On the Brauer group (Q2710699)

The author considers the Brauer group \(\text{Br}(V)\) and the cohomological Brauer group \(\text{Br}^\prime(V)\) of a smooth projective variety \(V\) over the perfect field \(k\). Let \(\ell\) be a prime. Assume that \(V\) has a \(k\)-rational point, so that \(\text{Br}(k) \subset \text{Br}^\prime(V)\). The author proves, under various restrictive hypotheses on \(V\), that for all sufficiently large \(\ell\) the \(\ell\)-primary component of \(\text{Br}^\prime(V)/\text{Br}(k)\) is finite. Moreover, if \(A\) is the ring of integers in a purely imaginary number field \(k\), and \(\pi:X \to \text{Spec}(A)\) is proper and flat, where \(X\) is a regular scheme and the generic fibre of \(\pi\) is one of the varieties \(V\) above, then, under some suitable conditions on \(X\), \(\pi\), and \(V\), the author proves that \(\text{Br}^\prime(X)\) has finite \(\ell\)-primary component. (M. Artin has conjectured that any proper scheme over \(\text{Spec}(\mathbb Z)\) has finite Brauer group.) The methods of proof are both (étale) cohomological and algebraic/geometric. Regarding the former, the author uses extensively a new exact sequence he derives of low degree ( \(\leq 3\)) terms for spectral sequences.