Introduction
In this chapter we discuss Grothendieck’s fundamental group of a scheme and applications. A foundational reference is [SGA1]. A nice introduction is [Lenstra]. Other references [Murre-lectures] and [Grothendieck-Murre].
Schemes étale over a point
In this section we describe schemes étale over the spectrum of a field. Before we state the result we introduce the category of \(G\)-sets for a topological group \(G\).
Definition
Let \(G\) be a topological group. A \(G\)-set, sometimes called a discrete \(G\)-set, is a set \(X\) endowed with a left action \(a : G \times X \to X\) such that \(a\) is continuous when \(X\) is given the discrete topology and \(G \times X\) the product topology. A morphism of \(G\)-sets \(f : X \to Y\) is simply any \(G\)-equivariant map from \(X\) to \(Y\). The category of \(G\)-sets is denoted \(G\textit{-Sets}\).
The condition that \(a : G \times X \to X\) is continuous signifies simply that the stabilizer of any \(x \in X\) is open in \(G\). If \(G\) is an abstract group \(G\) (i.e., a group but not a topological group) then this agrees with our preceding definition (see for example Sites, Example 00VK) provided we endow \(G\) with the discrete topology.
Recall that if \(L/K\) is an infinite Galois extension then the Galois group \(G = \text{Gal}(L/K)\) comes endowed with a canonical topology, see Fields, Section 0BMI.
Lemma
Let \(K\) be a field. Let \(K^{sep}\) be a separable closure of \(K\). Consider the profinite group \(G = \text{Gal}(K^{sep}/K)\). The functor \[\begin{matrix} \text{schemes \'etale over }K & \longrightarrow & G\textit{-Sets} \\ X/K & \longmapsto & \Mor_{\Spec(K)}(\Spec(K^{sep}), X) \end{matrix}\] is an equivalence of categories.
Proof
A scheme \(X\) over \(K\) is étale over \(K\) if and only if \(X \cong \coprod_{i\in I} \Spec(K_i)\) with each \(K_i\) a finite separable extension of \(K\) (Morphisms, Lemma 02GL). The functor of the lemma associates to \(X\) the \(G\)-set \[\coprod\nolimits_i \Hom_K(K_i, K^{sep})\] with its natural left \(G\)-action. Each element has an open stabilizer by definition of the topology on \(G\). Conversely, any \(G\)-set \(S\) is a disjoint union of its orbits. Say \(S = \coprod S_i\). Pick \(s_i \in S_i\) and denote \(G_i \subset G\) its open stabilizer. By Galois theory (Fields, Theorem 0BML) the fields \((K^{sep})^{G_i}\) are finite separable field extensions of \(K\), and hence the scheme \[\coprod\nolimits_i \Spec((K^{sep})^{G_i})\] is étale over \(K\). This gives an inverse to the functor of the lemma. Some details omitted.
Remark
Under the correspondence of Lemma 03QR, the coverings in the small étale site \(\Spec(K)_\etale\) of \(K\) correspond to surjective families of maps in \(G\textit{-Sets}\).
Galois categories
In this section we discuss some of the material the reader can find in [SGA1, Exposé V, Sections 4, 5, and 6].
Let \(F : \mathcal{C} \to \textit{Sets}\) be a functor. Recall that by our conventions categories have a set of objects and for any pair of objects a set of morphisms. There is a canonical injective map [0BS7]\[\begin{equation} \text{Aut}(F) \longrightarrow \prod\nolimits_{X \in \Ob(\mathcal{C})} \text{Aut}(F(X)) \end{equation}\] For a set \(E\) we endow \(\text{Aut}(E)\) with the compact open topology, see Topology, Example 0BMC. Of course this is the discrete topology when \(E\) is finite, which is the case of interest in this section1. We endow \(\text{Aut}(F)\) with the topology induced from the product topology on the right hand side of (0BS7). In particular, the action maps \[\text{Aut}(F) \times F(X) \longrightarrow F(X)\] are continuous when \(F(X)\) is given the discrete topology because this is true for the action maps \(\text{Aut}(E) \times E \to E\) for any set \(E\). The universal property of our topology on \(\text{Aut}(F)\) is the following: suppose that \(G\) is a topological group and \(G \to \text{Aut}(F)\) is a group homomorphism such that the induced actions \(G \times F(X) \to F(X)\) are continuous for all \(X \in \Ob(\mathcal{C})\) where \(F(X)\) has the discrete topology. Then \(G \to \text{Aut}(F)\) is continuous.
The following lemma tells us that the group of automorphisms of a functor to the category of finite sets is automatically a profinite group.
Lemma
Let \(\mathcal{C}\) be a category and let \(F : \mathcal{C} \to \textit{Sets}\) be a functor. The map (0BS7) identifies \(\text{Aut}(F)\) with a closed subgroup of \(\prod_{X \in \Ob(\mathcal{C})} \text{Aut}(F(X))\). In particular, if \(F(X)\) is finite for all \(X\), then \(\text{Aut}(F)\) is a profinite group.
Proof
Let \(\xi = (\gamma_X) \in \prod \text{Aut}(F(X))\) be an element not in \(\text{Aut}(F)\). Then there exists a morphism \(f : X \to X'\) of \(\mathcal{C}\) and an element \(x \in F(X)\) such that \(F(f)(\gamma_X(x)) \not = \gamma_{X'}(F(f)(x))\). Consider the open neighbourhood \(U = \{\gamma \in \text{Aut}(F(X)) \mid \gamma(x) = \gamma_X(x)\}\) of \(\gamma_X\) and the open neighbourhood \(U' = \{\gamma' \in \text{Aut}(F(X')) \mid \gamma'(F(f)(x)) = \gamma_{X'}(F(f)(x))\}\). Then \(U \times U' \times \prod_{X'' \not = X, X'} \text{Aut}(F(X''))\) is an open neighbourhood of \(\xi\) not meeting \(\text{Aut}(F)\). The final statement follows from the fact that \(\prod \text{Aut}(F(X))\) is a profinite space if each \(F(X)\) is finite.
Example
Let \(G\) be a topological group. An important example will be the forgetful functor [0BMT]\[\begin{equation} \textit{Finite-}G\textit{-Sets} \longrightarrow \textit{Sets} \end{equation}\] where \(\textit{Finite-}G\textit{-Sets}\) is the full subcategory of \(G\textit{-Sets}\) whose objects are the finite \(G\)-sets. The category \(G\textit{-Sets}\) of \(G\)-sets is defined in Definition 04JJ.
Let \(G\) be a topological group. The profinite completion of \(G\) will be the profinite group \[G^\wedge = \lim_{U \subset G\text{ open, normal, finite index}} G/U\] with its profinite topology. Observe that the limit is cofiltered as a finite intersection of open, normal subgroups of finite index is another. The universal property of the profinite completion is that any continuous map \(G \to H\) to a profinite group \(H\) factors canonically as \(G \to G^\wedge \to H\).
Lemma
Let \(G\) be a topological group. The automorphism group of the functor (0BMT) endowed with its profinite topology from Lemma 0BMR is the profinite completion of \(G\).
Proof
Denote \(F_G\) the functor (0BMT). Any morphism \(X \to Y\) in \(\textit{Finite-}G\textit{-Sets}\) commutes with the action of \(G\). Thus any \(g \in G\) defines an automorphism of \(F_G\) and we obtain a canonical homomorphism \(G \to \text{Aut}(F_G)\) of groups. Observe that any finite \(G\)-set \(X\) is a finite disjoint union of \(G\)-sets of the form \(G/H_i\) with canonical \(G\)-action where \(H_i \subset G\) is an open subgroup of finite index. Then \(U_i = \bigcap gH_ig^{-1}\) is open, normal, and has finite index. Moreover \(U_i\) acts trivially on \(G/H_i\) hence \(U = \bigcap U_i\) acts trivially on \(F_G(X)\). Hence the action \(G \times F_G(X) \to F_G(X)\) is continuous. By the universal property of the topology on \(\text{Aut}(F_G)\) the map \(G \to \text{Aut}(F_G)\) is continuous. By Lemma 0BMR and the universal property of profinite completion there is an induced continuous group homomorphism \[G^\wedge \longrightarrow \text{Aut}(F_G)\] Moreover, since \(G/U\) acts faithfully on \(G/U\) this map is injective. If the image is dense, then the map is surjective and hence a homeomorphism by Topology, Lemma 08YE.
Let \(\gamma \in \text{Aut}(F_G)\) and let \(X \in \Ob(\mathcal{C})\). We will show there is a \(g \in G\) such that \(\gamma\) and \(g\) induce the same action on \(F_G(X)\). This will finish the proof. As before we see that \(X\) is a finite disjoint union of \(G/H_i\). With \(U_i\) and \(U\) as above, the finite \(G\)-set \(Y = G/U\) surjects onto \(G/H_i\) for all \(i\) and hence it suffices to find \(g \in G\) such that \(\gamma\) and \(g\) induce the same action on \(F_G(G/U) = G/U\). Let \(e \in G\) be the neutral element and say that \(\gamma(eU) = g_0U\) for some \(g_0 \in G\). For any \(g_1 \in G\) the morphism \[R_{g_1} : G/U \longrightarrow G/U,\quad gU \longmapsto gg_1U\] of \(\textit{Finite-}G\textit{-Sets}\) commutes with the action of \(\gamma\). Hence \[\gamma(g_1U) = \gamma(R_{g_1}(eU)) = R_{g_1}(\gamma(eU)) = R_{g_1}(g_0U) = g_0g_1U\] Thus we see that \(g = g_0\) works.
