## 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 $D^{b}_{c}(Y,\mathbf{F}_{\ell})$.