The Brauer group of an noncomplete real algebraic surface (Q1585628)

The author proves that for a non-singular real algebraic surface \(X\) with Lefschetz number \(\rho(X)\), its Brauer group Br\((X)\) is isomorphic to \((\mathbb{Q}/ \mathbb{Z})^{\rho(X)} \oplus H^2(X (\mathbb{C}),G, \mathbb{Z})_{\text{tors}}\), where \(G\) is the Galois group \(G=\text{Gal} (\mathbb{C}|\mathbb{R})\) and the set \(\mathbb{Z}\) of integers is endowed with the structure of \(G\)-module where the complex conjugation acts as multiplication by \(-1\). This was proved previously by the author in case \(X\) is a projective surface [\textit{V. A. Krasnov}, Izv. Math. 60, No. 5, 933-962 (1996); translation from Izv. Ross. Akad. Nauk., Ser. Mat. 60, No. 5, 57-88 (1996; Zbl 0896.13003)], and so the goal of this article is to extend the result to non-complete surfaces. The proof differs substantially of the one in the complete case, and uses results by the author on étale cohomology [see \textit{V. A. Krasnov}, ``Analogue of the Harnack-Thom inequality for a real algebraic surface'', Izv. Math. 64, No. 5, 915-937 (2000); translation from Ross. Akad. Nauk, Ser. Mat. 64, No. 5, 45-68 (2000)]. Perhaps, the main ingredient in the proof is the existence of a (non-canonical) isomorphism \[ H^2_{\text{et}} (X,\mu_\infty) \cong(\mathbb{Q}/ \mathbb{Z})^\alpha\oplus H^3\bigl(X (\mathbb{C}),G, \mathbb{Z}\bigr)_{\text{tors}}, \] where \(\mu_\infty\) is the sheaf of roots of unity and \(\alpha\) is the dimension of \(H^2(X(\mathbb{C}), \mathbb{Q} \otimes \mathbb{Z})^G\). -- The paper is short and the proof is transparent.