Recall that an exact functor is one which commutes with all finite limits and finite colimits. In particular such a functor commutes with equalizers, coequalizers, fibred products, pushouts, etc.
Lemma
Let \(G\) be a topological group. Let \(F : \textit{Finite-}G\textit{-Sets} \to \textit{Sets}\) be an exact functor with \(F(X)\) finite for all \(X\). Then \(F\) is isomorphic to the functor (0BMT).
Proof
Let \(X\) be a nonempty object of \(\textit{Finite-}G\textit{-Sets}\). The diagram \[\xymatrix{ X \ar[r] \ar[d] & \{*\} \ar[d] \\ \{*\} \ar[r] & \{*\} }\] is cocartesian. Hence we conclude that \(F(X)\) is nonempty. Let \(U \subset G\) be an open, normal subgroup with finite index. Observe that \[G/U \times G/U = \coprod\nolimits_{gU \in G/U} G/U\] where the summand corresponding to \(gU\) corresponds to the orbit of \((eU, gU)\) on the left hand side. Then we see that \[F(G/U) \times F(G/U) = F(G/U \times G/U) = \coprod\nolimits_{gU \in G/U} F(G/U)\] Hence \(|F(G/U)| = |G/U|\) as \(F(G/U)\) is nonempty. Thus we see that \[\lim_{U \subset G\text{ open, normal, finite idex}} F(G/U)\] is nonempty (Categories, Lemma 086J). Pick \(\gamma = (\gamma_U)\) an element in this limit. Denote \(F_G\) the functor (0BMT). We can identify \(F_G\) with the functor \[X \longmapsto \colim_U \Mor(G/U, X)\] where \(f : G/U \to X\) corresponds to \(f(eU) \in X = F_G(X)\) (details omitted). Hence the element \(\gamma\) determines a well defined map \[t : F_G \longrightarrow F\] Namely, given \(x \in X\) choose \(U\) and \(f : G/U \to X\) sending \(eU\) to \(x\) and then set \(t_X(x) = F(f)(\gamma_U)\). We will show that \(t\) induces a bijective map \(t_{G/U} : F_G(G/U) \to F(G/U)\) for any \(U\). This implies in a straightforward manner that \(t\) is an isomorphism (details omitted). Since \(|F_G(G/U)| = |F(G/U)|\) it suffices to show that \(t_{G/U}\) is surjective. The image contains at least one element, namely \(t_{G/U}(eU) = F(\text{id}_{G/U})(\gamma_U) = \gamma_U\). For \(g \in G\) denote \(R_g : G/U \to G/U\) right multiplication. Then set of fixed points of \(F(R_g) : F(G/U) \to F(G/U)\) is equal to \(F(\emptyset) = \emptyset\) if \(g \not \in U\) because \(F\) commutes with equalizers. It follows that if \(g_1, \ldots, g_{|G/U|}\) is a system of representatives for \(G/U\), then the elements \(F(R_{g_i})(\gamma_U)\) are pairwise distinct and hence fill out \(F(G/U)\). Then \[t_{G/U}(g_iU) = F(R_{g_i})(\gamma_U)\] and the proof is complete.
Example
Let \(\mathcal{C}\) be a category and let \(F : \mathcal{C} \to \textit{Sets}\) be a functor such that \(F(X)\) is finite for all \(X \in \Ob(\mathcal{C})\). By Lemma 0BMR we see that \(G = \text{Aut}(F)\) comes endowed with the structure of a profinite topological group in a canonical manner. We obtain a functor [0BMX]\[\begin{equation} \mathcal{C} \longrightarrow \textit{Finite-}G\textit{-Sets},\quad X \longmapsto F(X) \end{equation}\] where \(F(X)\) is endowed with the induced action of \(G\). This action is continuous by our construction of the topology on \(\text{Aut}(F)\).
The purpose of defining Galois categories is to single out those pairs \((\mathcal{C}, F)\) for which the functor (0BMX) is an equivalence. Our definition of a Galois category is as follows.
Definition
Let \(\mathcal{C}\) be a category and let \(F : \mathcal{C} \to \textit{Sets}\) be a functor. The pair \((\mathcal{C}, F)\) is a Galois category if
\(\mathcal{C}\) has finite limits and finite colimits,
every object of \(\mathcal{C}\) is a finite (possibly empty) coproduct of connected objects,
\(F(X)\) is finite for all \(X \in \Ob(\mathcal{C})\), and
Here we say \(X \in \Ob(\mathcal{C})\) is connected if it is not initial and for any monomorphism \(Y \to X\) either \(Y\) is initial or \(Y \to X\) is an isomorphism.
Warning: This definition is not the same (although eventually we’ll see it is equivalent) as the definition given in most references. Namely, in [SGA1, Exposé V, Definition 5.1] a Galois category is defined to be a category equivalent to \(\textit{Finite-}G\textit{-Sets}\) for some profinite group \(G\). Then Grothendieck characterizes Galois categories by a list of axioms (G1) – (G6) which are weaker than our axioms above. The motivation for our choice is to stress the existence of finite limits and finite colimits and exactness of the functor \(F\). The price we’ll pay for this later is that we’ll have to work a bit harder to apply the results of this section.
Lemma
Let \((\mathcal{C}, F)\) be a Galois category. Let \(X \to Y \in \text{Arrows}(\mathcal{C})\). Then
\(F\) is faithful,
\(X \to Y\) is a monomorphism \(\Leftrightarrow F(X) \to F(Y)\) is injective,
\(X \to Y\) is an epimorphism \(\Leftrightarrow F(X) \to F(Y)\) is surjective,
an object \(A\) of \(\mathcal{C}\) is initial if and only if \(F(A) = \emptyset\),
an object \(Z\) of \(\mathcal{C}\) is final if and only if \(F(Z)\) is a singleton,
if \(X\) and \(Y\) are connected, then \(X \to Y\) is an epimorphism,
if \(X\) is connected and \(a, b : X \to Y\) are two morphisms then \(a = b\) as soon as \(F(a)\) and \(F(b)\) agree on one element of \(F(X)\),
if \(X = \coprod_{i = 1, \ldots, n} X_i\) and \(Y = \coprod_{j = 1, \ldots, m} Y_j\) where \(X_i\), \(Y_j\) are connected, then there is map \(\alpha : \{1, \ldots, n\} \to \{1, \ldots, m\}\) such that \(X \to Y\) comes from a collection of morphisms \(X_i \to Y_{\alpha(i)}\).
Proof
Proof of (1). Suppose \(a, b : X \to Y\) with \(F(a) = F(b)\). Let \(E\) be the equalizer of \(a\) and \(b\). Then \(F(E) = F(X)\) and we see that \(E = X\) because \(F\) reflects isomorphisms.
Proof of (2). This is true because \(F\) turns the morphism \(X \to X \times_Y X\) into the map \(F(X) \to F(X) \times_{F(Y)} F(X)\) and \(F\) reflects isomorphisms.
Proof of (3). This is true because \(F\) turns the morphism \(Y \amalg_X Y \to Y\) into the map \(F(Y) \amalg_{F(X)} F(Y) \to F(Y)\) and \(F\) reflects isomorphisms.
Proof of (4). There exists an initial object \(A\) and certainly \(F(A) = \emptyset\). On the other hand, if \(X\) is an object with \(F(X) = \emptyset\), then the unique map \(A \to X\) induces a bijection \(F(A) \to F(X)\) and hence \(A \to X\) is an isomorphism.
Proof of (5). There exists a final object \(Z\) and certainly \(F(Z)\) is a singleton. On the other hand, if \(X\) is an object with \(F(X)\) a singleton, then the unique map \(X \to Z\) induces a bijection \(F(X) \to F(Z)\) and hence \(X \to Z\) is an isomorphism.
Proof of (6). The equalizer \(E\) of the two maps \(Y \to Y \amalg_X Y\) is not an initial object of \(\mathcal{C}\) because \(X \to Y\) factors through \(E\) and \(F(X) \not = \emptyset\). Hence \(E = Y\) and we conclude.
Proof of (0BN1). The equalizer \(E\) of \(a\) and \(b\) comes with a monomorphism \(E \to X\) and \(F(E) \subset F(X)\) is the set of elements where \(F(a)\) and \(F(b)\) agree. To finish use that either \(E\) is initial or \(E = X\).
Proof of (8). For each \(i, j\) we see that \(E_{ij} = X_i \times_Y Y_j\) is either initial or equal to \(X_i\). Picking \(s \in F(X_i)\) we see that \(E_{ij} = X_i\) if and only if \(s\) maps to an element of \(F(Y_j) \subset F(Y)\), hence this happens for a unique \(j = \alpha(i)\).
By the lemma above we see that, given a connected object \(X\) of a Galois category \((\mathcal{C}, F)\), the automorphism group \(\text{Aut}(X)\) has order at most \(|F(X)|\). Namely, given \(s \in F(X)\) and \(g \in \text{Aut}(X)\) we see that \(F(g)(s) = s\) if and only if \(g = \text{id}_X\) by (0BN1). We say \(X\) is Galois if equality holds. Equivalently, \(X\) is Galois if it is connected and \(\text{Aut}(X)\) acts transitively on \(F(X)\).
Lemma
Let \((\mathcal{C}, F)\) be a Galois category. For any connected object \(X\) of \(\mathcal{C}\) there exists a Galois object \(Y\) and a morphism \(Y \to X\).
