Homomorphisms and Involutions of Unramified Henselian Division Algebras

Abstract

Let K be a Henselian field with a residue field $$ \overline{K} $$ , and let A1, A2 be finite-dimensional division unramified K-algebras with residue algebras Ā 1 and Ā 2 Further, let HomK(A1,A2) be the set of nonzero K-homomorphisms from A1 to A2. It is proved that there is a natural bijection between the set of nonzero $$ \overline{K} $$ -homomorphisms from Ā 1 to Ā 2 and the factor set of HomK and the factor set of HomK(A1,A2) under the equivalence relation: ϕ 1 ∼ ϕ 2 ⇔ there exists m ∈ 1 +MA2 such that ϕ2 = ϕ1 im, where im is the inner automorphism of A2 induced by m. A similar result is obtained for unramified algebras with involutions. Bibliography: 7 titles.