A better comparison of cdh- and ldh-cohomologies

TL;DR

This paper introduces pseudo-normalisation concepts to compare cdh- and ldh-cohomologies, providing a shorter proof of their equivalence for certain motivic sheaves, emphasizing invariance under universal homeomorphisms.

Contribution

It presents a novel approach using pseudo-normalisation to directly compare cdh- and ldh-cohomologies, simplifying previous proofs and highlighting invariance under universal homeomorphisms.

Findings

Proves $H^n_{cdh}(X, F) = H^n_{ldh}(X, F)$ for specific motivic sheaves.

Introduces pseudo-normalisation for non-Nagata rings.

Shows invariance of motivic cohomology under universal homeomorphisms.

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 and pseudo-normalisation. We use this notion to give a much more direct and shorter proof that $H^n_{cdh}(X, F) = H^n_{ldh}(X, F)$ 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 authors volume Ast\'erisque 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 $\mathbb{Z}[1/p]$-coefficients) is invariant not just for nilpotent thickenings, but for all universal homeomorphisms.