A stupid remark on cohomological dimensions

Let Y be a finite-dimensional Noetherian scheme, and let \ell be a prime invertible on Y. Gabber proved that if f:X \to Y is any finite type morphism, then there is some integer N such that R^n f_{\ast} F vanishes for all \ell-torsion etale sheaves F and all n > N, cf. Corollary XVIII.1.4 in the Travaux de Gabber volume. In particular, if the \ell-cohomological dimension \mathrm{cd}_{\ell}(Y) is finite, then so is \mathrm{cd}_{\ell}(X). It’s natural to ask for a quantitative form of this implication.

Claim. One has the bound \mathrm{cd}_{\ell}(X) \leq \mathrm{cd}_{\ell}(Y)+2\mathrm{dim}f+2\mathrm{dim}(X)+s(X).

Here \mathrm{dim}f is the supremum of the fibral dimensions of f, and s(X) \in \mathbf{Z}_{\geq 0} is defined to be one less than the minimal number of separated open subschemes required to cover X. In particular, s(X)=0 iff X is separated.

Can this be improved?

Anyway, here’s a cute consequence:

Corollary. Let Y be a finite-dimensional Notherian scheme, and let \ell be a prime invertible on Y such that \mathrm{cd}_{\ell}(Y) < \infty. Then all affine schemes X \in Y_{\mathrm{et}} have uniformly bounded \ell-cohomological dimension. In particular, the hypotheses of section 6.4 of Bhatt-Scholze hold, so the arguments therein show that D(Y,\mathbf{F}_{\ell}) is compactly generated and that its compact objects are exactly the objects of D^{b}_{c}(Y,\mathbf{F}_{\ell}).

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