Galois descent on the Brauer group (Q2856658)

The main object of the paper under review is the map NEWLINE\[NEWLINE \alpha: \mathrm{Br}(X)\to \mathrm{Br}(\overline X)^\Gamma, NEWLINE\]NEWLINE where \(X\) is a smooth, geometrically integral, projective variety \(X\) over a field \(k\) of characteristic zero, \(\overline X=X\times_k\bar k\), \(\Gamma=\mathrm{Gal}(\bar k/k)\). The main result states that the cokernel of \(\alpha\) is finite. In the case where \(H^1(X,O_X)=0\) or \(k\) is a number field, the authors provide explicit estimates for the order and exponent of this finite group. Particular applications include \(K3\) surfaces and products of two curves. Finally, the main finiteness theorem is extended to quasi-projective varieties, under the assumption that \(k\) is finitely generated over \(\mathbb Q\).