The Stacks project

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36.34 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 36.34.1. 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

  1. $f$ is smooth at $x$, and

  2. 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 36.21.1 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 28.31.15 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 28.32.14 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 10.59.12 (also $D_ s \subset U_ s$ is effective Cartier, see Divisors, Lemma 30.18.1) and then using Morphisms, Lemma 28.27.1. 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 28.32.14 again. After shrinking $U$ we get that $D \to S$ is smooth and we win. $\square$

Lemma 36.34.2. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point with image $s \in S$. Assume

  1. $f$ is smooth at $x$, and

  2. 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 10.130.8) we have $\Omega _{X_ s/s, x} = \Omega _{R/k}$. By Algebra, Lemma 10.130.9 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 36.34.1. $\square$

Remark 36.34.3. The second condition in Lemma 36.34.2 is necessary even if $x$ is a closed point of a positive dimensional fibre. An example is the following: Let $k$ be a field of characteristic $p > 0$ which is imperfect. Let $a \in k$ be an element which is not a $p$th power. Let $\mathfrak m = (x, y^ p - a) \subset k[x, y]$. This corresponds to a closed point $w$ of $X = \mathbf{A}^2_ k$. Set $S = \mathbf{A}^1_ k$ and let $f : X \to S$ be the morphism corresponding to $k[x] \to k[x, y]$. Then there does not exist any commutative diagram

\[ \xymatrix{ S' \ar[rr]_ h \ar[rd]_ g & & X \ar[ld]^ f \\ & S } \]

with $g$ étale and $w$ in the image of $h$. This is clear as the residue field extension $\kappa (f(w)) \subset \kappa (w)$ is purely inseparable, but for any $s' \in S'$ with $g(s') = f(w)$ the extension $\kappa (f(w)) \subset \kappa (s')$ would be separable.

If you assume the residue field extension is separable then the phenomenon of Remark 36.34.3 does not happen. Here is the precise result.

Lemma 36.34.4. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point with image $s \in S$. Assume

  1. $f$ is smooth at $x$,

  2. the residue field extension $\kappa (s) \subset \kappa (x)$ is separable, and

  3. $x$ 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 10.130.8) we have $\Omega _{X_ s/s, x} = \Omega _{R/k}$. By assumption (2) and Algebra, Lemma 10.138.4 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 36.34.1. $\square$

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 36.34.5. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point with image $s \in S$. Assume

  1. $f$ is smooth at $x$, and

  2. $x$ 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

  1. $Z \to S$ is étale, and

  2. $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 28.28.5. Hence there exists an affine open neighbourhood $U \subset X$ such that $U \to S$ is quasi-finite, see Morphisms, Lemma 28.53.2. 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 25.18.5). This proves the lemma in this case.

Next, assume $d > 0$. Note that because $x$ is a closed point of its fibre the extension $\kappa (s) \subset \kappa (x)$ is finite (by the Hilbert Nullstellensatz, see Morphisms, Lemma 28.19.3). Thus we see $\Omega _{\kappa (x)/\kappa (s)} = 0$ as this holds for algebraic separable field extensions. Thus we may apply Lemma 36.34.2 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 10.59.12 (also $D_ s \subset X_ s$ is effective Cartier, see Divisors, Lemma 30.18.1) and then using Morphisms, Lemma 28.27.1. 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 36.34.6. 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 36.34.6. By assumption $X_ s \not= \emptyset $. By Varieties, Lemma 32.25.6 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 36.34.5 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 36.34.6. Pick a point $x \in X$ with $f(x) = s$. Choose a diagram

\[ \xymatrix{ X \ar[d] & U \ar[l] \ar[d] \ar[r]_-\pi & \mathbf{A}^ d_ V \ar[ld] \\ Y & V \ar[l] } \]

with $\pi $ étale, $x \in U$ and $V = \mathop{\mathrm{Spec}}(R)$ affine, see Morphisms, Lemma 28.34.20. In particular $s \in V$. The morphism $\pi : U \to \mathbf{A}^ d_ V$ is open, see Morphisms, Lemma 28.34.13. Thus $W = \pi (V) \cap \mathbf{A}^ d_ s$ 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 32.25.5. By Algebra, Lemma 10.113.1 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 10.134.11 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 10.134.14. By construction the scheme theoretic fibre of $S' \to V$ over $s$ is $\mathop{\mathrm{Spec}}(\kappa (s'))$. Hence it follows from Morphisms, Lemma 28.34.15 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 28.34.4. 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 36.34.7. Let $S$ be a scheme. Let $\mathcal{U} = \{ S_ i \to S\} _{i \in I}$ be a smooth covering of $S$, see Topologies, Definition 33.5.1. Then there exists an étale covering $\mathcal{V} = \{ T_ j \to S\} _{j \in J}$ (see Topologies, Definition 33.4.1) 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 36.34.6 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 36.34.8. 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 32.25.6 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 36.34.5). 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 36.27. By Lemmas 36.27.6 and 36.27.4 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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