37.38 Slicing smooth morphisms
In this section we explain a result that roughly states that smooth coverings of a scheme S can be refined by étale coverings. The technique to prove this relies on a slicing argument.
Lemma 37.38.1. Let f : X \to S be a morphism of schemes. Let x \in X be a point with image s \in S. Let h \in \mathfrak m_ x \subset \mathcal{O}_{X, x}. Assume
f is smooth at x, and
the image \text{d}\overline{h} of \text{d}h in
\Omega _{X_ s/s, x} \otimes _{\mathcal{O}_{X_ s, x}} \kappa (x) = \Omega _{X/S, x} \otimes _{\mathcal{O}_{X, x}} \kappa (x)
is nonzero.
Then there exists an affine open neighbourhood U \subset X of x such that h comes from h \in \Gamma (U, \mathcal{O}_ U) and such that D = V(h) is an effective Cartier divisor in U with x \in D and D \to S smooth.
Proof.
As f is smooth at x we may assume, after replacing X by an open neighbourhood of x that f is smooth. In particular we see that f is flat and locally of finite presentation. By Lemma 37.23.1 we already know there exists an open neighbourhood U \subset X of x such that h comes from h \in \Gamma (U, \mathcal{O}_ U) and such that D = V(h) is an effective Cartier divisor in U with x \in D and D \to S flat and of finite presentation. By Morphisms, Lemma 29.32.15 we have a short exact sequence
\mathcal{C}_{D/U} \to i^*\Omega _{U/S} \to \Omega _{D/S} \to 0
where i : D \to U is the closed immersion and \mathcal{C}_{D/U} is the conormal sheaf of D in U. As D is an effective Cartier divisor cut out by h \in \Gamma (U, \mathcal{O}_ U) we see that \mathcal{C}_{D/U} = h \cdot \mathcal{O}_ S. Since U \to S is smooth the sheaf \Omega _{U/S} is finite locally free, hence its pullback i^*\Omega _{U/S} is finite locally free also. The first arrow of the sequence maps the free generator h to the section \text{d}h|_ D of i^*\Omega _{U/S} which has nonzero value in the fibre \Omega _{U/S, x} \otimes \kappa (x) by assumption. By right exactness of \otimes \kappa (x) we conclude that
\dim _{\kappa (x)} \left( \Omega _{D/S, x} \otimes \kappa (x) \right) = \dim _{\kappa (x)} \left( \Omega _{U/S, x} \otimes \kappa (x) \right) - 1.
By Morphisms, Lemma 29.34.14 we see that \Omega _{U/S, x} \otimes \kappa (x) can be generated by at most \dim _ x(U_ s) elements. By the displayed formula we see that \Omega _{D/S, x} \otimes \kappa (x) can be generated by at most \dim _ x(U_ s) - 1 elements. Note that \dim _ x(D_ s) = \dim _ x(U_ s) - 1 for example because \dim (\mathcal{O}_{D_ s, x}) = \dim (\mathcal{O}_{U_ s, x}) - 1 by Algebra, Lemma 10.60.13 (also D_ s \subset U_ s is effective Cartier, see Divisors, Lemma 31.18.1) and then using Morphisms, Lemma 29.28.1. Thus we conclude that \Omega _{D/S, x} \otimes \kappa (x) can be generated by at most \dim _ x(D_ s) elements and we conclude that D \to S is smooth at x by Morphisms, Lemma 29.34.14 again. After shrinking U we get that D \to S is smooth and we win.
\square
Lemma 37.38.2. Let f : X \to S be a morphism of schemes. Let x \in X be a point with image s \in S. Assume
f is smooth at x, and
the map
\Omega _{X_ s/s, x} \otimes _{\mathcal{O}_{X_ s, x}} \kappa (x) \longrightarrow \Omega _{\kappa (x)/\kappa (s)}
has a nonzero kernel.
Then there exists an affine open neighbourhood U \subset X of x and an effective Cartier divisor D \subset U containing x such that D \to S is smooth.
Proof.
Write k = \kappa (s) and R = \mathcal{O}_{X_ s, x}. Denote \mathfrak m the maximal ideal of R and \kappa = R/\mathfrak m so that \kappa = \kappa (x). As formation of modules of differentials commutes with localization (see Algebra, Lemma 10.131.8) we have \Omega _{X_ s/s, x} = \Omega _{R/k}. By Algebra, Lemma 10.131.9 there is an exact sequence
\mathfrak m/\mathfrak m^2 \xrightarrow {\text{d}} \Omega _{R/k} \otimes _ R \kappa \to \Omega _{\kappa /k} \to 0.
Hence if (2) holds, there exists an element \overline{h} \in \mathfrak m such that \text{d}\overline{h} is nonzero. Choose a lift h \in \mathcal{O}_{X, x} of \overline{h} and apply Lemma 37.38.1.
\square
If you assume the residue field extension is separable then the phenomenon of Remark 37.38.3 does not happen. Here is the precise result.
