Maximal orders in unramified central simple algebras (Q5964492)

Let \(A\) be a central simple algebra over the function field \(K\) of a regular noetherian integral scheme \(X\). Assume that the Brauer class of \(A\) belongs to the image of the map \(\mathrm{Br}(X)\to\mathrm{Br}(K)\). \textit{M. Auslander} and \textit{O. Goldman} [Trans. Am. Math. Soc. 97, 1--24 (1960; Zbl 0117.02506)] have shown that maximal orders in \(A\) are Azumaya, provided that the dimension of \(X\) is at most two. The authors prove, in the more general context of Japanese integral noetherian schemes, that this result does not hold in higher dimensions if the degree of \(A\) is greater than one. Moreover, they prove that a two-dimensional integral noetherian scheme \(X\) with function field \(K\) is regular if and only if every maximal order over \(X\) in a central simple \(K\)-algebra with unramified Brauer class is Azumaya. The proof makes use of a concept of depth for coherent sheaves on an algebraic stack.