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