The \(\operatorname{arc}\)-topology
type: Item
instance of: scholarly article
title: The \(\operatorname{arc}\)-topology
zbMATH Open document ID: 1478.14036
DOI: 10.1215/00127094-2020-0088
author: Bhargav Bhatt
author: Akhil Mathew
published in: Duke Mathematical Journal
publication date: 2021-07-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1807.04725
review text: The paper under review aims to study a new Grothendieck topology on the category of quasi-compact quasi-separated schemes, the so-called arc-topology. It is a slight refinement of the v-topology and h-topology for more general non-noetherian schemes. Covers in the arc-topology are tested via rank \(\leq 1 \) valuation rings. Examples of arc sheaves include étale cohomology with torsion coefficients and perfect complexes on perfect \(\mathbb{F}_p\)-schemes. Using the arc-desent, Bhatt (the first author) and Scholze established the étale comparison theorem for prismatic cohomology. As applications, the authors reproved two classical results in étale cohomology: (i) the Gabber-Huber affine analog of proper base change theorem; (ii) the Fujiwara-Gabber theorem for punctured henselian pairs. They also gave an application to Artin-Grothendieck's vanishing theorem in rigid analytic geometry, which strengthens the results of \textit{D. Hansen} [Compos. Math. 156, No. 2, 299--324 (2020; Zbl 1441.14085)].
reviewed by: Yigeng Zhao
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 14F06
Mathematics Subject Classification ID: 14G22
zbMATH DE Number: 7369864
zbMATH Keywords: Artin-Grothendieck vanishing theorem
zbMATH Keywords: étale cohomology
zbMATH Keywords: excision
zbMATH Keywords: Grothendieck topologies
zbMATH Keywords: proper base change
zbMATH Keywords: rigid analytic geometry
zbMATH Keywords: valuation rings
MaRDI profile type: Publication
OpenAlex ID: W3152570866
arXiv ID: 1807.04725
cites work: Seminar of algebraic geometry du Bois-Marie 1963--1964. Topos theory and étale cohomology of schemes (SGA 4). Vol. 1: Topos theory. Exp. I--IV
cites work: Affine representability results in \(\mathbb{A}^1\)-homotopy theory. I: Vector bundles.
cites work: Q4852332
cites work: Algebraization and Tannaka duality
cites work: Projectivity of the Witt vector affine Grassmannian
cites work: Q3325851
cites work: Q3413653
cites work: Model theory.
cites work: Étale motives
cites work: The obstruction to excision in \(K\)-theory and in cyclic homology
cites work: Smoothness, semi-stability and alterations
cites work: Cohomologie étale. Seminaire de géométrie algébrique du Bois-Marie SGA 4 1/2 par P. Deligne, avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier
cites work: Q3349023
cites work: Solutions d'équations à coefficients dans un anneau hensélien
cites work: Cdh descent, cdarc descent, and Milnor excision
cites work: Theory of tubular neighborhood in étale topology
cites work: Q4010222
cites work: Affine analog of the proper base change theorem
cites work: Points in algebraic geometry
cites work: On the \(p\)-dimension of fields
cites work: Almost ring theory
cites work: Bi-relative algebraic \(K\)-theory and topological cyclic homology
cites work: Q4092470
cites work: Q5612510
cites work: A cohomological bound for the \(h\)-topology
cites work: Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas (Quatrième partie). Rédigé avec la colloboration de J. Dieudonné
cites work: Vanishing and comparison theorems in rigid analytic geometry
cites work: Differential forms in the h-topology
cites work: Differential forms in positive characteristic. II: cdh-descent via functorial Riemann-Zariski spaces
cites work: Continuous valuations
cites work: Étale cohomology of Henselian rings and cohomology of abstract Riemann surfaces of fields
cites work: Q2785517
cites work: Q3191546
cites work: Q4357478
cites work: Q3711309
cites work: Q4117922
cites work: On the \(K\)-theory of pullbacks
cites work: The six operations for sheaves on Artin stacks. I: Finite coefficients
cites work: Higher Topos Theory (AM-170)
cites work: The Galois group of a stable homotopy theory
cites work: Q3871006
cites work: Introduction to Algebraic K-Theory. (AM-72)
cites work: Submersion et descente. (Submersion and descent)
cites work: Submersions and effective descent of étale morphisms
cites work: Noetherian approximation of algebraic spaces and stacks
cites work: Totally ordered sets and the prime spectra of rings
cites work: \(p\)-adic Hodge theory for rigid-analytic varieties
cites work: Perfectoid Spaces: A survey
cites work: Excision in algebraic \(K\)-theory
cites work: Q3356420
cites work: Homology of schemes
cites work: Basically bounded functors and flat sheaves
description: scientific article