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A BETTER COMPARISON OF $\operatorname{cdh}$- AND $l\operatorname{dh}$-COHOMOLOGIES

Part of: (Co)homology theory Algebraic geometry: Foundations Categories and geometry

Published online by Cambridge University Press:  13 September 2019

SHANE KELLY Open the ORCID record for SHANE KELLY [Opens in a new window]
SHANE KELLY*
Affiliation:
Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan email [email protected]
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Abstract

In order to work with non-Nagata rings which are Nagata “up-to-completely-decomposed-universal-homeomorphism,” specifically finite rank Hensel valuation rings, we introduce the notions of pseudo-integral closure, pseudo-normalization, and pseudo-Hensel valuation ring. We use this notion to give a shorter and more direct proof that $H_{\operatorname{cdh}}^{n}(X,F_{\operatorname{cdh}})=H_{l\operatorname{dh}}^{n}(X,F_{l\operatorname{dh}})$ for homotopy sheaves $F$ of modules over the $\mathbb{Z}_{(l)}$-linear motivic Eilenberg–Maclane spectrum. This comparison is an alternative to the first half of the author’s volume Astérisque 391 whose main theorem is a cdh-descent result for Voevodsky motives. The motivating new insight is really accepting that Voevodsky’s motivic cohomology (with $\mathbb{Z}[\frac{1}{p}]$-coefficients) is invariant not just for nilpotent thickenings, but for all universal homeomorphisms.

MSC classification

Primary: 14A05: Relevant commutative algebra 18F10: Grothendieck topologies 14F42: Motivic cohomology; motivic homotopy theory
Type
Article
Information
Nagoya Mathematical Journal , Volume 236: Celebrating the 60th Birthday of Shuji Saito , December 2019 , pp. 183 - 213
Copyright
© 2019 Foundation Nagoya Mathematical Journal  

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