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Homomorphisms and Involutions of Unramified Henselian Division Algebras

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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.

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Authors and Affiliations

  1. Belarus State University, Minsk, Belarus

    S. V. Tikhonov

  2. The Institute of Mathematics, Belarus National Academy of Sciences, Minsk, Belarus

    V. I. Yanchevskii

Authors
  1. S. V. Tikhonov
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  2. V. I. Yanchevskii
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Correspondence to S. V. Tikhonov.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 423, 2014, pp. 264–275.

Translated by N. B. Lebedinskaya.

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Tikhonov, S.V., Yanchevskii, V.I. Homomorphisms and Involutions of Unramified Henselian Division Algebras. J Math Sci 209, 657–664 (2015). https://doi.org/10.1007/s10958-015-2519-x

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