Proof
We will use the results of Lemma 0BN0 without further mention. Let \(n = |F(X)|\). Consider \(X^n\) endowed with its natural action of \(S_n\). Let \[X^n = \coprod\nolimits_{t \in T} Z_t\] be the decomposition into connected objects. Pick a \(t\) such that \(F(Z_t)\) contains \((s_1, \ldots, s_n)\) with \(s_i\) pairwise distinct. If \((s'_1, \ldots, s'_n) \in F(Z_t)\) is another element, then we claim \(s'_i\) are pairwise distinct as well. Namely, if not, say \(s'_i = s'_j\), then \(Z_t\) is the image of an connected component of \(X^{n - 1}\) under the diagonal morphism \[\Delta_{ij} : X^{n - 1} \longrightarrow X^n\] Since morphisms of connected objects are epimorphisms and induce surjections after applying \(F\) it would follow that \(s_i = s_j\) which is not the case.
Let \(G \subset S_n\) be the subgroup of elements with \(g(Z_t) = Z_t\). Looking at the action of \(S_n\) on \[F(X)^n = F(X^n) = \coprod\nolimits_{t' \in T} F(Z_{t'})\] we see that \(G = \{g \in S_n \mid g(s_1, \ldots, s_n) \in F(Z_t)\}\). Now pick a second element \((s'_1, \ldots, s'_n) \in F(Z_t)\). Above we have seen that \(s'_i\) are pairwise distinct. Thus we can find a \(g \in S_n\) with \(g(s_1, \ldots, s_n) = (s'_1, \ldots, s'_n)\). In other words, the action of \(G\) on \(F(Z_t)\) is transitive and the proof is complete.
Here is a key lemma.
Lemma
Let \((\mathcal{C}, F)\) be a Galois category. Let \(G = \text{Aut}(F)\) be as in Example 0BMW. For any connected \(X\) in \(\mathcal{C}\) the action of \(G\) on \(F(X)\) is transitive.
Proof
We will use the results of Lemma 0BN0 without further mention. Let \(I\) be the set of isomorphism classes of Galois objects in \(\mathcal{C}\). For each \(i \in I\) let \(X_i\) be a representative of the isomorphism class. Choose \(\gamma_i \in F(X_i)\) for each \(i \in I\). We define a partial ordering on \(I\) by setting \(i \geq i'\) if and only if there is a morphism \(f_{ii'} : X_i \to X_{i'}\). Given such a morphism we can post-compose by an automorphism \(X_{i'} \to X_{i'}\) to assure that \(F(f_{ii'})(\gamma_i) = \gamma_{i'}\). With this normalization the morphism \(f_{ii'}\) is unique. Observe that \(I\) is a directed partially ordered set: (Categories, Definition 00D3) if \(i_1, i_2 \in I\) there exists a Galois object \(Y\) and a morphism \(Y \to X_{i_1} \times X_{i_2}\) by Lemma 0BN2 applied to a connected component of \(X_{i_1} \times X_{i_2}\). Then \(Y \cong X_i\) for some \(i \in I\) and \(i \geq i_1\), \(i \geq I_2\).
We claim that the functor \(F\) is isomorphic to the functor \(F'\) which sends \(X\) to \[F'(X) = \colim_I \Mor_\mathcal{C}(X_i, X)\] via the transformation of functors \(t : F' \to F\) defined as follows: given \(f : X_i \to X\) we set \(t_X(f) = F(f)(\gamma_i)\). Using (0BN1) we find that \(t_X\) is injective. To show surjectivity, let \(\gamma \in F(X)\). Then we can immediately reduce to the case where \(X\) is connected by the definition of a Galois category. Then we may assume \(X\) is Galois by Lemma 0BN2. In this case \(X\) is isomorphic to \(X_i\) for some \(i\) and we can choose the isomorphism \(X_i \to X\) such that \(\gamma_i\) maps to \(\gamma\) (by definition of Galois objects). We conclude that \(t\) is an isomorphism.
Set \(A_i = \text{Aut}(X_i)\). We claim that for \(i \geq i'\) there is a canonical map \(h_{ii'} : A_i \to A_{i'}\) such that for all \(a \in A_i\) the diagram \[\xymatrix{ X_i \ar[d]_a \ar[r]_{f_{ii'}} & X_{i'} \ar[d]^{h_{ii'}(a)} \\ X_i \ar[r]^{f_{ii'}} & X_{i'} }\] commutes. Namely, just let \(h_{ii'}(a) = a' : X_{i'} \to X_{i'}\) be the unique automorphism such that \(F(a')(\gamma_{i'}) = F(f_{ii'} \circ a)(\gamma_i)\). As before this makes the diagram commute and moreover the choice is unique. It follows that \(h_{i'i''} \circ h_{ii'} = h_{ii''}\) if \(i \geq i' \geq i''\). Since \(F(X_i) \to F(X_{i'})\) is surjective we see that \(A_i \to A_{i'}\) is surjective. Taking the inverse limit we obtain a group \[A = \lim_I A_i\] This is a profinite group since the automorphism groups are finite. The map \(A \to A_i\) is surjective for all \(i\) by Categories, Lemma 086J.
Since elements of \(A\) act on the inverse system \(X_i\) we get an action of \(A\) (on the right) on \(F'\) by pre-composing. In other words, we get a homomorphism \(A^{opp} \to G\). Since \(A \to A_i\) is surjective we conclude that \(G\) acts transitively on \(F(X_i)\) for all \(i\). Since every connected object is dominated by one of the \(X_i\) we conclude the lemma is true.
Proposition
Let \((\mathcal{C}, F)\) be a Galois category. Let \(G = \text{Aut}(F)\) be as in Example 0BMW. The functor \(F : \mathcal{C} \to \textit{Finite-}G\textit{-Sets}\) (0BMX) an equivalence.
Proof
We will use the results of Lemma 0BN0 without further mention. In particular we know the functor is faithful. By Lemma 0BN3 we know that for any connected \(X\) the action of \(G\) on \(F(X)\) is transitive. Hence for general \(X\) the functor \(F\) sends the connected components of \(X\) (which exist by an axiom of a Galois category) to the \(G\)-orbits of \(F(X)\). Let \(X\) and \(Y\) be objects and let \(s : F(X) \to F(Y)\) be a map. Then the graph \(\Gamma_s \subset F(X) \times F(Y)\) of \(s\) is a union of connected components. Hence there exists a union of connected components \(Z\) of \(X \times Y\), which comes equipped with a monomorphism \(Z \to X \times Y\), with \(F(Z) = \Gamma_s\). Since \(F(Z) \to F(X)\) is bijective we see that \(Z \to X\) is an isomorphism and we conclude that \(s = F(f)\) where \(f : X \cong Z \to Y\) is the composition. Hence \(F\) is fully faithful.
To finish the proof we show that \(F\) is essentially surjective. It suffices to show that \(G/H\) is in the essential image for any open subgroup \(H \subset G\) of finite index. By definition of the topology on \(G\) there exists a finite collection of objects \(X_i\) such that \[\Ker(G \longrightarrow \prod\nolimits_i \text{Aut}(F(X_i)))\] is contained in \(H\). We may assume \(X_i\) is connected for all \(i\). We can choose a Galois object \(Y\) mapping to a connected component of \(\prod X_i\) using Lemma 0BN2. Choose an isomorphism \(F(Y) = G/U\) in \(G\textit{-sets}\) for some open subgroup \(U \subset G\). As \(Y\) is Galois, the group \(\text{Aut}(Y) = \text{Aut}_{G\textit{-Sets}}(G/U)\) acts transitively on \(F(Y) = G/U\). This implies that \(U\) is normal. Since \(F(Y)\) surjects onto \(F(X_i)\) for each \(i\) we see that \(U \subset H\). Let \(M \subset \text{Aut}(Y)\) be the finite subgroup corresponding to \[(H/U)^{opp} \subset (G/U)^{opp} = \text{Aut}_{G\textit{-Sets}}(G/U) = \text{Aut}(Y).\] Set \(X = Y/M\), i.e., \(X\) is the coequalizer of the arrows \(m : Y \to Y\), \(m \in M\). Since \(F\) is exact we see that \(F(X) = G/H\) and the proof is complete.
Lemma
Let \((\mathcal{C}, F)\) and \((\mathcal{C}', F')\) be Galois categories. Let \(H : \mathcal{C} \to \mathcal{C}'\) be an exact functor. There exists an isomorphism \(t : F' \circ H \to F\). The choice of \(t\) determines a continuous homomorphism \(h : G' = \text{Aut}(F') \to \text{Aut}(F) = G\) and a \(2\)-commutative diagram \[\xymatrix{ \mathcal{C} \ar[r]_H \ar[d] & \mathcal{C}' \ar[d] \\ \textit{Finite-}G\textit{-Sets} \ar[r]^h & \textit{Finite-}G'\textit{-Sets} }\] The map \(h\) is independent of \(t\) up to an inner automorphism of \(G\). Conversely, given a continuous homomorphism \(h : G' \to G\) there is an exact functor \(H : \mathcal{C} \to \mathcal{C}'\) and an isomorphism \(t\) recovering \(h\) as above.
Proof
By Proposition 0BN4 and Lemma 0BMU we may assume \(\mathcal{C} = \textit{Finite-}G\textit{-Sets}\) and \(F\) is the forgetful functor and similarly for \(\mathcal{C}'\). Thus the existence of \(t\) follows from Lemma 0BMV. The map \(h\) comes from transport of structure via \(t\). The commutativity of the diagram is obvious. Uniqueness of \(h\) up to inner conjugation by an element of \(G\) comes from the fact that the choice of \(t\) is unique up to an element of \(G\). The final statement is straightforward.
