The Stacks project

Example 85.23.2. Let $S$ be one of the following types of schemes

  1. the spectrum of a finite field,

  2. the spectrum of a separably closed field,

  3. the spectrum of a strictly henselian Noetherian local ring,

  4. the spectrum of a henselian Noetherian local ring with finite residue field,

  5. add more here.

Let $\Lambda $ be a finite ring whose order is invertible on $S$. Let $\mathcal{C} \subset (\mathit{Sch}/S)_{\acute{e}tale}$ be the full subcategory consisting of schemes locally of finite type over $S$ endowed with the étale topology. Let $\mathcal{O}_\mathcal {C} = \underline{\Lambda }$ be the constant sheaf. Set $\mathcal{A}_ U = \textit{Mod}(\mathcal{O}_ U)$, in other words, we consider all étale sheaves of $\Lambda $-modules. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be the set of quasi-compact objects. For $V \in \mathcal{B}$ set

\[ d_ V = 1 + 2\dim (S) + \sup \nolimits _{v \in V}(\text{trdeg}_{\kappa (s)}(\kappa (v)) + 2 \dim \mathcal{O}_{V, v}) \]

and let $\text{Cov}_ V$ denote the étale coverings $\{ V_ i \to V\} $ with $V_ i$ quasi-compact for all $i$. Our choice of bound $d_ V$ comes from Gabber's theorem on cohomological dimension. To see that condition (3) holds with this choice, use [Exposé VIII-A, Corollary 1.2 and Lemma 2.2, Traveaux] plus elementary arguments on cohomological dimensions of fields. We add $1$ to the formula because our list contains cases where we allow $S$ to have finite residue field. We will come back to this example later (insert future reference).

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