Automorphisms of finitary incidence rings
Abstract
Let P be a quasiordered set, R an associative unital ring, C(P,R) a partially ordered category associated with the pair (P,R)[6], FI(P,R) a finitary incidence ring of C(P,R)[6]. We prove that the group OutFI of outer automorphisms of FI(P,R) is isomorphic to the group OutC of outer automorphisms of 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=⋃i∈IPi is the decomposition of P into the disjoint union of the connected components, then OutFI≅(H1(¯P,C(R)∗)⋊. 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.
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