The Stacks project

55.28 Purity of ramification locus

In this section we discuss the analogue of purity of branch locus for generically finite morphisms. Apparently, this result is due to Gabber. A special case is van der Waerden's purity theorem for the locus where a birational morphism from a normal variety to a smooth variety is not an isomorphism.

Lemma 55.28.1. Let $A$ be a Noetherian normal local domain of dimension $2$. Assume $A$ is Nagata, has a dualizing module $\omega _ A$, and has a resolution of singularities $f : X \to \mathop{\mathrm{Spec}}(A)$. Let $\omega _ X$ be as in Resolution of Surfaces, Remark 53.7.7. If $\omega _ X \cong \mathcal{O}_ X(E)$ for some effective Cartier divisor $E \subset X$ supported on the exceptional fibre, then $A$ defines a rational singularity. If $f$ is a minimal resolution, then $E = 0$.

Proof. There is a trace map $Rf_*\omega _ X \to \omega _ A$, see Duality for Schemes, Section 48.7. By Grauert-Riemenschneider (Resolution of Surfaces, Proposition 53.7.8) we have $R^1f_*\omega _ X = 0$. Thus the trace map is a map $f_*\omega _ X \to \omega _ A$. Then we can consider

\[ \mathcal{O}_{\mathop{\mathrm{Spec}}(A)} = f_*\mathcal{O}_ X \to f_*\omega _ X \to \omega _ A \]

where the first map comes from the map $\mathcal{O}_ X \to \mathcal{O}_ X(E) = \omega _ X$ which is assumed to exist in the statement of the lemma. The composition is an isomorphism by Divisors, Lemma 30.2.11 as it is an isomorphism over the punctured spectrum of $A$ (by the assumption in the lemma and the fact that $f$ is an isomorphism over the punctured spectrum) and $A$ and $\omega _ A$ are $A$-modules of depth $2$ (by Algebra, Lemma 10.151.4 and Dualizing Complexes, Lemma 47.17.5). Hence $f_*\omega _ X \to \omega _ A$ is surjective whence an isomorphism. Thus $Rf_*\omega _ X = \omega _ A$ which by duality implies $Rf_*\mathcal{O}_ X = \mathcal{O}_{\mathop{\mathrm{Spec}}(A)}$. Whence $H^1(X, \mathcal{O}_ X) = 0$ which implies that $A$ defines a rational singularity (see discussion in Resolution of Surfaces, Section 53.8 in particular Lemmas 53.8.7 and 53.8.1). If $f$ is minimal, then $E = 0$ because the map $f^*\omega _ A \to \omega _ X$ is surjective by a repeated application of Resolution of Surfaces, Lemma 53.9.7 and $\omega _ A \cong A$ as we've seen above. $\square$

Lemma 55.28.2. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a finite type morphism. Let $x \in X$ be a point. Assume

  1. $A$ is an excellent regular local ring,

  2. $\mathcal{O}_{X, x}$ is normal of dimension $2$,

  3. $f$ is étale outside of $\overline{\{ x\} }$.

Then $f$ is étale at $x$.

Proof. We first replace $X$ by an affine open neighbourhood of $x$. Observe that $\mathcal{O}_{X, x}$ is an excellent local ring (More on Algebra, Lemma 15.51.2). Thus we can choose a minimal resolution of singularities $W \to \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$, see Resolution of Surfaces, Theorem 53.14.5. After possibly replacing $X$ by an affine open neighbourhood of $x$ we can find a proper morphism $b : X' \to X$ such that $X' \times _ X \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}) = W$, see Limits, Lemma 31.18.1. After shrinking $X$ further, we may assume $X'$ is regular. Namely, we know $W$ is regular and $X'$ is excellent and the regular locus of the spectrum of an excellent ring is open. Since $W \to \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$ is projective (as a sequence of normalized blowing ups), we may assume after shrinking $X$ that $b$ is projective (details omitted). Let $U = X \setminus \overline{\{ x\} }$. Since $W \to \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$ is an isomorphism over the punctured spectrum, we may assume $b : X' \to X$ is an isomorphism over $U$. Thus we may and will think of $U$ as an open subscheme of $X'$ as well. Set $f' = f \circ b : X' \to \mathop{\mathrm{Spec}}(A)$.

Since $A$ is regular we see that $\mathcal{O}_ Y$ is a dualizing complex for $Y$. Hence $f^!\mathcal{O}_ Y$ is a dualzing complex on $X$ (Duality for Schemes, Lemma 48.17.6). The Cohen-Macaulay locus of $X$ is open by Duality for Schemes, Lemma 48.23.1 (this can also be proven using excellency). Since $\mathcal{O}_{X, x}$ is Cohen-Macaulay, after shrinking $X$ we may assume $X$ is Cohen-Macaulay. Observe that an étale morphism is a local complete intersection. Thus Duality for Schemes, Lemma 48.28.3 applies with $r = 0$ and we get a map

\[ \mathcal{O}_ X \longrightarrow \omega _{X/Y} = H^0(f^!\mathcal{O}_ Y) \]

which is an isomorphism over $X \setminus \overline{\{ x\} }$. Since $\omega _{X/Y}$ is $(S_2)$ by Duality for Schemes, Lemma 48.21.5 we find this map is an isomorphism by Divisors, Lemma 30.2.11. This already shows that $X$ and in particular $\mathcal{O}_{X, x}$ is Gorenstein.

