The relative Brauer group and generalized cyclic crossed products for a ramified covering
The crossed product theorem states that every central simple algebra over a field \(K\) split by a Galois extension \(L\) of \(K\) is Brauer equivalent to a crossed product, otherwise stated, the relative Brauer group \(B(L/K)\) is isomorphic to the second cohomology group \(H^2(G,L^*)\). If \(T\) is a Galois extension of a commutative ring \(A\), then we have a homomorphism \(H^2(G,T^*)\to B(T/A)\) that is in general not surjective or injective. This homomomorphism fits into a 7-term exact sequence, due to Chase, Harrison and Rosenberg [\textit{S. U. Chase} et al., Mem. Am. Math. Soc. 52, 15--33 (1965; Zbl 0143.05902)]. At the level of algebras, this can restated in the following form: every \(A\)-Azumaya algebra split by \(T\) is Brauer equivalent to a \textsl{generalized} crossed product, a statement originally due to \textit{T. Kanzaki} [Osaka J. Math. 5, 175--188 (1968; Zbl 0202.04202)]. The generalization comes from the fact that such a generalized crossed product Azumaya algebra is isomorphic as an \(T\)-module to a direct sum of \(|G|\) rank one \textsl{projective} \(T\)-modules, instead of rank one \textsl{free} modules. The starting point of the present paper is the following. Let \(T/A\) be a ramified extension of noetherian integrally closed integral domains for which the extension of quotient fields L/K is Galois with cyclic group \(G=\langle\sigma\rangle\). Now define generalized cyclic crossed modules starting from rank one \textsl{reflexive} \(T\)-modules, or, equivalently, reflexive fractional ideals of \(T\) in \(L\). The fact that \(G\) is cyclic entails that the generalized crossed product is determined by a single reflexive \(T\)-module, allowing the author to make explicit computations. \textit{D. S. Rim} [Proc. Am. Math. Soc. 16, 837--840 (1965; Zbl 0166.30604)] presented a divisorial version of the Chase-Harrison-Rosenberg exact sequence, where the group replacing the relative Brauer group is described in cohomological terms. In some cases, this group equals the relative reflexive Brauer group. Results in this direction that exist in the literature are under the assumption that \(T/A\) is unramified, see for example [\textit{S. Yuan}, J. Algebra 32, 405--417 (1974; Zbl 0297.13010)]. The main result of the present paper, Theorem 1.1, gives new results in the ramified case. Under some regularity conditions, the reflexive Brauer group can be replaced by the Brauer group. One of the attractive features of the paper is the final Section 5, where explicit computations in some specific situations are given.
scientific article; zbMATH DE number 6537406
scientific article