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57.11 Fundamental groups of normal schemes

Let $X$ be an integral, geometrically unibranch scheme. In the previous section we have seen that the fundamental group of $X$ is a quotient of the Galois group of the function field $K$ of $X$. Since the map is continuous the kernel is a normal closed subgroup of the Galois group. Hence this kernel corresponds to a Galois extension $M/K$ by Galois theory (Fields, Theorem 9.22.4). In this section we will determine $M$ when $X$ is a normal integral scheme.

Let $X$ be an integral normal scheme with function field $K$. Let $K \subset L$ be a finite extension. Consider the normalization $Y \to X$ of $X$ in the morphism $\mathop{\mathrm{Spec}}(L) \to X$ as defined in Morphisms, Section 29.53. We will say (in this setting) that $X$ is unramified in $L$ if $Y \to X$ is an unramified morphism of schemes. In Lemma 57.13.4 we will elucidate this condition. Observe that the scheme theoretic fibre of $Y \to X$ over $\mathop{\mathrm{Spec}}(K)$ is $\mathop{\mathrm{Spec}}(L)$. Hence the field extension $L/K$ is separable if $X$ is unramified in $L$, see Morphisms, Lemmas 29.35.11.

Lemma 57.11.1. In the situation above the following are equivalent

  1. $X$ is unramified in $L$,

  2. $Y \to X$ is étale, and

  3. $Y \to X$ is finite étale.

Proof. Observe that $Y \to X$ is an integral morphism. In each case the morphism $Y \to X$ is locally of finite type by definition. Hence we find that in each case $Y \to X$ is finite by Morphisms, Lemma 29.44.4. In particular we see that (2) is equivalent to (3). An étale morphism is unramified, hence (2) implies (1).

Conversely, assume $Y \to X$ is unramified. Let $x \in X$. We can choose an étale neighbourhood $(U, u) \to (X, x)$ such that

\[ Y \times _ X U = \coprod V_ j \longrightarrow U \]

is a disjoint union of closed immersions, see Étale Morphisms, Lemma 41.17.3. Shrinking we may assume $U$ is quasi-compact. Then $U$ has finitely many irreducible components (Descent, Lemma 35.13.3). Since $U$ is normal (Descent, Lemma 35.15.2) the irreducible components of $U$ are open and closed (Properties, Lemma 28.7.5) and we may assume $U$ is irreducible. Then $U$ is an integral scheme whose generic point $\xi $ maps to the generic point of $X$. On the other hand, we know that $Y \times _ X U$ is the normalization of $U$ in $\mathop{\mathrm{Spec}}(L) \times _ X U$ by More on Morphisms, Lemma 37.17.2. Every point of $\mathop{\mathrm{Spec}}(L) \times _ X U$ maps to $\xi $. Thus every $V_ j$ contains a point mapping to $\xi $ by Morphisms, Lemma 29.53.9. Thus $V_ j \to U$ is an isomorphism as $U = \overline{\{ \xi \} }$. Thus $Y \times _ X U \to U$ is étale. By Descent, Lemma 35.20.29 we conclude that $Y \to X$ is étale over the image of $U \to X$ (an open neighbourhood of $x$). $\square$

Lemma 57.11.2. Let $X$ be a normal integral scheme with function field $K$. Let $Y \to X$ be a finite étale morphism. If $Y$ is connected, then $Y$ is an integral normal scheme and $Y$ is the normalization of $X$ in the function field of $Y$.

Proof. The scheme $Y$ is normal by Descent, Lemma 35.15.2. Since $Y \to X$ is flat every generic point of $Y$ maps to the generic point of $X$ by Morphisms, Lemma 29.25.9. Since $Y \to X$ is finite we see that $Y$ has a finite number of irreducible components. Thus $Y$ is the disjoint union of a finite number of integral normal schemes by Properties, Lemma 28.7.5. Thus if $Y$ is connected, then $Y$ is an integral normal scheme.

Let $L$ be the function field of $Y$ and let $Y' \to X$ be the normalization of $X$ in $L$. By Morphisms, Lemma 29.53.4 we obtain a factorization $Y' \to Y \to X$ and $Y' \to Y$ is the normalization of $Y$ in $L$. Since $Y$ is normal it is clear that $Y' = Y$ (this can also be deduced from Morphisms, Lemma 29.54.8). $\square$

Proposition 57.11.3. Let $X$ be a normal integral scheme with function field $K$. Then the canonical map (57.10.6.1)

\[ \text{Gal}(K^{sep}/K) = \pi _1(\eta , \overline{\eta }) \longrightarrow \pi _1(X, \overline{\eta }) \]

is identified with the quotient map $\text{Gal}(K^{sep}/K) \to \text{Gal}(M/K)$ where $M \subset K^{sep}$ is the union of the finite subextensions $L$ such that $X$ is unramified in $L$.

