigcup\limits_{i\in I}P_i\) is the decomposition of \(P\) into the disjoint union of the connected components, then \({\rm Out}FI\cong (H^1(\overline P,C(R)^*)\rtimes\prod\limits_{i\in I}{\rm Out}R)\rtimes{\rm Out}P\). Here \(H^1(\overline P,C(R)^*)\) is the first cohomology group of the order complex of the induced poset \(\overline P\) with the values in the multiplicative group of central invertible elements of \(R\). As a consequences, Theorem 2 [9], Theorem 5 [2 ] and Theorem 1.2  [8] are obtained.