The Stacks project

37.41 Étale localization of quasi-finite morphisms

Now we come to a series of lemmas around the theme “quasi-finite morphisms become finite after étale localization”. The general idea is the following. Suppose given a morphism of schemes $f : X \to S$ and a point $s \in S$. Let $\varphi : (U, u) \to (S, s)$ be an étale neighbourhood of $s$ in $S$. Consider the fibre product $X_ U = U \times _ S X$ and the basic diagram

37.41.0.1
\begin{equation} \label{more-morphisms-equation-basic-diagram} \vcenter { \xymatrix{ V \ar[r] \ar[dr] & X_ U \ar[d] \ar[r] & X \ar[d]^ f \\ & U \ar[r]^\varphi & S } } \end{equation}

where $V \subset X_ U$ is open. Is there some standard model for the morphism $f_ U : X_ U \to U$, or for the morphism $V \to U$ for suitable opens $V$? Of course the answer is no in general. But for quasi-finite morphisms we can say something.

Lemma 37.41.1. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. Set $s = f(x)$. Assume that

  1. $f$ is locally of finite type, and

  2. $x \in X_ s$ is isolated1.

Then there exist

  1. an elementary étale neighbourhood $(U, u) \to (S, s)$,

  2. an open subscheme $V \subset X_ U$ (see 37.41.0.1)

such that

  1. $V \to U$ is a finite morphism,

  2. there is a unique point $v$ of $V$ mapping to $u$ in $U$, and

  3. the point $v$ maps to $x$ under the morphism $X_ U \to X$, inducing $\kappa (x) = \kappa (v)$.

Moreover, for any elementary étale neighbourhood $(U', u') \to (U, u)$ setting $V' = U' \times _ U V \subset X_{U'}$ the triple $(U', u', V')$ satisfies the properties (i), (ii), and (iii) as well.

Proof. Let $Y \subset X$, $W \subset S$ be affine opens such that $f(Y) \subset W$ and such that $x \in Y$. Note that $x$ is also an isolated point of the fibre of the morphism $f|_ Y : Y \to W$. If we can prove the theorem for $f|_ Y : Y \to W$, then the theorem follows for $f$. Hence we reduce to the case where $f$ is a morphism of affine schemes. This case is Algebra, Lemma 10.145.2. $\square$

In the preceding and following lemma we do not assume that the morphism $f$ is separated. This means that the opens $V$, $V_ i$ created in them are not necessarily closed in $X_ U$. Moreover, if we choose the neighbourhood $U$ to be affine, then each $V_ i$ is affine, but the intersections $V_ i \cap V_ j$ need not be affine (in the nonseparated case).

Lemma 37.41.2. Let $f : X \to S$ be a morphism of schemes. Let $x_1, \ldots , x_ n \in X$ be points having the same image $s$ in $S$. Assume that

  1. $f$ is locally of finite type, and

  2. $x_ i \in X_ s$ is isolated for $i = 1, \ldots , n$.

Then there exist

  1. an elementary étale neighbourhood $(U, u) \to (S, s)$,

  2. for each $i$ an open subscheme $V_ i \subset X_ U$,

such that for each $i$ we have

  1. $V_ i \to U$ is a finite morphism,

  2. there is a unique point $v_ i$ of $V_ i$ mapping to $u$ in $U$, and

  3. the point $v_ i$ maps to $x_ i$ in $X$ and $\kappa (x_ i) = \kappa (v_ i)$.

