Adelic points and unmramified Brauer approximation for classifying stacks

Abstract

Let $k$ be a number field and let $G$ be a connected linear algebraic group over $k$. We compare three approaches to strong approximation for the classifying stack $BG$ with respect to its full Brauer group: the homogeneous-space method of \cite{DhillonClassifying}, Kottwitz's local--global sequence, and, for reductive groups, Borovoi's localization theorem. Under the natural identification \[ Br(BG)/Br(k)\simeq Pic(G), \] we identify the Kottwitz and Borovoi obstruction maps with Brauer evaluation. The three approaches therefore give the same exact description of the localization image as the projected full Brauer--Manin set. We then study strong approximation with respect to the ordinary unramified Brauer group. We prove \[ Br^{un}(BG)/ Br(k)\simeq Sha^1_{\mathrm{cyc}}(k,\widehat G), \qquad \widehat G=X^*(G_{\bar{k}}), \] and give an exact local criterion for unramified Brauer approximation off a finite set of places. Examples show that this approximation can fail even when a finite place is omitted, and exhibit a torus for which the unramified Brauer group cuts out the global image as a proper subset of the adelic space.