Functors and homomorphisms
Let \((\mathcal{C}, F)\), \((\mathcal{C}', F')\), \((\mathcal{C}'', F'')\) be Galois categories. Set \(G = \text{Aut}(F)\), \(G' = \text{Aut}(F')\), and \(G'' = \text{Aut}(F'')\). Let \(H : \mathcal{C} \to \mathcal{C}'\) and \(H' : \mathcal{C}' \to \mathcal{C}''\) be exact functors. Let \(h : G' \to G\) and \(h' : G'' \to G'\) be the corresponding continuous homomorphism as in Lemma 0BN5. In this section we consider the corresponding \(2\)-commutative diagram [0BTR]\[\begin{equation} \vcenter{ \xymatrix{ \mathcal{C} \ar[r]_H \ar[d] & \mathcal{C}' \ar[r]_{H'} \ar[d] & \mathcal{C}'' \ar[d] \\ \textit{Finite-}G\textit{-Sets} \ar[r]^h & \textit{Finite-}G'\textit{-Sets} \ar[r]^{h'} & \textit{Finite-}G''\textit{-Sets} } } \end{equation}\] and we relate exactness properties of the sequence \(1 \to G'' \to G' \to G \to 1\) to properties of the functors \(H\) and \(H'\).
Lemma
In diagram (0BTR) the following are equivalent
\(h : G' \to G\) is surjective,
\(H : \mathcal{C} \to \mathcal{C}'\) is fully faithful,
if \(X \in \Ob(\mathcal{C})\) is connected, then \(H(X)\) is connected,
if \(X \in \Ob(\mathcal{C})\) is connected and there is a morphism \(*' \to H(X)\) in \(\mathcal{C}'\), then there is a morphism \(* \to X\), and
for any object \(X\) of \(\mathcal{C}\) the map \(\Mor_\mathcal{C}(*, X) \to \Mor_{\mathcal{C}'}(*', H(X))\) is bijective.
Here \(*\) and \(*'\) are final objects of \(\mathcal{C}\) and \(\mathcal{C}'\).
Proof
The implications (5) \(\Rightarrow\) (4) and (2) \(\Rightarrow\) (5) are clear.
Assume (3). Let \(X\) be a connected object of \(\mathcal{C}\) and let \(*' \to H(X)\) be a morphism. Since \(H(X)\) is connected by (3) we see that \(*' \to H(X)\) is an isomorphism. Hence the \(G'\)-set corresponding to \(H(X)\) has exactly one element, which means the \(G\)-set corresponding to \(X\) has one element which means \(X\) is isomorphic to the final object of \(\mathcal{C}\), in particular there is a map \(* \to X\). In this way we see that (3) \(\Rightarrow\) (4).
If (1) is true, then the functor \(\textit{Finite-}G\textit{-Sets} \to \textit{Finite-}G'\textit{-Sets}\) is fully faithful: in this case a map of \(G\)-sets commutes with the action of \(G\) if and only if it commutes with the action of \(G'\). Thus (1) \(\Rightarrow\) (2).
If (1) is true, then for a \(G\)-set \(X\) the \(G\)-orbits and \(G'\)-orbits agree. Thus (1) \(\Rightarrow\) (3).
To finish the proof it suffices to show that (4) implies (1). If (1) is false, i.e., if \(h\) is not surjective, then there is an open subgroup \(U \subset G\) containing \(h(G')\) which is not equal to \(G\). Then the finite \(G\)-set \(M = G/U\) has a transitive action but \(G'\) has a fixed point. The object \(X\) of \(\mathcal{C}\) corresponding to \(M\) would contradict (3). In this way we see that (3) \(\Rightarrow\) (1) and the proof is complete.
Lemma
In diagram (0BTR) the following are equivalent
\(h \circ h'\) is trivial, and
the image of \(H' \circ H\) consists of objects isomorphic to finite coproducts of final objects.
Proof
We may replace \(H\) and \(H'\) by the canonical functors \(\textit{Finite-}G\textit{-Sets} \to \textit{Finite-}G'\textit{-Sets} \to \textit{Finite-}G''\textit{-Sets}\) determined by \(h\) and \(h'\). Then we are saying that the action of \(G''\) on every \(G\)-set is trivial if and only if the homomorphism \(G'' \to G\) is trivial. This is clear.
Lemma
In diagram (0BTR) the following are equivalent
the sequence \(G'' \xrightarrow{h'} G' \xrightarrow{h} G \to 1\) is exact in the following sense: \(h\) is surjective, \(h \circ h'\) is trivial, and \(\Ker(h)\) is the smallest closed normal subgroup containing \(\Im(h')\),
\(H\) is fully faithful and an object \(X'\) of \(\mathcal{C}'\) is in the essential image of \(H\) if and only if \(H'(X')\) is isomorphic to a finite coproduct of final objects, and
\(H\) is fully faithful, \(H \circ H'\) sends every object to a finite coproduct of final objects, and for an object \(X'\) of \(\mathcal{C}'\) such that \(H'(X')\) is a finite coproduct of final objects there exists an object \(X\) of \(\mathcal{C}\) and an epimorphism \(H(X) \to X'\).
Proof
By Lemmas 0BN6 and 0BS8 we may assume that \(H\) is fully faithful, \(h\) is surjective, \(H' \circ H\) maps objects to disjoint unions of the final object, and \(h \circ h'\) is trivial. Let \(N \subset G'\) be the smallest closed normal subgroup containing the image of \(h'\). It is clear that \(N \subset \Ker(h)\). We may assume the functors \(H\) and \(H'\) are the canonical functors \(\textit{Finite-}G\textit{-Sets} \to \textit{Finite-}G'\textit{-Sets} \to \textit{Finite-}G''\textit{-Sets}\) determined by \(h\) and \(h'\).
Suppose that (2) holds. This means that for a finite \(G'\)-set \(X'\) such that \(G''\) acts trivially, the action of \(G'\) factors through \(G\). Apply this to \(X' = G'/U'N\) where \(U'\) is a small open subgroup of \(G'\). Then we see that \(\Ker(h) \subset U'N\) for all \(U'\). Since \(N\) is closed this implies \(\Ker(h) \subset N\), i.e., (1) holds.
Suppose that (1) holds. This means that \(N = \Ker(h)\). Let \(X'\) be a finite \(G'\)-set such that \(G''\) acts trivially. This means that \(\Ker(G' \to \text{Aut}(X'))\) is a closed normal subgroup containing \(\Im(h')\). Hence \(N = \Ker(h)\) is contained in it and the \(G'\)-action on \(X'\) factors through \(G\), i.e., (2) holds.
Suppose that (3) holds. This means that for a finite \(G'\)-set \(X'\) such that \(G''\) acts trivially, there is a surjection of \(G'\)-sets \(X \to X'\) where \(X\) is a \(G\)-set. Clearly this means the action of \(G'\) on \(X'\) factors through \(G\), i.e., (2) holds.
The implication (2) \(\Rightarrow\) (3) is immediate. This finishes the proof.
Lemma
In diagram (0BTR) the following are equivalent
\(h'\) is injective, and
for every connected object \(X''\) of \(\mathcal{C}''\) there exists an object \(X'\) of \(\mathcal{C}'\) and a diagram \[X'' \leftarrow Y'' \rightarrow H(X')\] in \(\mathcal{C}''\) where \(Y'' \to X''\) is an epimorphism and \(Y'' \to H(X')\) is a monomorphism.
Proof
We may replace \(H'\) by the corresponding functor between the categories of finite \(G'\)-sets and finite \(G''\)-sets.
Assume \(h' : G'' \to G'\) is injective. Let \(H'' \subset G''\) be an open subgroup. Since the topology on \(G''\) is the induced topology from \(G'\) there exists an open subgroup \(H' \subset G'\) such that \((h')^{-1}(H') \subset H''\). Then the desired diagram is \[G''/H'' \leftarrow G''/(h')^{-1}(H') \rightarrow G'/H'\] Conversely, assume (2) holds for the functor \(\textit{Finite-}G'\textit{-Sets} \to \textit{Finite-}G''\textit{-Sets}\). Let \(g'' \in \Ker(h')\). Pick any open subgroup \(H'' \subset G''\). By assumption there exists a finite \(G'\)-set \(X'\) and a diagram \[G''/H'' \leftarrow Y'' \rightarrow X'\] of \(G''\)-sets with the left arrow surjective and the right arrow injective. Since \(g''\) is in the kernel of \(h'\) we see that \(g''\) acts trivially on \(X'\). Hence \(g''\) acts trivially on \(Y''\) and hence trivially on \(G''/H''\). Thus \(g'' \in H''\). As this holds for all open subgroups we conclude that \(g''\) is the identity element as desired.
Lemma
In diagram (0BTR) the following are equivalent
the image of \(h'\) is normal, and
for every connected object \(X'\) of \(\mathcal{C}'\) such that there is a morphism from the final object of \(\mathcal{C}''\) to \(H'(X')\) we have that \(H'(X')\) is isomorphic to a finite coproduct of final objects.
Proof
This translates into the following statement for the continuous group homomorphism \(h' : G'' \to G'\): the image of \(h'\) is normal if and only if every open subgroup \(U' \subset G'\) which contains \(h'(G'')\) also contains every conjugate of \(h'(G'')\). The result follows easily from this; some details omitted.
