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15 June 2021 The arc-topology
Bhargav Bhatt, Akhil Mathew
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Duke Math. J. 170(9): 1899-1988 (15 June 2021). DOI: 10.1215/00127094-2020-0088

Abstract

We study a Grothendieck topology on schemes which we call the arc-topology. This topology is a refinement of the v-topology (the pro-version of Voevodsky’s h-topology), where covers are tested via rank 1 valuation rings. Functors which are arc-sheaves are forced to satisfy a variety of gluing conditions such as excision in the sense of algebraic K-theory. We show that étale cohomology is an arc-sheaf, and we deduce various pullback squares in étale cohomology. Using arc-descent, we re-prove the Gabber–Huber affine analogue of proper base change (in a large class of examples), as well as the Fujiwara–Gabber base change theorem on the étale cohomology of the complement of a Henselian pair. As a final application, we prove a rigid analytic version of the Artin–Grothendieck vanishing theorem, extending results of Hansen.

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Bhargav Bhatt. Akhil Mathew. "The arc-topology." Duke Math. J. 170 (9) 1899 - 1988, 15 June 2021. https://doi.org/10.1215/00127094-2020-0088

Information

Received: 31 August 2018; Revised: 23 November 2020; Published: 15 June 2021
First available in Project Euclid: 23 March 2021

Digital Object Identifier: 10.1215/00127094-2020-0088

Subjects:
Primary: 14F20
Secondary: 14F06 , 14G22

Keywords: Artin–Grothendieck vanishing theorem , étale cohomology , excision , Grothendieck topologies , proper base change , rigid analytic geometry , valuation rings

Rights: Copyright © 2021 Duke University Press

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Vol.170 • No. 9 • 15 June 2021
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