  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.
