Let be a finite-dimensional Noetherian scheme, and let be a prime invertible on . Gabber proved that if is any finite type morphism, then there is some integer such that vanishes for all -torsion etale sheaves and all , cf. Corollary XVIII.1.4 in the Travaux de Gabber volume. In particular, if the -cohomological dimension is finite, then so is . It’s natural to ask for a quantitative form of this implication.
Claim. One has the bound .
Here is the supremum of the fibral dimensions of , and is defined to be one less than the minimal number of separated open subschemes required to cover . In particular, iff is separated.
Can this be improved?
Anyway, here’s a cute consequence:
Corollary. Let be a finite-dimensional Notherian scheme, and let be a prime invertible on such that . Then all affine schemes have uniformly bounded -cohomological dimension. In particular, the hypotheses of section 6.4 of Bhatt-Scholze hold, so the arguments therein show that is compactly generated and that its compact objects are exactly the objects of .