Finite étale morphisms
In this section we prove enough basic results on finite étale morphisms to be able to construct the étale fundamental group.
Let \(X\) be a scheme. We will use the notation \(\textit{F\'Et}_X\) to denote the category of schemes finite and étale over \(X\). Thus
an object of \(\textit{F\'Et}_X\) is a finite étale morphism \(Y \to X\) with target \(X\), and
a morphism in \(\textit{F\'Et}_X\) from \(Y \to X\) to \(Y' \to X\) is a morphism \(Y \to Y'\) making the diagram \[\xymatrix{ Y \ar[rr] \ar[rd] & & Y' \ar[ld] \\ & X }\] commute.
We will often call an object of \(\textit{F\'Et}_X\) a finite étale cover of \(X\) (even if \(Y\) is empty). It turns out that there is a stack \(p : \textit{F\'Et} \to \Sch\) over the category of schemes whose fibre over \(X\) is the category \(\textit{F\'Et}_X\) just defined. See Examples of Stacks, Section 0BLY.
Example
Let \(k\) be an algebraically closed field and \(X = \Spec(k)\). In this case \(\textit{F\'Et}_X\) is equivalent to the category of finite sets. This works more generally when \(k\) is separably algebraically closed. The reason is that a scheme étale over \(k\) is the disjoint union of spectra of fields finite separable over \(k\), see Morphisms, Lemma 02GL.
Lemma
Let \(X\) be a scheme. The category \(\textit{F\'Et}_X\) has finite limits and finite colimits and for any morphism \(X' \to X\) the base change functor \(\textit{F\'Et}_X \to \textit{F\'Et}_{X'}\) is exact.
Proof
Finite limits and left exactness. By Categories, Lemma 002O it suffices to show that \(\textit{F\'Et}_X\) has a final object and fibred products. This is clear because the category of all schemes over \(X\) has a final object (namely \(X\)) and fibred products. Also, fibred products of schemes finite étale over \(X\) are finite étale over \(X\). Moreover, it is clear that base change commutes with these operations and hence base change is left exact (Categories, Lemma 0035).
Finite colimits and right exactness. By Categories, Lemma 002Q it suffices to show that \(\textit{F\'Et}_X\) has finite coproducts and coequalizers. Finite coproducts are given by disjoint unions (the empty coproduct is the empty scheme). Let \(a, b : Z \to Y\) be two morphisms of \(\textit{F\'Et}_X\). Since \(Z \to X\) and \(Y \to X\) are finite étale we can write \(Z = \underline{\Spec}(\mathcal{C})\) and \(Y = \underline{\Spec}(\mathcal{B})\) for some finite locally free \(\mathcal{O}_X\)-algebras \(\mathcal{C}\) and \(\mathcal{B}\). The morphisms \(a, b\) induce two maps \(a^\sharp, b^\sharp : \mathcal{B} \to \mathcal{C}\). Let \(\mathcal{A} = \text{Eq}(a^\sharp, b^\sharp)\) be their equalizer. If \[\underline{\Spec}(\mathcal{A}) \longrightarrow X\] is finite étale, then it is clear that this is the coequalizer (after all we can write any object of \(\textit{F\'Et}_X\) as the relative spectrum of a sheaf of \(\mathcal{O}_X\)-algebras). This we may do after replacing \(X\) by the members of an étale covering (Descent, Lemmas 02LA and 02VN). Thus by Étale Morphisms, Lemma 04HN we may assume that \(Y = \coprod_{i = 1, \ldots, n} X\) and \(Z = \coprod_{j = 1, \ldots, m} X\). Then \[\mathcal{C} = \prod\nolimits_{1 \leq j \leq m} \mathcal{O}_X \quad\text{and}\quad \mathcal{B} = \prod\nolimits_{1 \leq i \leq n} \mathcal{O}_X\] After a further replacement by the members of an open covering we may assume that \(a, b\) correspond to maps \(a_s, b_s : \{1, \ldots, m\} \to \{1, \ldots, n\}\), i.e., the summand \(X\) of \(Z\) corresponding to the index \(j\) maps into the summand \(X\) of \(Y\) corresponding to the index \(a_s(j)\), resp. \(b_s(j)\) under the morphism \(a\), resp. \(b\). Let \(\{1, \ldots, n\} \to T\) be the coequalizer of \(a_s, b_s\). Then we see that \[\mathcal{A} = \prod\nolimits_{t \in T} \mathcal{O}_X\] whose spectrum is certainly finite étale over \(X\). We omit the verification that this is compatible with base change. Thus base change is a right exact functor.
Remark
Let \(X\) be a scheme. Consider the natural functors \(F_1 : \textit{F\'Et}_X \to \Sch\) and \(F_2 : \textit{F\'Et}_X \to \Sch/X\). Then
The functors \(F_1\) and \(F_2\) commute with finite colimits.
The functor \(F_2\) commutes with finite limits,
The functor \(F_1\) commutes with connected finite limits, i.e., with equalizers and fibre products.
The results on limits are immediate from the discussion in the proof of Lemma 0BN9 and Categories, Lemma 002T. It is clear that \(F_1\) and \(F_2\) commute with finite coproducts. By the dual of Categories, Lemma 0035 we need to show that \(F_1\) and \(F_2\) commute with coequalizers. In the proof of Lemma 0BN9 we saw that coequalizers in \(\textit{F\'Et}_X\) look étale locally like this \[\xymatrix{ \coprod_{j \in J} U \ar@<1ex>[r]^a \ar@<-1ex>[r]_b & \coprod_{i \in I} U \ar[r] & \coprod_{t \in \text{Coeq}(a, b)} U }\] which is certainly a coequalizer in the category of schemes. Hence the statement follows from the fact that being a coequalizer is fpqc local as formulated precisely in Descent, Lemma 0BMN.
Lemma
Let \(X\) be a scheme. Given \(U, V\) finite étale over \(X\) there exists a scheme \(W\) finite étale over \(X\) such that \[\Mor_X(X, W) = \Mor_X(U, V)\] and such that the same remains true after any base change.
Proof
By More on Morphisms, Lemma 0BL5 there exists a scheme \(W\) representing \(\mathit{Mor}_X(U, V)\). (Use that an étale morphism is locally quasi-finite by Morphisms, Lemmas 03WS and that a finite morphism is separated.) This scheme clearly satisfies the formula after any base change. To finish the proof we have to show that \(W \to X\) is finite étale. This we may do after replacing \(X\) by the members of an étale covering (Descent, Lemmas 02LA and 02KU). Thus by Étale Morphisms, Lemma 04HN we may assume that \(U = \coprod_{i = 1, \ldots, n} X\) and \(V = \coprod_{j = 1, \ldots, m} X\). In this case \(W = \coprod_{\alpha : \{1, \ldots, n\} \to \{1, \ldots, m\}} X\) by inspection (details omitted) and the proof is complete.
Let \(X\) be a scheme. A geometric point of \(X\) is a morphism \(\Spec(k) \to X\) where \(k\) is algebraically closed. Such a point is usually denoted \(\overline{x}\), i.e., by an overlined small case letter. We often use \(\overline{x}\) to denote the scheme \(\Spec(k)\) as well as the morphism, and we use \(\kappa(\overline{x})\) to denote \(k\). We say \(\overline{x}\) lies over \(x\) to indicate that \(x \in X\) is the image of \(\overline{x}\). We will discuss this further in Étale Cohomology, Section 03PN. Given \(\overline{x}\) and an étale morphism \(U \to X\) we can consider \[|U_{\overline{x}}| : \text{the underlying set of points of the scheme }U_{\overline{x}} = U \times_X \overline{x}\] Since \(U_{\overline{x}}\) as a scheme over \(\overline{x}\) is a disjoint union of copies of \(\overline{x}\) (Morphisms, Lemma 02GL) we can also describe this set as \[|U_{\overline{x}}| = \left\{ \begin{matrix} \text{commutative} \\ \text{diagrams} \end{matrix} \vcenter{ \xymatrix{ \overline{x} \ar[rd]_{\overline{x}} \ar[r]_{\overline{u}} & U \ar[d] \\ & X } } \right\}\] The assignment \(U \mapsto |U_{\overline{x}}|\) is a functor which is often denoted \(F_{\overline{x}}\).
Lemma
Let \(X\) be a connected scheme. Let \(\overline{x}\) be a geometric point. The functor \[F_{\overline{x}} : \textit{F\'Et}_X \longrightarrow \textit{Sets},\quad Y \longmapsto |Y_{\overline{x}}|\] defines a Galois category (Definition 0BMY).
Proof
After identifying \(\textit{F\'Et}_{\overline{x}}\) with the category of finite sets (Example 0BN8) we see that our functor \(F_{\overline{x}}\) is nothing but the base change functor for the morphism \(\overline{x} \to X\). Thus we see that \(\textit{F\'Et}_X\) has finite limits and finite colimits and that \(F_{\overline{x}}\) is exact by Lemma 0BN9. We will also use that finite limits in \(\textit{F\'Et}_X\) agree with the corresponding finite limits in the category of schemes over \(X\), see Remark 0BNA.