Proof. The normal scheme $X$ is geometrically unibranch (Properties, Lemma 28.15.2). Hence Lemma 57.10.7 applies to $X$. Thus $\pi _1(\eta , \overline{\eta }) \to \pi _1(X, \overline{\eta })$ is surjective and top horizontal arrow of the commutative diagram

\[ \xymatrix{ \textit{FÉt}_ X \ar[r] \ar[d] \ar[rd]_ c & \textit{FÉt}_\eta \ar[d] \\ \textit{Finite-}\pi _1(X, \overline{\eta })\textit{-sets} \ar[r] & \textit{Finite-}\text{Gal}(K^{sep}/K)\textit{-sets} } \]

is fully faithful. The left vertical arrow is the equivalence of Theorem 57.6.2 and the right vertical arrow is the equivalence of Lemma 57.6.3. The lower horizontal arrow is induced by the map of the proposition. By Lemmas 57.11.1 and 57.11.2 we see that the essential image of $c$ consists of $\text{Gal}(K^{sep}/K)\textit{-Sets}$ isomorphic to sets of the form

\[ S = \mathop{\mathrm{Hom}}\nolimits _ K(\prod \nolimits _{i = 1, \ldots , n} L_ i, K^{sep}) = \coprod \nolimits _{i = 1, \ldots , n} \mathop{\mathrm{Hom}}\nolimits _ K(L_ i, K^{sep}) \]

with $L_ i/K$ finite separable such that $X$ is unramified in $L_ i$. Thus if $M \subset K^{sep}$ is as in the statement of the lemma, then $\text{Gal}(K^{sep}/M)$ is exactly the subgroup of $\text{Gal}(K^{sep}/K)$ acting trivially on every object in the essential image of $c$. On the other hand, the essential image of $c$ is exactly the category of $S$ such that the $\text{Gal}(K^{sep}/K)$-action factors through the surjection $\text{Gal}(K^{sep}/K) \to \pi _1(X, \overline{\eta })$. We conclude that $\text{Gal}(K^{sep}/M)$ is the kernel. Hence $\text{Gal}(K^{sep}/M)$ is a normal subgroup, $M/K$ is Galois, and we have a short exact sequence

\[ 1 \to \text{Gal}(K^{sep}/M) \to \text{Gal}(K^{sep}/K) \to \text{Gal}(M/K) \to 1 \]

by Galois theory (Fields, Theorem 9.22.4 and Lemma 9.22.5). The proof is done. $\square$

Lemma 57.11.4. Let $(A, \mathfrak m)$ be a normal local ring. Set $X = \mathop{\mathrm{Spec}}(A)$. Let $A^{sh}$ be the strict henselization of $A$. Let $K$ and $K^{sh}$ be the fraction fields of $A$ and $A^{sh}$. Then the sequence

\[ \pi _1(\mathop{\mathrm{Spec}}(K^{sh})) \to \pi _1(\mathop{\mathrm{Spec}}(K)) \to \pi _1(X) \to 1 \]

is exact in the sense of Lemma 57.4.3 part (1).

Proof. Note that $A^{sh}$ is a normal domain, see More on Algebra, Lemma 15.45.6. The map $\pi _1(\mathop{\mathrm{Spec}}(K)) \to \pi _1(X)$ is surjective by Proposition 57.11.3.

Write $X^{sh} = \mathop{\mathrm{Spec}}(A^{sh})$. Let $Y \to X$ be a finite étale morphism. Then $Y^{sh} = Y \times _ X X^{sh} \to X^{sh}$ is a finite étale morphism. Since $A^{sh}$ is strictly henselian we see that $Y^{sh}$ is isomorphic to a disjoint union of copies of $X^{sh}$. Thus the same is true for $Y \times _ X \mathop{\mathrm{Spec}}(K^{sh})$. It follows that the composition $\pi _1(\mathop{\mathrm{Spec}}(K^{sh})) \to \pi _1(X)$ is trivial, see Lemma 57.4.2.

To finish the proof, it suffices according to Lemma 57.4.3 to show the following: Given a finite étale morphism $V \to \mathop{\mathrm{Spec}}(K)$ such that $V \times _{\mathop{\mathrm{Spec}}(K)} \mathop{\mathrm{Spec}}(K^{sh})$ is a disjoint union of copies of $\mathop{\mathrm{Spec}}(K^{sh})$, we can find a finite étale morphism $Y \to X$ with $V \cong Y \times _ X \mathop{\mathrm{Spec}}(K)$ over $\mathop{\mathrm{Spec}}(K)$. Write $V = \mathop{\mathrm{Spec}}(L)$, so $L$ is a finite product of finite separable extensions of $K$. Let $B \subset L$ be the integral closure of $A$ in $L$. If $A \to B$ is étale, then we can take $Y = \mathop{\mathrm{Spec}}(B)$ and the proof is complete. By Algebra, Lemma 10.147.4 (and a limit argument we omit) we see that $B \otimes _ A A^{sh}$ is the integral closure of $A^{sh}$ in $L^{sh} = L \otimes _ K K^{sh}$. Our assumption is that $L^{sh}$ is a product of copies of $K^{sh}$ and hence $B^{sh}$ is a product of copies of $A^{sh}$. Thus $A^{sh} \to B^{sh}$ is étale. As $A \to A^{sh}$ is faithfully flat it follows that $A \to B$ is étale (Descent, Lemma 35.20.29) as desired. $\square$


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