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arXiv · 1401.7782

A cohomological Hasse principle over two-dimensional local rings

Yong Hu
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Abstract

Let $K$ be the fraction field of a two-dimensional henselian, excellent, equi-characteristic local domain. We prove a local-global principle for Galois cohomology with finite coefficients over $K$. We use classical machinery from étale cohomology theory, drawing upon an idea in Saito's work on two-dimensional local class field theory. This approach works equally well over the function field of a curve over an equi-characteristic henselian discrete valuation field, thereby giving a different proof of (a slightly generalized version of) a recent result of Harbater, Hartmann and Krashen. We also present two applications. One is the Hasse principle for torsors under quasi-split semisimple simply connected groups without $E_8$ factor. The other gives an explicit upper bound for the Pythagoras number of a Laurent series field in three variables. This bound is sharper than earlier estimates.

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Yong Hu. 2016-06-13. A cohomological Hasse principle over two-dimensional local rings. https://doi.org/10.1093/imrn%2Frnw149

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