# Constructible étale sheaves on X are étale algebraic spaces over X

I saw the following statement in a paper of Bhatt-Mathew:

Let $$X$$ be a quasicompact quasiseparated scheme. Then there is an equivalence of categories between constructible étale sheaves (of sets) on $$X$$ and algebraic spaces $$\mathcal{Y}\to X$$ over $$X$$ whose structure map is étale.

I'm familiar with the proof that lcc étale sheaves of sets $$F$$ are equivalent to finite étale schemes $$Y\to X$$, which works by showing that constant finite sheaves of sets on $$X$$ correspond to trivial finite étale surjections over $$X$$ then using a descent argument (namely, if $$F$$ is the constant étale sheaf with value $$S$$ on $$X$$, it is representable by the étale $$X$$-scheme $$X\times S$$. Then the locally constant constructible case follows by unwinding the sheaf data to glue the $$U_i \times S_i$$ where the $$U_i$$ form a cover of $$X$$ such that the restriction of $$F$$ to $$U_i$$ is the constant sheaf valued in the set $$S_i$$.

For the case of an algebraic space $$\mathcal{Y}\to X$$ étale over $$X$$, I don't understand why there is a stratification $$\{X_i\}$$ of $$X$$ such that the pullback $$Y_i:=\mathcal{Y}\times_X X_i \to X_i$$ to each stratum is finite étale, and conversely, given a sheaf that is constructible with respect to a particular stratification $$\{X_i\}$$ of $$X$$, I don't see why this helps us build an algebraic space from the corresponding family of finite étale schemes $$Y_i \to X_i$$.

• Locally constant constructible sheaves are not finite étale over X. They are just étale schemes of finite presentation. What is true is that any algebraic space has a finite stratification by locally closed subspaces which are genuine schemes. Such stratifications induce stratifications on the base scheme by étaleness. Conversely, for any locally constant constructible sheaf, there is a finite stratification of the base scheme such that the sheaf becomes locally constant over each stratum. Jan 25, 2020 at 9:55
• @Denis-CharlesCisinski Thanks! I wonder why finite étale things show up then. Is this just because étale-locally, every finite presentation étale map is finite étale (since étale maps are locally quasifinite)? Jan 25, 2020 at 10:32
• If the base scheme is of dimension zero, then all étale coverings of finite presentation are finite. This implies that any étale map of finite presentation is finite over a dense open subset of the base. Therefore, we may find stratifications over which your étale maps becomes finite. Jan 25, 2020 at 10:43
• @Denis-CharlesCisinski Sorry, one last question: Once you find a stratification of the base scheme in the converse such that the sheaf becomes locally constant over each stratum, this gives an étale scheme over each stratum. How do you show that these paste back together to an algebraic space? Jan 25, 2020 at 10:54
• To complete the argument: SGA1 shows descent for the v-topology of Bhatt and Mathew (descent along universal epimophisms). Both étale algebraic spaces and sheaves on the small étale sites form sheaves of categories (stacks) for the v-topplogy. Moreover both stacks take filtered limits of schemes with affine transition maps to filtered colimits of categories. To establish the equivalence, we may assume that the base scheme if affine with separably closed residue fields. My previous comments are then a proof that the equivalence holds. Jan 26, 2020 at 11:49

Proposition 2.7 Let $$X$$ be a quasicompact and quasiseparated scheme, and let $$F$$ be a sheaf of sets. For $$F$$ to be constructible, it is necessary and sufficient that it be isomorphic to the coequalizer of a pair of morphisms $$H\rightrightarrows G$$, where $$H$$ and $$G$$ are sheaves of sets representable by by étale schemes of finite presentation over $$X$$.
• Ok, I see now! So e.g. for a closed immersion $i\colon Y\to X$ and $F$ the constant sheaf of two-element sets on $Y$ (representable by two copies of $Y$), $i_* F$ is representable by two copies of $X$ with two copies of $X\setminus Y$ identified... Jan 25, 2020 at 16:18