A better comparison of cdh- and ldh-cohomologies

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 $H_{cdh}^n(X,F)=H_{ldh}^n(X,F)$ 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 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 $\mathbb{Z}[1/p]$-coefficients) is invariant not just for nilpotent thickenings, but for all universal homeomorphisms.

arXiv commentMain theorem hypothesis (G1) or (G2) changed to (G1) *and* (G2). Proof remains essentially the same

References in corpus (8)

• Triangulated categories of mixed motives
• Submersions and effective descent of etale morphisms
• Perfection in motivic homotopy theory
• Relative Riemann-Zariski spaces
• Triangulated categories of motives in positive characteristic
• Vanishing of negative $K$-theory in positive characteristic
• $K$-theory of valuation rings
• Differential forms in positive characteristic II: cdh-descent via functorial Riemann-Zariski spaces

Cited by in corpus (2)

• $K$-theory of valuation rings
• Modulus sheaves with transfers