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Transactions of the American Mathematical Society

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Brauer $ p$-dimension of complete discretely valued fields


Authors: Nivedita Bhaskhar and Bastian Haase
Journal: Trans. Amer. Math. Soc. 373 (2020), 3709-3732
MSC (2010): Primary 16K50, 11R58
DOI: https://doi.org/10.1090/tran/8038
Published electronically: February 20, 2020
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Abstract: Let $ K$ be a complete discretely valued field of characteristic 0 with residue field $ \kappa $ of characteristic $ p$. Let $ n$ be the $ p$-rank of $ \kappa $, i.e., $ p^n=[\kappa :\kappa ^p]$. It was proved by Parimala and Suresh [Invent. Math. 197 (2014), no. 1, pp. 215-235] that the Brauer $ p$-dimension of $ K$ lies between $ n/2$ and $ 2n$. For $ n\leq 3$, we improve the upper bound to $ n+1$ and provide examples to show that our bound is sharp. For $ n \leq 2$, we also improve the lower bound to $ n$. For general $ n$, we construct a family of fields $ K_n$ with residue fields of $ p$-rank $ n$ such that $ K_n$ admits a central simple algebra $ D_n$ of period $ p$ and index $ p^{n+1}$. Our sharp lower bounds for $ n\leq 2$ and upper bounds for $ n\leq 3$ in combination with the nature of these examples motivate us to conjecture that the Brauer $ p$-dimension of such fields always lies between $ n$ and $ n+1$.


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Additional Information

Nivedita Bhaskhar
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
Email: bhaskhar@usc.edu

Bastian Haase
Affiliation: Department of Mathematics and Computer Science, Emory University, 400 Dowman Drive NE, Atlanta, Georgia 30322
Email: bhaase@emory.edu

DOI: https://doi.org/10.1090/tran/8038
Keywords: Brauer group, Brauer $p$-dimension, complete discretely valued field, $p$-rank, Milnor $k$-groups, Kato's filtration.
Received by editor(s): November 15, 2016
Received by editor(s) in revised form: February 25, 2019, and September 30, 2019
Published electronically: February 20, 2020
Additional Notes: The authors were partially supported by National Science Foundation grants DMS-1401319 and DMS-1463882.
Article copyright: © Copyright 2020 American Mathematical Society