If \(Y' \to Y\) is a monomorphism in \(\textit{F\'Et}_X\) then we see that \(Y' \to Y' \times_Y Y'\) is an isomorphism, and hence \(Y' \to Y\) is a monomorphism of schemes. It follows that \(Y' \to Y\) is an open immersion (Étale Morphisms, Theorem 025G). Since \(Y'\) is finite over \(X\) and \(Y\) separated over \(X\), the morphism \(Y' \to Y\) is finite (Morphisms, Lemma 035D), hence closed (Morphisms, Lemma 01WN), hence it is the inclusion of an open and closed subscheme of \(Y\). It follows that \(Y\) is a connected object of the category \(\textit{F\'Et}_X\) (as in Definition 0BMY) if and only if \(Y\) is connected as a scheme. Then it follows from Topology, Lemma 07VB that \(Y\) is a finite coproduct of its connected components both as a scheme and in the sense of Definition 0BMY.
Let \(Y \to Z\) be a morphism in \(\textit{F\'Et}_X\) which induces a bijection \(F_{\overline{x}}(Y) \to F_{\overline{x}}(Z)\). We have to show that \(Y \to Z\) is an isomorphism. By the above we may assume \(Z\) is connected. Since \(Y \to Z\) is finite étale and hence finite locally free it suffices to show that \(Y \to Z\) is finite locally free of degree \(1\). This is true in a neighbourhood of any point of \(Z\) lying over \(\overline{x}\) and since \(Z\) is connected and the degree is locally constant we conclude.
Fundamental groups
In this section we define Grothendieck’s algebraic fundamental group. The following definition makes sense thanks to Lemma 0BNB.
Definition
Let \(X\) be a connected scheme. Let \(\overline{x}\) be a geometric point of \(X\). The fundamental group of \(X\) with base point \(\overline{x}\) is the group \[\pi_1(X, \overline{x}) = \text{Aut}(F_{\overline{x}})\] of automorphisms of the fibre functor \(F_{\overline{x}} : \textit{F\'Et}_X \to \textit{Sets}\) endowed with its canonical profinite topology from Lemma 0BMR.
Combining the above with the material from Section 0BMQ we obtain the following theorem.
Theorem
Let \(X\) be a connected scheme. Let \(\overline{x}\) be a geometric point of \(X\).
The fibre functor \(F_{\overline{x}}\) defines an equivalence of categories \[\textit{F\'Et}_X \longrightarrow \textit{Finite-}\pi_1(X, \overline{x})\textit{-Sets}\]
Given a second geometric point \(\overline{x}'\) of \(X\) there exists an isomorphism \(t : F_{\overline{x}} \to F_{\overline{x}'}\). This gives an isomorphism \(\pi_1(X, \overline{x}) \to \pi_1(X, \overline{x}')\) compatible with the equivalences in (1). This isomorphism is independent of \(t\) up to inner conjugation.
Given a morphism \(f : X \to Y\) of connected schemes denote \(\overline{y} = f \circ \overline{x}\). There is a canonical continuous homomorphism \[f_* : \pi_1(X, \overline{x}) \to \pi_1(Y, \overline{y})\] such that the diagram \[\xymatrix{ \textit{F\'Et}_Y \ar[r]_{\text{base change}} \ar[d]_{F_{\overline{y}}} & \textit{F\'Et}_X \ar[d]^{F_{\overline{x}}} \\ \textit{Finite-}\pi_1(Y, \overline{y})\textit{-Sets} \ar[r]^{f_*} & \textit{Finite-}\pi_1(X, \overline{x})\textit{-Sets} }\] is commutative.
Proof
Part (1) follows from Lemma 0BNB and Proposition 0BN4. Part (2) is a special case of Lemma 0BN5. For part (3) observe that the diagram \[\xymatrix{ \textit{F\'Et}_Y \ar[r] \ar[d]_{F_{\overline{y}}} & \textit{F\'Et}_X \ar[d]^{F_{\overline{x}}} \\ \textit{Sets} \ar@{=}[r] & \textit{Sets} }\] is commutative (actually commutative, not just \(2\)-commutative) because \(\overline{y} = f \circ \overline{x}\). Hence we can apply Lemma 0BN5 with the implied transformation of functors to get (3).
Lemma
Let \(K\) be a field and set \(X = \Spec(K)\). Let \(\overline{K}\) be an algebraic closure and denote \(\overline{x} : \Spec(\overline{K}) \to X\) the corresponding geometric point. Let \(K^{sep} \subset \overline{K}\) be the separable algebraic closure.
The functor of Lemma 03QR induces an equivalence \[\textit{F\'Et}_X \longrightarrow \textit{Finite-}\text{Gal}(K^{sep}/K)\textit{-Sets}.\] compatible with \(F_{\overline{x}}\) and the functor \(\textit{Finite-}\text{Gal}(K^{sep}/K)\textit{-Sets} \to \textit{Sets}\).
This induces a canonical isomorphism \[\text{Gal}(K^{sep}/K) \longrightarrow \pi_1(X, \overline{x})\] of profinite topological groups.
Proof
The functor of Lemma 03QR is the same as the functor \(F_{\overline{x}}\) because for any \(Y\) étale over \(X\) we have \[\Mor_X(\Spec(\overline{K}), Y) = \Mor_X(\Spec(K^{sep}), Y)\] Namely, as seen in the proof of Lemma 03QR we have \(Y = \coprod_{i \in I} \Spec(L_i)\) with \(L_i/K\) finite separable over \(K\). Hence any \(K\)-algebra homomorphism \(L_i \to \overline{K}\) factors through \(K^{sep}\). Also, note that \(F_{\overline{x}}(Y)\) is finite if and only if \(I\) is finite if and only if \(Y \to X\) is finite étale. This proves (1).
Part (2) is a formal consequence of (1), Lemma 0BN5, and Lemma 0BMU. (Please also see the remark below.)
Remark
In the situation of Lemma 0BNE let us give a more explicit construction of the isomorphism \(\text{Gal}(K^{sep}/K) \to \pi_1(X, \overline{x}) = \text{Aut}(F_{\overline{x}})\). Observe that \(\text{Gal}(K^{sep}/K) = \text{Aut}(\overline{K}/K)\) as \(\overline{K}\) is the perfection of \(K^{sep}\). Since \(F_{\overline{x}}(Y) = \Mor_X(\Spec(\overline{K}), Y)\) we may consider the map \[\text{Aut}(\overline{K}/K) \times F_{\overline{x}}(Y) \to F_{\overline{x}}(Y), \quad (\sigma, \overline{y}) \mapsto \sigma \cdot \overline{y} = \overline{y} \circ \Spec(\sigma)\] This is an action because \[\sigma\tau \cdot \overline{y} = \overline{y} \circ \Spec(\sigma\tau) = \overline{y} \circ \Spec(\tau) \circ \Spec(\sigma) = \sigma \cdot (\tau \cdot \overline{y})\] The action is functorial in \(Y \in \textit{F\'Et}_X\) and we obtain the desired map.
Galois covers of connected schemes
Let \(X\) be a connected scheme with geometric point \(\overline{x}\). Since \(F_{\overline{x}} : \textit{F\'Et}_X \to \textit{Sets}\) is a Galois category (Lemma 0BNB) the material in Section 0BMQ applies. In this section we explicitly transfer some of the terminology and results to the setting of schemes and finite étale morphisms.
We will say a finite étale morphism \(Y \to X\) is a Galois cover if \(Y\) defines a Galois object of \(\textit{F\'Et}_X\). For a finite étale morphism \(Y \to X\) with \(G = \text{Aut}_X(Y)\) the following are equivalent
\(Y\) is a Galois cover of \(X\),
\(Y\) is connected and \(|G|\) is equal to the degree of \(Y \to X\),
\(Y\) is connected and \(G\) acts transitively on \(F_{\overline{x}}(Y)\), and
\(Y\) is connected and \(G\) acts simply transitively on \(F_{\overline{x}}(Y)\).
This follows immediately from the discussion in Section 0BMQ.
For any finite étale morphism \(f : Y \to X\) with \(Y\) connected, there is a finite étale Galois cover \(Y' \to X\) which dominates \(Y\) (Lemma 0BN2).
The Galois objects of \(\textit{F\'Et}_X\) correspond, via the equivalence \[F_{\overline{x}} : \textit{F\'Et}_X \to \textit{Finite-}\pi_1(X, \overline{x})\textit{-Sets}\] of Theorem 0BND, with the finite \(\pi_1(X, \overline{x})\textit{-Sets}\) of the form \(G = \pi_1(X, \overline{x})/H\) where \(H\) is a normal open subgroup. Equivalently, if \(G\) is a finite group and \(\pi_1(X, \overline{x}) \to G\) is a continuous surjection, then \(G\) viewed as a \(\pi_1(X, \overline{x})\)-set corresponds to a Galois covering.
If \(Y_i \to X\), \(i = 1, 2\) are finite étale Galois covers with Galois groups \(G_i\), then there exists a finite étale Galois cover \(Y \to X\) whose Galois group is a subgroup of \(G_1 \times G_2\). Namely, take the corresponding continuous homomorphisms \(\pi_1(X, \overline{x}) \to G_i\) and let \(G\) be the image of the induced continuous homomorphism \(\pi_1(X, \overline{x}) \to G_1 \times G_2\).
Topological invariance of the fundamental group
The main result of this section is that a universal homeomorphism of connected schemes induces an isomorphism on fundamental groups. See Proposition 0BQN.
Instead of directly proving two schemes have the same fundamental group, we often prove that their categories of finite étale coverings are the same. This of course implies that their fundamental groups are equal provided they are connected.
Lemma
Let \(f : X \to Y\) be a morphism of quasi-compact and quasi-separated schemes such that the base change functor \(\textit{F\'Et}_Y \to \textit{F\'Et}_X\) is an equivalence of categories. In this case
\(f\) induces a homeomorphism \(\pi_0(X) \to \pi_0(Y)\),
if \(X\) or equivalently \(Y\) is connected, then \(\pi_1(X, \overline{x}) = \pi_1(Y, \overline{y})\).
