A better comparison of cdh- and ldh-cohomologies

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 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 $\mathbb{Z}[1\mathrm{/}p]$-coefficients) is invariant not just for nilpotent thickenings, but for all universal homeomorphisms.