On $p$-torsion in etale cohomology and in the Brauer group

Abstract:

If X is an affine scheme in characteristic $p > 0$, then ${\text {Br}}(X)(p)\tilde \to H_{{\text {et}}}^2(X,{{\mathbf {G}}_m})(p)$ and $H_{{\text {et}}}^n(X,{{\mathbf {G}}_m})(p) = 0$ for $n \geqslant 3$. This gives a partial answer to the conjecture that the Brauer group of any scheme X is canonically isomorphic to the torsion part of $H_{{\text {et}}}^2(X,{{\mathbf {G}}_m})$. This result is then applied to prove that ${\text {Br}}(R)(p)$ is p-divisible where R is a commutative ring of characteristic $p > 0$ (theorem of Knus, Ojanguren and Saltman), and also to construct examples of domains R of characteristic $p > 0$ with large ${\operatorname {Ker}}({\text {Br}}(R)(p) \to {\text {Br}}(Q)(p))$, where Q is the ring of fractions of R.