Lemma 37.38.4. Let f : X \to S be a morphism of schemes. Let x \in X be a point with image s \in S. Assume
f is smooth at x,
the residue field extension \kappa (x)/\kappa (s) is separable, and
x is not a generic point of X_ s.
Then there exists an affine open neighbourhood U \subset X of x and an effective Cartier divisor D \subset U containing x such that D \to S is smooth.
Proof.
Write k = \kappa (s) and R = \mathcal{O}_{X_ s, x}. Denote \mathfrak m the maximal ideal of R and \kappa = R/\mathfrak m so that \kappa = \kappa (x). As formation of modules of differentials commutes with localization (see Algebra, Lemma 10.131.8) we have \Omega _{X_ s/s, x} = \Omega _{R/k}. By assumption (2) and Algebra, Lemma 10.140.4 the map
\text{d} : \mathfrak m/\mathfrak m^2 \longrightarrow \Omega _{R/k} \otimes _ R \kappa (\mathfrak m)
is injective. Assumption (3) implies that \mathfrak m/\mathfrak m^2 \not= 0. Thus there exists an element \overline{h} \in \mathfrak m such that \text{d}\overline{h} is nonzero. Choose a lift h \in \mathcal{O}_{X, x} of \overline{h} and apply Lemma 37.38.1.
\square
The subscheme Z constructed in the following lemma is really a complete intersection in an affine open neighbourhood of x. If we ever need this we will explicitly formulate a separate lemma stating this fact.
Lemma 37.38.5. Let f : X \to S be a morphism of schemes. Let x \in X be a point with image s \in S. Assume
f is smooth at x, and
x is a closed point of X_ s and \kappa (s) \subset \kappa (x) is separable.
Then there exists an immersion Z \to X containing x such that
Z \to S is étale, and
Z_ s = \{ x\} set theoretically.
Proof.
We may and do replace S by an affine open neighbourhood of s. We may and do replace X by an affine open neighbourhood of x such that X \to S is smooth. We will prove the lemma for smooth morphisms of affines by induction on d = \dim _ x(X_ s).
The case d = 0. In this case we show that we may take Z to be an open neighbourhood of x. Namely, if d = 0, then X \to S is quasi-finite at x, see Morphisms, Lemma 29.29.5. Hence there exists an affine open neighbourhood U \subset X such that U \to S is quasi-finite, see Morphisms, Lemma 29.56.2. Thus after replacing X by U we see that X is quasi-finite and smooth over S, hence smooth of relative dimension 0 over S, hence étale over S. Moreover, the fibre X_ s is a finite discrete set. Hence after replacing X by a further affine open neighbourhood of X we see that f^{-1}(\{ s\} ) = \{ x\} (because the topology on X_ s is induced from the topology on X, see Schemes, Lemma 26.18.5). This proves the lemma in this case.
Next, assume d > 0. Note that because x is a closed point of its fibre the extension \kappa (x)/\kappa (s) is finite (by the Hilbert Nullstellensatz, see Morphisms, Lemma 29.20.3). Thus we see \Omega _{\kappa (x)/\kappa (s)} = 0 as this holds for algebraic separable field extensions. Thus we may apply Lemma 37.38.2 to find a diagram
\xymatrix{ D \ar[r] \ar[rrd] & U \ar[r] \ar[rd] & X \ar[d] \\ & & S }
with x \in D. Note that \dim _ x(D_ s) = \dim _ x(X_ s) - 1 for example because \dim (\mathcal{O}_{D_ s, x}) = \dim (\mathcal{O}_{X_ s, x}) - 1 by Algebra, Lemma 10.60.13 (also D_ s \subset X_ s is effective Cartier, see Divisors, Lemma 31.18.1) and then using Morphisms, Lemma 29.28.1. Thus the morphism D \to S is smooth with \dim _ x(D_ s) = \dim _ x(X_ s) - 1 = d - 1. By induction hypothesis we can find an immersion Z \to D as desired, which finishes the proof.
\square
Lemma 37.38.6. Let f : X \to S be a smooth morphism of schemes. Let s \in S be a point in the image of f. Then there exists an étale neighbourhood (S', s') \to (S, s) and a S-morphism S' \to X.
First proof of Lemma 37.38.6.
By assumption X_ s \not= \emptyset . By Varieties, Lemma 33.25.6 there exists a closed point x \in X_ s such that \kappa (x) is a finite separable field extension of \kappa (s). Hence by Lemma 37.38.5 there exists an immersion Z \to X such that Z \to S is étale and such that x \in Z. Take (S' , s') = (Z, x).
\square
Second proof of Lemma 37.38.6.
