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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On $p$-torsion in etale cohomology and in the Brauer group
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by Robert Treger PDF
Proc. Amer. Math. Soc. 78 (1980), 189-192 Request permission

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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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 78 (1980), 189-192
  • MSC: Primary 14F20; Secondary 16A16
  • DOI: https://doi.org/10.1090/S0002-9939-1980-0550491-8
  • MathSciNet review: 550491