Proof
Let \(Y = Y_0 \amalg Y_1\) be a decomposition into nonempty open and closed subschemes. We claim that \(f(X)\) meets both \(Y_i\). Namely, if not, say \(f(X) \subset Y_1\), then we can consider the finite étale morphism \(V = Y_1 \to Y\). This is not an isomorphism but \(V \times_Y X \to X\) is an isomorphism, which is a contradiction.
Suppose that \(X = X_0 \amalg X_1\) is a decomposition into open and closed subschemes. Consider the finite étale morphism \(U = X_1 \to X\). Then \(U = X \times_Y V\) for some finite étale morphism \(V \to Y\). The degree of the morphism \(V \to Y\) is locally constant, hence we obtain a decomposition \(Y = \coprod_{d \geq 0} Y_d\) into open and closed subschemes such that \(V \to Y\) has degree \(d\) over \(Y_d\). Since \(f^{-1}(Y_d) = \emptyset\) for \(d > 1\) we conclude that \(Y_d = \emptyset\) for \(d > 1\) by the above. And we conclude that \(f^{-1}(Y_i) = X_i\) for \(i = 0, 1\).
It follows that \(f^{-1}\) induces a bijection between the set of open and closed subsets of \(Y\) and the set of open and closed subsets of \(X\). Note that \(X\) and \(Y\) are spectral spaces, see Properties, Lemma 094L. By Topology, Lemma 005F the lattice of open and closed subsets of a spectral space determines the set of connected components. Hence \(\pi_0(X) \to \pi_0(Y)\) is bijective. Since \(\pi_0(X)\) and \(\pi_0(Y)\) are profinite spaces (Topology, Lemma 0900) we conclude that \(\pi_0(X) \to \pi_0(Y)\) is a homeomorphism by Topology, Lemma 08YE. This proves (1). Part (2) is immediate.
The following lemma tells us that the fundamental group of a henselian pair is the fundamental group of the closed subset.
Lemma
Let \((A, I)\) be a henselian pair. Set \(X = \Spec(A)\) and \(Z = \Spec(A/I)\). The functor \[\textit{F\'Et}_X \longrightarrow \textit{F\'Et}_Z,\quad U \longmapsto U \times_X Z\] is an equivalence of categories.
Proof
This is a translation of More on Algebra, Lemma 09ZL.
The following lemma tells us that the fundamental group of a thickening is the same as the fundamental group of the original. We will use this in the proof of the strong proposition concerning universal homeomorphisms below.
Lemma
Let \(X \subset X'\) be a thickening of schemes. The functor \[\textit{F\'Et}_{X'} \longrightarrow \textit{F\'Et}_X,\quad U' \longmapsto U' \times_{X'} X\] is an equivalence of categories.
Proof
For a discussion of thickenings see More on Morphisms, Section 04EW. Let \(U' \to X'\) be an étale morphism such that \(U = U' \times_{X'} X \to X\) is finite étale. Then \(U' \to X'\) is finite étale as well. This follows for example from More on Morphisms, Lemma 0BPG. Now, if \(X \subset X'\) is a finite order thickening then this remark combined with Étale Morphisms, Theorem 039R proves the lemma. Below we will prove the lemma for general thickenings, but we suggest the reader skip the proof.
Let \(X' = \bigcup X_i'\) be an affine open covering. Set \(X_i = X \times_{X'} X_i'\), \(X_{ij}' = X'_i \cap X'_j\), \(X_{ij} = X \times_{X'} X_{ij}'\), \(X_{ijk}' = X'_i \cap X'_j \cap X'_k\), \(X_{ijk} = X \times_{X'} X_{ijk}'\). Suppose that we can prove the theorem for each of the thickenings \(X_i \subset X'_i\), \(X_{ij} \subset X_{ij}'\), and \(X_{ijk} \subset X_{ijk}'\). Then the result follows for \(X \subset X'\) by relative glueing of schemes, see Constructions, Section 01LG. Observe that the schemes \(X_i'\), \(X_{ij}'\), \(X_{ijk}'\) are each separated as open subschemes of affine schemes. Repeating the argument one more time we reduce to the case where the schemes \(X'_i\), \(X_{ij}'\), \(X_{ijk}'\) are affine.
In the affine case we have \(X' = \Spec(A')\) and \(X = \Spec(A'/I')\) where \(I'\) is a locally nilpotent ideal. Then \((A', I')\) is a henselian pair (More on Algebra, Lemma 0ALI) and the result follows from Lemma 09ZS (which is much easier in this case).
The “correct” way to prove the following proposition would be to deduce it from the invariance of the étale site, see Étale Cohomology, Theorem 04DZ.
Proposition
Let \(f : X \to Y\) be a universal homeomorphism of schemes. Then \[\textit{F\'Et}_Y \longrightarrow \textit{F\'Et}_X,\quad V \longmapsto V \times_Y X\] is an equivalence. Thus if \(X\) and \(Y\) are connected, then \(f\) induces an isomorphism \(\pi_1(X, \overline{x}) \to \pi_1(Y, \overline{y})\) of fundamental groups.
Proof
Recall that a universal homeomorphism is the same thing as an integral, universally injective, surjective morphism, see Morphisms, Lemma 04DF. In particular, the diagonal \(\Delta : X \to X \times_Y X\) is a thickening by Morphisms, Lemma 01S4. Thus by Lemma 0BQB we see that given a finite étale morphism \(U \to X\) there is a unique isomorphism \[\varphi : U \times_Y X \to X \times_Y U\] of schemes finite étale over \(X \times_Y X\) which pulls back under \(\Delta\) to \(\text{id} : U \to U\) over \(X\). Since \(X \to X \times_Y X \times_Y X\) is a thickening as well (it is bijective and a closed immersion) we conclude that \((U, \varphi)\) is a descent datum relative to \(X/Y\). By Étale Morphisms, Proposition 0BTP we conclude that \(U = X \times_Y V\) for some \(V \to Y\) quasi-compact, separated, and étale. We omit the proof that \(V \to Y\) is finite (hints: the morphism \(U \to V\) is surjective and \(U \to Y\) is integral). We conclude that \(\textit{F\'Et}_Y \to \textit{F\'Et}_X\) is essentially surjective.
Arguing in the same manner as above we see that given \(V_1 \to Y\) and \(V_2 \to Y\) in \(\textit{F\'Et}_Y\) any morphism \(a : X \times_Y V_1 \to X \times_Y V_2\) over \(X\) is compatible with the canonical descent data. Thus \(a\) descends to a morphism \(V_1 \to V_2\) over \(Y\) by Étale Morphisms, Lemma 0BTL.
Comparison over the complex numbers
Let \(X\) be a scheme of finite type over \(\mathbf{C}\). We write \(X^{an}\) for the associated complex analytic space and \(X(\mathbf{C})\) for its underlying set of complex points with the classical topology. Analytification sends a finite étale morphism of schemes to a finite covering in the analytic topology. The converse is the content of the following theorem.
Theorem
Let \(X\) be a scheme of finite type over \(\mathbf{C}\). The analytification functor is an equivalence between
schemes finite étale over \(X\), and
finite étale analytic spaces over \(X^{an}\).
Equivalently, finite étale covers of \(X\) are the same as finite topological covering spaces of \(X(\mathbf{C})\).
Proof
This is the Riemann existence theorem; see [SGA1, Exposé XII, Théorème 5.1]. In particular, no normality hypothesis on \(X\) is needed. The passage from a reduced scheme to an arbitrary scheme is also visible algebraically: \(X_{red} \to X\) is a universal homeomorphism, so Proposition 0BQN identifies their categories of finite étale covers, while \(X_{red}(\mathbf{C}) = X(\mathbf{C})\).
Lemma
Let \(X\) be a connected scheme of finite type over \(\mathbf{C}\), let \(x \in X(\mathbf{C})\), and let \(\overline{x}\) be the corresponding geometric point. There is a canonical isomorphism \[\widehat{\pi_1^{top}(X(\mathbf{C}), x)} \longrightarrow \pi_1(X, \overline{x}),\] where the left hand side is the profinite completion of the topological fundamental group.
Proof
Apply Theorem pione-theorem-riemann-existence to the fibre functors at \(x\) and use the classification of finite covering spaces; equivalently, this is [SGA1, Exposé XII, Corollaire 5.2].
Finite étale covers of proper schemes
In this section we show that the fundamental group of a connected proper scheme over a henselian local ring is the same as the fundamental group of its special fibre. We also prove a variant of this result for a henselian pair.
We also show that the fundamental group of a connected proper scheme over an algebraically closed field \(k\) does not change if we replace \(k\) by an algebraically closed extension.
Instead of stating and proving the results in the connected case we prove the results in general and we leave it to the reader to deduce the result for fundamental groups using Lemma 0BQA.
Lemma
Let \(A\) be a henselian local ring. Let \(X\) be a proper scheme over \(A\) with closed fibre \(X_0\). Then the functor \[\textit{F\'Et}_X \to \textit{F\'Et}_{X_0},\quad U \longmapsto U_0 = U \times_X X_0\] is an equivalence of categories.
Proof
The proof given here is an example of applying algebraization and approximation. We proceed in a number of stages.
