Homomorphisms and Involutions of Unramified Henselian Division Algebras

Let K be a Henselian field with a residue field K ¯ , and let A 1 , A 2 be finite-dimensional division unramified K-algebras with residue algebras Ā 1 and Ā 2 Further, let Hom K (A 1 ,A 2 ) be the set of nonzero K-homomorphisms from A 1 to A 2 . It is proved that there is a natural bijection between...

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Published inJournal of mathematical sciences (New York, N.Y.) Vol. 209; no. 4; pp. 657 - 664
Main Authors Tikhonov, S. V., Yanchevskii, V. I.
Format Journal Article
LanguageEnglish
Published New York Springer US 02.09.2015
Springer
Subjects
Algebra
Mathematics
Mathematics and Statistics
Central Simple Algebra
Ring Versus
Identity Automorphism
Division Algebra
Central Division
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Summary:Let K be a Henselian field with a residue field K ¯ , and let A 1 , A 2 be finite-dimensional division unramified K-algebras with residue algebras Ā 1 and Ā 2 Further, let Hom K (A 1 ,A 2 ) be the set of nonzero K-homomorphisms from A 1 to A 2 . It is proved that there is a natural bijection between the set of nonzero K ¯ -homomorphisms from Ā 1 to Ā 2 and the factor set of Hom K and the factor set of Hom K (A 1 ,A 2 ) under the equivalence relation: ϕ 1  ∼  ϕ 2  ⇔ there exists m ∈ 1 +M A2 such that ϕ 2 = ϕ 1 i m, where im is the inner automorphism of A 2 induced by m. A similar result is obtained for unramified algebras with involutions. Bibliography: 7 titles.
ISSN:1072-3374
1573-8795
DOI:10.1007/s10958-015-2519-x