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zbMATH Open
Article . 2015
Data sources: zbMATH Open
https://doi.org/10.4171/dms/7/...
Part of book or chapter of book . 2015 . Peer-reviewed
Data sources: Crossref
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Wedderburn’s theorem for regular local rings

Wedderburn's theorem for regular local rings
Authors: Ojanguren, Manuel;

Wedderburn’s theorem for regular local rings

Abstract

Let \(R\) be a regular local ring containing a field of characteristic zero, \(K\) its field of fractions and \((V, \Phi)\) a quadratic space over \(R\). \textit{I. Panin} proved that if \((V, \Phi) \otimes_RK\) is isotropic over \(K\), then \((V, \Phi)\) is isotropic over \(R\) [Invent. Math. 176, No. 2, 397--403 (2009; Zbl 1173.11025)]. Using the same argument of the above result, the author extends the Wedderburn theorem to a large class of regular local rings. Theorem. Let \(R\) be a regular local ring containing a field of characteristic zero and \(A\) an Azumaya \(R\)-algebra. If \(A \otimes_RK \cong M_n(D)\), a matrix ring of order \(n\) where \(D\) is a central division algebra over \(K\), then \(A \cong M_n(\Delta)\), where \(\Delta\) is a maximal (unramified) \(R\)-order of \(D\). In other words, every class of the Brauer group of \(R\) is represented by an Azumaya algebra \(\Delta\) such that \(\Delta \otimes_RK\) is a division \(K\)-algebra. It is also pointed out that the above theorem has been generalized to arbitrary semi-local regular rings by \textit{B. Antieau} and \textit{B. Williams} [Doc. Math., J. DMV 20, 333--355 (2015; Zbl 1349.14068)].

Keywords

regular local ring, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), division ring, Azumaya algebra, Noncommutative local and semilocal rings, perfect rings

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
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