
doi: 10.4171/dms/7/19
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)].
regular local ring, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), division ring, Azumaya algebra, Noncommutative local and semilocal rings, perfect rings
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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