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

Author: Robert Treger
Journal: Proc. Amer. Math. Soc. 78 (1980), 189-192
MSC: Primary 14F20; Secondary 16A16
MathSciNet review: 550491
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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.

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