Pick a point x \in X with f(x) = s. Choose a diagram
\xymatrix{ X \ar[d] & U \ar[l] \ar[d] \ar[r]_-\pi & \mathbf{A}^ d_ V \ar[ld] \\ S & V \ar[l] }
with \pi étale, x \in U and V = \mathop{\mathrm{Spec}}(R) affine, see Morphisms, Lemma 29.36.20. In particular s \in V. The morphism \pi : U \to \mathbf{A}^ d_ V is open, see Morphisms, Lemma 29.36.13. Thus W = \pi (U) \cap \mathbf{A}^ d_ s is a nonempty open subset of \mathbf{A}^ d_ s. Let w \in W be a point with \kappa (s) \subset \kappa (w) finite separable, see Varieties, Lemma 33.25.5. By Algebra, Lemma 10.114.1 there exist d elements \overline{f}_1, \ldots , \overline{f}_ d \in \kappa (s)[x_1, \ldots , x_ d] which generate the maximal ideal corresponding to w in \kappa (s)[x_1, \ldots , x_ d]. After replacing R by a principal localization we may assume there are f_1, \ldots , f_ d \in R[x_1, \ldots , x_ d] which map to \overline{f}_1, \ldots , \overline{f}_ d \in \kappa (s)[x_1, \ldots , x_ d]. Consider the R-algebra
R' = R[x_1, \ldots , x_ d]/(f_1, \ldots , f_ d)
and set S' = \mathop{\mathrm{Spec}}(R'). By construction we have a closed immersion j : S' \to \mathbf{A}^ d_ V over V. By construction the fibre of S' \to V over s is a single point s' whose residue field is finite separable over \kappa (s). Let \mathfrak q' \subset R' be the corresponding prime. By Algebra, Lemma 10.136.10 we see that (R')_ g is a relative global complete intersection over R for some g \in R', g \not\in \mathfrak q. Thus S' \to V is flat and of finite presentation in a neighbourhood of s', see Algebra, Lemma 10.136.13. By construction the scheme theoretic fibre of S' \to V over s is \mathop{\mathrm{Spec}}(\kappa (s')). Hence it follows from Morphisms, Lemma 29.36.15 that S' \to S is étale at s'. Set
S'' = U \times _{\pi , \mathbf{A}^ d_ V, j} S'.
By construction there exists a point s'' \in S'' which maps to s' via the projection p : S'' \to S'. Note that p is étale as the base change of the étale morphism \pi , see Morphisms, Lemma 29.36.4. Choose a small affine neighbourhood S''' \subset S'' of s'' which maps into the nonempty open neighbourhood of s' \in S' where the morphism S' \to S is étale. Then the étale neighbourhood (S''', s'') \to (S, s) is a solution to the problem posed by the lemma.
\square
The following lemma shows that sheaves for the smooth topology are the same thing as sheaves for the étale topology.
Lemma 37.38.7. Let S be a scheme. Let \mathcal{U} = \{ S_ i \to S\} _{i \in I} be a smooth covering of S, see Topologies, Definition 34.5.1. Then there exists an étale covering \mathcal{V} = \{ T_ j \to S\} _{j \in J} (see Topologies, Definition 34.4.1) which refines (see Sites, Definition 7.8.1) \mathcal{U}.
Proof.
For every s \in S there exists an i \in I such that s is in the image of S_ i \to S. By Lemma 37.38.6 we can find an étale morphism g_ s : T_ s \to S such that s \in g_ s(T_ s) and such that g_ s factors through S_ i \to S. Hence \{ T_ s \to S\} is an étale covering of S that refines \mathcal{U}.
\square
Lemma 37.38.8. Let f : X \to S be a smooth morphism of schemes. Then there exists an étale covering \{ U_ i \to X\} _{i \in I} such that U_ i \to S factors as U_ i \to V_ i \to S where V_ i \to S is étale and U_ i \to V_ i is a smooth morphism of affine schemes, which has a section, and has geometrically connected fibres.
Proof.
Let s \in S. By Varieties, Lemma 33.25.6 the set of closed points x \in X_ s such that \kappa (x)/\kappa (s) is separable is dense in X_ s. Thus it suffices to construct an étale morphism U \to X with x in the image such that U \to S factors in the manner described in the lemma. To do this, choose an immersion Z \to X passing through x such that Z \to S is étale (Lemma 37.38.5). After replacing S by Z and X by Z \times _ S X we see that we may assume X \to S has a section \sigma : S \to X with \sigma (s) = x. Then we can first replace S by an affine open neighbourhood of s and next replace X by an affine open neighbourhood of x. Then finally, we consider the subset X^0 \subset X of Section 37.29. By Lemmas 37.29.6 and 37.29.4 this is a retrocompact open subscheme containing \sigma such that the fibres X^0 \to S are geometrically connected. If X^0 is not affine, then we choose an affine open U \subset X^0 containing x. Since X^0 \to S is smooth, the image of U is open. Choose an affine open neighbourhood V \subset S of s contained in \sigma ^{-1}(U) and in the image of U \to S. Finally, the reader sees that U \cap f^{-1}(V) \to V has all the desired properties. For example U \cap f^{-1}(V) is equal to U \times _ S V is affine as a fibre product of affine schemes. Also, the geometric fibres of U \cap f^{-1}(V) \to V are nonempty open subschemes of the irreducible fibres of X^0 \to S and hence connected. Some details omitted.
\square
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