Automorphisms of finitary incidence rings

Nikolay Khripchenko

Algebra and discrete mathematics, Tome 9 (2010) no. 2, pp. 78-97

Résumé

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 $\mathrm{Out}FI$ of outer automorphisms of $FI(P,R)$ is isomorphic to the group $\mathrm{Out}\mathcal C$ 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 $P=\bigcup_{i\in I}P_i$ is the decomposition of $P$ into the disjoint union of the connected components, then $\mathrm{Out}FI\cong (H^1(\overline P,C(R)^*)\rtimes\prod_{i\in I}\mathrm{Out}R)\rtimes\mathrm{Out}P$. 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.

Détail

Keywords: finitary incidence algebra, partially ordered category, quasiordered set
Mots-clés : automorphism.