Lemma 97.9.1 (05XQ)—The Stacks project

Lemma 97.9.1. Let $Z \to U$ be a finite morphism of schemes. Let $W$ be an algebraic space and let $W \to Z$ be a surjective étale morphism. Then there exists a surjective étale morphism $U' \to U$ and a section

\[ \sigma : Z_{U'} \to W_{U'} \]

of the morphism $W_{U'} \to Z_{U'}$.

Proof. We may choose a separated scheme $W'$ and a surjective étale morphism $W' \to W$. Hence after replacing $W$ by $W'$ we may assume that $W$ is a separated scheme. Write $f : W \to Z$ and $\pi : Z \to U$. Note that $f \circ \pi : W \to U$ is separated as $W$ is separated (see Schemes, Lemma 26.21.13). Let $u \in U$ be a point. Clearly it suffices to find an étale neighbourhood $(U', u')$ of $(U, u)$ such that a section $\sigma $ exists over $U'$. Let $z_1, \ldots , z_ r$ be the points of $Z$ lying above $u$. For each $i$ choose a point $w_ i \in W$ which maps to $z_ i$. We may pick an étale neighbourhood $(U', u') \to (U, u)$ such that the conclusions of More on Morphisms, Lemma 37.41.5 hold for both $Z \to U$ and the points $z_1, \ldots , z_ r$ and $W \to U$ and the points $w_1, \ldots , w_ r$. Hence, after replacing $(U, u)$ by $(U', u')$ and relabeling, we may assume that all the field extensions $\kappa (z_ i)/\kappa (u)$ and $\kappa (w_ i)/\kappa (u)$ are purely inseparable, and moreover that there exist disjoint union decompositions

\[ Z = V_1 \amalg \ldots \amalg V_ r \amalg A, \quad W = W_1 \amalg \ldots \amalg W_ r \amalg B \]

by open and closed subschemes with $z_ i \in V_ i$, $w_ i \in W_ i$ and $V_ i \to U$, $W_ i \to U$ finite. After replacing $U$ by $U \setminus \pi (A)$ we may assume that $A = \emptyset $, i.e., $Z = V_1 \amalg \ldots \amalg V_ r$. After replacing $W_ i$ by $W_ i \cap f^{-1}(V_ i)$ and $B$ by $B \cup \bigcup W_ i \cap f^{-1}(Z \setminus V_ i)$ we may assume that $f$ maps $W_ i$ into $V_ i$. Then $f_ i = f|_{W_ i} : W_ i \to V_ i$ is a morphism of schemes finite over $U$, hence finite (see Morphisms, Lemma 29.44.14). It is also étale (by assumption), $f_ i^{-1}(\{ z_ i\} ) = w_ i$, and induces an isomorphism of residue fields $\kappa (z_ i) = \kappa (w_ i)$ (because both are purely inseparable extensions of $\kappa (u)$ and $\kappa (w_ i)/\kappa (z_ i)$ is separable as $f$ is étale). Hence by Étale Morphisms, Lemma 41.14.2 we see that $f_ i$ is an isomorphism in a neighbourhood $V_ i'$ of $z_ i$. Since $\pi : Z \to U$ is closed, after shrinking $U$, we may assume that $W_ i \to V_ i$ is an isomorphism. This proves the lemma. $\square$

Comments (9)

Comment #4923 by Robot0079 on February 10, 2020 at 03:19

Here is a conceptual proof.

We call an etale sheaf $F$ over scheme $S$ surjective, if the structure map $F \to S$ is surjective. Here we identify etale sheaves with etale algebraic spaces over $S$ . Note that this is equivalent to require stalks of $F$ is nonempty at every (geometric) point. Another equivalent condition is $F/S$ has sections locally.

Then our lemma says that direct image of finite morphism preserve surjectivity.

Now that we have formula for stalk of finite direct image functor, this is obvious.

Comment #5191 by Johan on June 10, 2020 at 12:45

@#4932: No, I don't think this argument works. The problem is to find a section of $W \to Z$ \'etale locally on $U$ . Your argument tells us that \'etale locally on $Z$ we can do this. Or maybe I misunderstood what you were saying?

Comment #6308 by Robot0079 on July 05, 2021 at 00:45

@#5191: A etale morphism locally has sections is equivalent to surjectivity, thus is also equivalent to having a global section after base changing along a surjective etale map.

Let $f$ be $Z \to W$ . The proposition amounts to say that $u: f_*W \to Z$ has a U'-global section. As I said, u is surjective (etale). So we can just choose U' to be $f_*W$ (or an atalas of it, if you prefer scheme), since diagonal is always a section.

Comment #6309 by Robot0079 on July 05, 2021 at 00:52

Sorry, I mistyped some symbols. An etale morphism locally having sections is equivalent to surjectivity, thus is also equivalent to having a global section after base change along a surjective etale map.

Let $f$ be $Z \to U$ . The proposition amounts to say that $u: f_*W \to U$ has a U'-global section. As I said, u is surjective (etale). So we can just choose U' to be $f_*W$ (or an atlas of it, if you prefer a scheme), since diagonal is always a section.

Comment #6310 by Laurent Moret-Bailly on July 05, 2021 at 04:23

@#6309: this works if we know that $f_*W$ is an algebraic space, but this is proved only in the next proposition, right?

Comment #6311 by Laurent Moret-Bailly on July 05, 2021 at 04:24

Perhaps a simpler approach is to reduce (by a limit argument) to the case where $U$ is local and strictly henselian: then $Z$ is a sum of strictly henselian local schemes, so clearly $W\to Z$ has a section.

Comment #6312 by Robot0079 on July 05, 2021 at 17:59

@#6310: No, the direct image here is defined by first regarding W as etale sheaf over Z, then applying the direct image functor for etale sheaves and identifying this etale sheaf over U as the desired etale algebraic space $f_*W$ . Here we use the equivalence between etale algebraic spaces and etale sheaves, which can be proved by direct verification.

So from this point of view, the next proposition is just the proper base change theorem for (non abelian) etale sheaves. And using Lemma 59.91.5 we can just assume $Z \to U$ to be finite.

And yes, I think what you says is exactly unwrapping this procedure, so that works as well.

Comment #6314 by Johan on July 05, 2021 at 18:18

@Robot0079: OK, the thing with the diagonal, thinking of $f_*W$ as an algebraic space (via Lemma 66.27.3), and choosing an atlas works. Also, the reduction to strictly henselian local rings works too (with some additional effort). For me the argument as given is fine as well. The Stacks project has many, many arguments using etale localization of quasi-finite morphisms -- kind of like the $\epsilon$ - $\delta$ arguments in analysis.

The next time I go through all the comments I might add one or both of your arguments, but please feel free to code your arguments with details in latex and send it to me.

Comment #6420 by Johan on July 19, 2021 at 12:25

OK, I am going to leave this as is. Others can contribute alternative proofs if they so desire and this is welcomed.

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