A BETTER COMPARISON OF $\operatorname{cdh}$- AND $l\operatorname{dh}$-COHOMOLOGIES. (13th December 2019)

• Record Type: Journal Article
• Authors: KELLY, SHANE
• Is Part Of: Nagoya mathematical journal. Volume 236(2019)
• Page Start: 183
• Page End: 213
• Publication Date: 2019-12-13
• Subjects: 14A05, -- 18F10, -- 14F42; Mathematics -- Periodicals; Mathématiques -- Périodiques; Mathematics; Wiskunde; Electronic journals; Periodicals; 510.5
• DOI: 10.1017/nmj.2019.24
• Languages: English
• ISSNs: 0027-7630

Abstract

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.