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arXiv · 1706.08976

Skolem-Noether algebras

Matej Brešar·Christoph Hanselka·Igor Klep·Jurij Volčič
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Abstract

An algebra $S$ is called a Skolem-Noether algebra (SN algebra for short) if for every central simple algebra $R$, every homomorphism $R\to R\otimes S$ extends to an inner automorphism of $R\otimes S$. One of the important properties of such an algebra is that each automorphism of a matrix algebra over $S$ is the composition of an inner automorphism with an automorphism of $S$. The bulk of the paper is devoted to finding properties and examples of SN algebras. The classical Skolem-Noether theorem implies that every central simple algebra is SN. In this article it is shown that actually so is every semilocal, and hence every finite-dimensional algebra. Not every domain is SN, but, for instance, unique factorization domains, polynomial algebras and free algebras are. Further, an algebra $S$ is SN if and only if the power series algebra $S[[\xi]]$ is SN.

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Matej Brešar, Christoph Hanselka, Igor Klep, Jurij Volčič. 2017-06-27. Skolem-Noether algebras. https://doi.org/10.1016/j.jalgebra.2017.11.045

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