Proof. We will use induction on $n$. Namely, suppose $(U, u) \to (S, s)$ and $V_ i \subset X_ U$, $i = 1, \ldots , n - 1$ work for $x_1, \ldots , x_{n - 1}$. Since $\kappa (s) = \kappa (u)$ the fibre $(X_ U)_ u = X_ s$. Hence there exists a unique point $x'_ n \in X_ u \subset X_ U$ corresponding to $x_ n \in X_ s$. Also $x'_ n$ is isolated in $X_ u$. Hence by Lemma 37.41.1 there exists an elementary étale neighbourhood $(U', u') \to (U, u)$ and an open $V_ n \subset X_{U'}$ which works for $x'_ n$ and hence for $x_ n$. By the final assertion of Lemma 37.41.1 the open subschemes $V'_ i = U'\times _ U V_ i$ for $i = 1, \ldots , n - 1$ still work with respect to $x_1, \ldots , x_{n - 1}$. Hence we win. $\square$

If we allow a nontrivial field extension $\kappa (u)/\kappa (s)$, i.e., general étale neighbourhoods, then we can split the points as follows.

Lemma 37.41.3. Let $f : X \to S$ be a morphism of schemes. Let $x_1, \ldots , x_ n \in X$ be points having the same image $s$ in $S$. Assume that

  1. $f$ is locally of finite type, and

  2. $x_ i \in X_ s$ is isolated for $i = 1, \ldots , n$.

Then there exist

  1. an étale neighbourhood $(U, u) \to (S, s)$,

  2. for each $i$ an integer $m_ i$ and open subschemes $V_{i, j} \subset X_ U$, $j = 1, \ldots , m_ i$

such that we have

  1. each $V_{i, j} \to U$ is a finite morphism,

  2. there is a unique point $v_{i, j}$ of $V_{i, j}$ mapping to $u$ in $U$ with $\kappa (u) \subset \kappa (v_{i, j})$ finite purely inseparable,

  3. if $v_{i, j} = v_{i', j'}$, then $i = i'$ and $j = j'$, and

  4. the points $v_{i, j}$ map to $x_ i$ in $X$ and no other points of $(X_ U)_ u$ map to $x_ i$.

Proof. This proof is a variant of the proof of Algebra, Lemma 10.145.4 in the language of schemes. By Morphisms, Lemma 29.20.6 the morphism $f$ is quasi-finite at each of the points $x_ i$. Hence $\kappa (s) \subset \kappa (x_ i)$ is finite for each $i$ (Morphisms, Lemma 29.20.5). For each $i$, let $\kappa (s) \subset L_ i \subset \kappa (x_ i)$ be the subfield such that $L_ i/\kappa (s)$ is separable, and $\kappa (x_ i)/L_ i$ is purely inseparable. Choose a finite Galois extension $L/\kappa (s)$ such that there exist $\kappa (s)$-embeddings $L_ i \to L$ for $i = 1, \ldots , n$. Choose an étale neighbourhood $(U, u) \to (S, s)$ such that $L \cong \kappa (u)$ as $\kappa (s)$-extensions (Lemma 37.35.2).

Let $y_{i, j}$, $j = 1, \ldots , m_ i$ be the points of $X_ U$ lying over $x_ i \in X$ and $u \in U$. By Schemes, Lemma 26.17.5 these points $y_{i, j}$ correspond exactly to the primes in the rings $\kappa (u) \otimes _{\kappa (s)} \kappa (x_ i)$. This also explains why there are finitely many; in fact $m_ i = [L_ i : \kappa (s)]$ but we do not need this. By our choice of $L$ (and elementary field theory) we see that $\kappa (u) \subset \kappa (y_{i, j})$ is finite purely inseparable for each pair $i, j$. Also, by Morphisms, Lemma 29.20.13 for example, the morphism $X_ U \to U$ is quasi-finite at the points $y_{i, j}$ for all $i, j$.

Apply Lemma 37.41.2 to the morphism $X_ U \to U$, the point $u \in U$ and the points $y_{i, j} \in (X_ U)_ u$. This gives an étale neighbourhood $(U', u') \to (U, u)$ with $\kappa (u) = \kappa (u')$ and opens $V_{i, j} \subset X_{U'}$ with the properties (i), (ii), and (iii) of that lemma. We claim that the étale neighbourhood $(U', u') \to (S, s)$ and the opens $V_{i, j} \subset X_{U'}$ are a solution to the problem posed by the lemma. We omit the verifications. $\square$

Lemma 37.41.4. Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Let $x_1, \ldots , x_ n \in X_ s$. Assume that

  1. $f$ is locally of finite type,

  2. $f$ is separated, and

  3. $x_1, \ldots , x_ n$ are pairwise distinct isolated points of $X_ s$.

