Unramified division algebras do not always contain Azumaya maximal orders. (Q398672)

In their fundamental 1960 paper on Brauer groups [Trans. Am. Math. Soc. 97, 367-409 (1961; Zbl 0100.26304)], \textit{M. Auslander} and \textit{O. Goldman} prove that over a regular domain \(R\) of Krull dimension at most two, the Brauer group satisfies \(B(R)=\bigcap B(R_{\mathfrak p})\), where \(\mathfrak p\) ranges over the prime ideals of height one. Having shown that a maximal order \(\Delta\) is reflexive, they need the condition \(\dim R\leq 2\) just to ensure that \(\Delta\) is projective. The question remained open whether the theorem remains true in higher dimensions. In the present paper, the authors confirm the necessity of \(\dim R\leq 2\) by viewing the existence problem of Azumaya algebras in the broader context where \(\mathrm{Spec\,}R\) is replaced by a regular Noetherian integral scheme. As a result, they exhibit a 6-dimensional smooth complex affine variety \(X\) and a Brauer class in \(\mathrm{Br}(X)\) such that the skew-field over the generic point of \(X\) has no Azumaya maximal order over \(X\).