Set $\omega _{X'/Y} = H^0((f')^!\mathcal{O}_ Y)$. Arguing in exactly the same manner as above we find that $(f')^!\mathcal{O}_ Y = \omega _{X'/Y}[0]$ is a dualizing complex for $X'$. Since $X'$ is regular the morphism $X' \to Y$ is a local complete intersection morphism, see More on Morphisms, Lemma 36.54.11. By Duality for Schemes, Lemma 48.28.2 there exists a map

\[ \mathcal{O}_{X'} \longrightarrow \omega _{X'/Y} \]

which is an isomorphism over $U$. We conclude $\omega _{X'/Y} = \mathcal{O}_{X'}(E)$ for some effective Cartier divisor $E \subset X'$ disjoint from $U$.

Since $\omega _{X/Y} = \mathcal{O}_ Y$ we see that $\omega _{X'/Y} = b^! f^!\mathcal{O}_ Y = b^!\mathcal{O}_ X$. Returning to $W \to \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$ we see that $\omega _ W = \mathcal{O}_ W(E|_ W)$. By Lemma 55.28.1 we find $E|_ W = 0$. This means that $f' : X' \to Y$ is étale by (the already used) Duality for Schemes, Lemma 48.28.2. This immediately finishes the proof, as étaleness of $f'$ forces $b$ to be an isomorphism. $\square$

reference

Lemma 55.28.3 (Purity of ramification locus). Let $f : X \to Y$ be a morphism of locally Noetherian schemes. Let $x \in X$ and set $y = f(x)$. Assume

  1. $\mathcal{O}_{X, x}$ is normal of dimension $\geq 1$,

  2. $\mathcal{O}_{Y, y}$ is regular,

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

  4. for specializations $x' \leadsto x$ with $\dim (\mathcal{O}_{X, x'}) = 1$ our $f$ is étale at $x'$.

Then $f$ is étale at $x$.

Proof. We will prove the lemma by induction on $d = \dim (\mathcal{O}_{X, x})$.

An uninteresting case is $d = 1$ since in that case the morphism $f$ is étale at $x$ by assumption. Assume $d \geq 2$.

We can base change by $\mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y}) \to Y$ without affecting the conclusion of the lemma, see Morphisms, Lemma 28.34.17. Thus we may assume $Y = \mathop{\mathrm{Spec}}(A)$ where $A$ is a regular local ring and $y$ corresponds to the maximal ideal $\mathfrak m$ of $A$.

Let $x' \leadsto x$ be a specialization with $x' \not= x$. Then $\mathcal{O}_{X, x'}$ is normal as a localization of $\mathcal{O}_{X, x}$. If $x'$ is not a generic point of $X$, then $1 \leq \dim (\mathcal{O}_{X, x'}) < d$ and we conclude that $f$ is étale at $x'$ by induction hypothesis. Thus we may assume that $f$ is étale at all points specializing to $x$. Since the set of points where $f$ is étale is open in $X$ (by definition) we may after replacing $X$ by an open neighbourhood of $x$ assume that $f$ is étale away from $\overline{\{ x\} }$. In particular, we see that $f$ is étale except at points lying over the closed point $y \in Y = \mathop{\mathrm{Spec}}(A)$.

Let $X' = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A^\wedge )$. Let $x' \in X'$ be the unique point lying over $x$. By the above we see that $X'$ is étale over $\mathop{\mathrm{Spec}}(A^\wedge )$ away from the closed fibre and hence $X'$ is normal away from the closed fibre. Since $X$ is normal we conclude that $X'$ is normal by Resolution of Surfaces, Lemma 53.11.6. Then if we can show $X' \to \mathop{\mathrm{Spec}}(A^\wedge )$ is étale at $x'$, then $f$ is étale at $x$ (by the aforementioned Morphisms, Lemma 28.34.17). Thus we may and do assume $A$ is a regular complete local ring.

The case $d = 2$ now follows from Lemma 55.28.2.

Assume $d > 2$. Let $t \in \mathfrak m$, $t \not\in \mathfrak m^2$. Set $Y_0 = \mathop{\mathrm{Spec}}(A/tA)$ and $X_0 = X \times _ Y Y_0$. Then $X_0 \to Y_0$ is étale away from the fibre over the closed point. Since $d > 2$ we have $\dim (\mathcal{O}_{X_0, x}) = d - 1$ is $\geq 2$. The normalization $X_0' \to X_0$ is surjective and finite (as we're working over a complete local ring and such rings are Nagata). Let $x' \in X_0'$ be a point mapping to $x$. By induction hypothesis the morphism $X'_0 \to Y$ is étale at $x'$. From the inclusions $\kappa (y) \subset \kappa (x) \subset \kappa (x')$ we conclude that $\kappa (x)$ is finite over $\kappa (y)$. Hence $x$ is a closed point of the fibre of $X \to Y$ over $y$. But since $x$ is also a generic point of this fibre, we conclude that $f$ is quasi-finite at $x$ and we reduce to the case of purity of branch locus, see Lemma 55.21.4. $\square$


Comments (2)

Comment #4187 by typo_bot on

In the beginning of the section 'apparantly' should be 'apparently.'


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