Then there exists an elementary étale neighbourhood $(U, u) \to (S, s)$ and a decomposition

\[ U \times _ S X = W \amalg V_1 \amalg \ldots \amalg V_ n \]

into open and closed subschemes such that the morphisms $V_ i \to U$ are finite, the fibres of $V_ i \to U$ over $u$ are singletons $\{ v_ i\} $, each $v_ i$ maps to $x_ i$ with $\kappa (x_ i) = \kappa (v_ i)$, and the fibre of $W \to U$ over $u$ contains no points mapping to any of the $x_ i$.

Proof. Choose $(U, u) \to (S, s)$ and $V_ i \subset X_ U$ as in Lemma 37.41.2. Since $X_ U \to U$ is separated (Schemes, Lemma 26.21.12) and $V_ i \to U$ is finite hence proper (Morphisms, Lemma 29.44.11) we see that $V_ i \subset X_ U$ is closed by Morphisms, Lemma 29.41.7. Hence $V_ i \cap V_ j$ is a closed subset of $V_ i$ which does not contain $v_ i$. Hence the image of $V_ i \cap V_ j$ in $U$ is a closed set (because $V_ i \to U$ proper) not containing $u$. After shrinking $U$ we may therefore assume that $V_ i \cap V_ j = \emptyset $ for all $i, j$. This gives the decomposition as in the lemma. $\square$

Here is the variant where we reduce to purely inseparable field extensions.

Lemma 37.41.5. Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Let $x_1, \ldots , x_ n \in X_ s$. Assume that

  1. $f$ is locally of finite type,

  2. $f$ is separated, and

  3. $x_1, \ldots , x_ n$ are pairwise distinct isolated points of $X_ s$.

Then there exists an étale neighbourhood $(U, u) \to (S, s)$ and a decomposition

\[ U \times _ S X = W \amalg \ \coprod \nolimits _{i = 1, \ldots , n} \ \coprod \nolimits _{j = 1, \ldots , m_ i} V_{i, j} \]

into open and closed subschemes such that the morphisms $V_{i, j} \to U$ are finite, the fibres of $V_{i, j} \to U$ over $u$ are singletons $\{ v_{i, j}\} $, each $v_{i, j}$ maps to $x_ i$, $\kappa (u) \subset \kappa (v_{i, j})$ is purely inseparable, and the fibre of $W \to U$ over $u$ contains no points mapping to any of the $x_ i$.

Proof. This is proved in exactly the same way as the proof of Lemma 37.41.4 except that it uses Lemma 37.41.3 instead of Lemma 37.41.2. $\square$

The following version may be a little easier to parse.

Lemma 37.41.6. Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Assume that

  1. $f$ is locally of finite type,

  2. $f$ is separated, and

  3. $X_ s$ has at most finitely many isolated points.

Then there exists an elementary étale neighbourhood $(U, u) \to (S, s)$ and a decomposition

\[ U \times _ S X = W \amalg V \]

into open and closed subschemes such that the morphism $V \to U$ is finite, and the fibre $W_ u$ of the morphism $W \to U$ contains no isolated points. In particular, if $f^{-1}(s)$ is a finite set, then $W_ u = \emptyset $.

Proof. This is clear from Lemma 37.41.4 by choosing $x_1, \ldots , x_ n$ the complete set of isolated points of $X_ s$ and setting $V = \bigcup V_ i$. $\square$

[1] In the presence of (1) this means that $f$ is quasi-finite at $x$, see Morphisms, Lemma 29.20.6.

Comments (2)

Comment #2349 by Eric Ahlqvist on

\romannumeral (several places)

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