Brauer groups and Galois cohomology for a Krull scheme
In a previous paper the authors introduced the class group \(Cl(\mathcal C(X))\) and the Brauer group \(Br(\mathcal C(X))\), where \(\mathcal C(X)\) is a suitable category of divisorial lattices over a Krull scheme. In this paper it is shown that there are exact sequences of Galois cohomology which relate the class groups and Brauer groups so defined. The main exact sequence obtained is \[ \begin{multlined} 1 \to H^1(G,\mathcal O_ Y(Y)^*) \to Cl(\mathcal C(Y)) \to H^0(G,Cl(\mathcal C(Y))) \to H^2(G,\mathcal O_ Y(Y)^*) \to \\ \to Br(\mathcal C(Y/X)) \to H^1(G,Cl(\mathcal C(Y))) \to H^3(G,\mathcal O_ Y(Y)^*)), \end{multlined} \] where \(G\) is a finite group and \(Y/X\) is a \(\mathcal C\)-Galois covering of Krull schemes with group \(G\). (This condition agrees with being an ordinary Galois covering when \(\mathcal C(W)\) is the category of locally free \(\mathcal O_ W\)- modules of finite type, but for other categories of interest the condition may be less restrictive.) By choosing \(\mathcal C(X)\) properly a number of previously known exact sequences of Galois cohomology can be obtained: one presented by Rim, involving Brauer groups and class groups of normal domains; one considered by Chase and Rosenberg, by DeMeyer and Ingraham and by Auslander and Brumer, involving Brauer groups and Picard groups of commutative rings; one considered by Yuan, dealing with class groups and modified Brauer groups of a noetherian normal domain; and a sequence of Lichtenbaum, involving divisor class groups and Brauer groups of a projective curve. When \(\mathcal C(X)\) is the category of \(X\)-lattices the exact sequence above is also obtained for \(G\) a profinite group, and is used to compute the ordinary Brauer group of some projective varieties.
scientific article; zbMATH DE number 3924943
scientific article