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.