Essential surjectivity when \(A\) is a complete local Noetherian ring. Let \(X_n = X \times_{\Spec(A)} \Spec(A/\mathfrak m^{n + 1})\). By Étale Morphisms, Theorem 039R the inclusions \[X_0 \to X_1 \to X_2 \to \ldots\] induce equivalence of categories between the category of schemes étale over \(X_0\) and the category of schemes étale over \(X_n\). Moreover, if \(U_n \to X_n\) corresponds to a finite étale morphism \(U_0 \to X_0\), then \(U_n \to X_n\) is finite too, for example by More on Morphisms, Lemma 09ZW. In this case the morphism \(U_0 \to \Spec(A/\mathfrak m)\) is proper as \(X_0\) is proper over \(A/\mathfrak m\). Thus we may apply Grothendieck’s algebraization theorem (in the form of Cohomology of Schemes, Lemma 09ZT) to see that there is a finite morphism \(U \to X\) whose restriction to \(X_0\) recovers \(U_0\). By More on Morphisms, Lemma 0A43 we see that \(U \to X\) is étale at every point of \(U_0\). However, since every point of \(U\) specializes to a point of \(U_0\) (as \(U\) is proper over \(A\)), we conclude that \(U \to X\) is étale. In this way we conclude the functor is essentially surjective.
Fully faithfulness when \(A\) is a complete local Noetherian ring. Let \(U \to X\) and \(V \to X\) be finite étale morphisms and let \(\varphi_0 : U_0 \to V_0\) be a morphism over \(X_0\). Look at the morphism \[\Gamma_{\varphi_0} : U_0 \longrightarrow U_0 \times_{X_0} V_0\] This morphism is both finite étale and a closed immersion. By essential surjectivity applied to \(X = U \times_X V\) we find a finite étale morphism \(W \to U \times_X V\) whose special fibre is isomorphic to \(\Gamma_{\varphi_0}\). Consider the projection \(W \to U\). It is finite étale and an isomorphism over \(U_0\) by construction. By Étale Morphisms, Lemma 04DH \(W \to U\) is an isomorphism in an open neighbourhood of \(U_0\). Thus it is an isomorphism and the composition \(\varphi : U \cong W \to V\) is the desired lift of \(\varphi_0\).
Essential surjectivity when \(A\) is a henselian local Noetherian G-ring. Let \(U_0 \to X_0\) be a finite étale morphism. Let \(A^\wedge\) be the completion of \(A\) with respect to the maximal ideal. Let \(X^\wedge\) be the base change of \(X\) to \(A^\wedge\). By the result above there exists a finite étale morphism \(V \to X^\wedge\) whose special fibre is \(U_0\). Write \(A^\wedge = \colim A_i\) with \(A \to A_i\) of finite type. By Limits, Lemma 01ZM there exists an \(i\) and a finitely presented morphism \(U_i \to X_{A_i}\) whose base change to \(X^\wedge\) is \(V\). After increasing \(i\) we may assume that \(U_i \to X_{A_i}\) is finite and étale (Limits, Lemmas 01ZO and 07RP). Writing \[A_i = A[x_1, \ldots, x_n]/(f_1, \ldots, f_m)\] the ring map \(A_i \to A^\wedge\) can be reinterpreted as a solution \((a_1, \ldots, a_n)\) in \(A^\wedge\) for the system of equations \(f_j = 0\). By Smoothing Ring Maps, Theorem 07QY we can approximate this solution (to order \(11\) for example) by a solution \((b_1, \ldots, b_n)\) in \(A\). Translating back we find an \(A\)-algebra map \(A_i \to A\) which gives the same closed point as the original map \(A_i \to A^\wedge\) (as \(11 > 1\)). The base change \(U \to X\) of \(V \to X_{A_i}\) by this ring map will therefore be a finite étale morphism whose special fibre is isomorphic to \(U_0\).
Fully faithfulness when \(A\) is a henselian local Noetherian G-ring. This can be deduced from essential surjectivity in exactly the same manner as was done in the case that \(A\) is complete Noetherian.
General case. Let \((A, \mathfrak m)\) be a henselian local ring. Set \(S = \Spec(A)\) and denote \(s \in S\) the closed point. By Limits, Lemma 0A0P we can write \(X \to \Spec(A)\) as a cofiltered limit of proper morphisms \(X_i \to S_i\) with \(S_i\) of finite type over \(\mathbf{Z}\). For each \(i\) let \(s_i \in S_i\) be the image of \(s\). Since \(S = \lim S_i\) and \(A = \mathcal{O}_{S, s}\) we have \(A = \colim \mathcal{O}_{S_i, s_i}\). The ring \(A_i = \mathcal{O}_{S_i, s_i}\) is a Noetherian local G-ring (More on Algebra, Proposition 07PX). By More on Algebra, Lemma 0A04 we see that \(A = \colim A_i^h\). By More on Algebra, Lemma 07QR the rings \(A_i^h\) are G-rings. Thus we see that \(A = \colim A_i^h\) and \[X = \lim (X_i \times_{S_i} \Spec(A_i^h))\] as schemes. The category of schemes finite étale over \(X\) is the limit of the category of schemes finite étale over \(X_i \times_{S_i} \Spec(A_i^h)\) (by Limits, Lemmas 01ZM, 01ZO, and 07RP) The same thing is true for schemes finite étale over \(X_0 = \lim (X_i \times_{S_i} s_i)\). Thus we formally deduce the result for \(X / \Spec(A)\) from the result for the \((X_i \times_{S_i} \Spec(A_i^h)) / \Spec(A_i^h)\) which we dealt with above.
Lemma
Let \((A, I)\) be a henselian pair. Let \(X\) be a proper scheme over \(A\). Set \(X_0 = X \times_{\Spec(A)} \Spec(A/I)\). Then the functor \[\textit{F\'Et}_X \to \textit{F\'Et}_{X_0},\quad U \longmapsto U_0 = U \times_X X_0\] is an equivalence of categories.
Proof
The proof of this lemma is exactly the same as the proof of Lemma 0A48.
Essential surjectivity when \(A\) is Noetherian and \(I\)-adically complete. Let \(X_n = X \times_{\Spec(A)} \Spec(A/I^{n + 1})\). By Étale Morphisms, Theorem 039R the inclusions \[X_0 \to X_1 \to X_2 \to \ldots\] induce equivalence of categories between the category of schemes étale over \(X_0\) and the category of schemes étale over \(X_n\). Moreover, if \(U_n \to X_n\) corresponds to a finite étale morphism \(U_0 \to X_0\), then \(U_n \to X_n\) is finite too, for example by More on Morphisms, Lemma 09ZW. In this case the morphism \(U_0 \to \Spec(A/I)\) is proper as \(X_0\) is proper over \(A/I\). Thus we may apply Grothendieck’s algebraization theorem (in the form of Cohomology of Schemes, Lemma 09ZT) to see that there is a finite morphism \(U \to X\) whose restriction to \(X_0\) recovers \(U_0\). By More on Morphisms, Lemma 0A43 we see that \(U \to X\) is étale at every point of \(U_0\). However, since every point of \(U\) specializes to a point of \(U_0\) (as \(U\) is proper over \(A\)), we conclude that \(U \to X\) is étale. In this way we conclude the functor is essentially surjective.
Fully faithfulness when \(A\) is Noetherian and \(I\)-adically complete. Let \(U \to X\) and \(V \to X\) be finite étale morphisms and let \(\varphi_0 : U_0 \to V_0\) be a morphism over \(X_0\). Look at the morphism \[\Gamma_{\varphi_0} : U_0 \longrightarrow U_0 \times_{X_0} V_0\] This morphism is both finite étale and a closed immersion. By essential surjectivity applied to \(X = U \times_X V\) we find a finite étale morphism \(W \to U \times_X V\) whose special fibre is isomorphic to \(\Gamma_{\varphi_0}\). Consider the projection \(W \to U\). It is finite étale and an isomorphism over \(U_0\) by construction. By Étale Morphisms, Lemma 04DH \(W \to U\) is an isomorphism in an open neighbourhood of \(U_0\). Thus it is an isomorphism and the composition \(\varphi : U \cong W \to V\) is the desired lift of \(\varphi_0\).
Essential surjectivity when \((A, I)\) is a henselian pair and \(A\) is a Noetherian G-ring. Let \(U_0 \to X_0\) be a finite étale morphism. Let \(A^\wedge\) be the completion of \(A\) with respect to \(I\). Observe that \(A^\wedge\) is a Noetherian ring which is \(IA^\wedge\)-adically complete, see Algebra, Lemmas 031C and 0316. Let \(X^\wedge\) be the base change of \(X\) to \(A^\wedge\). By the result above there exists a finite étale morphism \(V \to X^\wedge\) whose special fibre is \(U_0\). Write \(A^\wedge = \colim A_i\) with \(A \to A_i\) of finite type. By Limits, Lemma 01ZM there exists an \(i\) and a finitely presented morphism \(U_i \to X_{A_i}\) whose base change to \(X^\wedge\) is \(V\). After increasing \(i\) we may assume that \(U_i \to X_{A_i}\) is finite and étale (Limits, Lemmas 01ZO and 07RP). Writing \[A_i = A[x_1, \ldots, x_n]/(f_1, \ldots, f_m)\] the ring map \(A_i \to A^\wedge\) can be reinterpreted as a solution \((a_1, \ldots, a_n)\) in \(A^\wedge\) for the system of equations \(f_j = 0\). By Smoothing Ring Maps, Lemma 0AH5 we can approximate this solution (to order \(11\) for example) by a solution \((b_1, \ldots, b_n)\) in \(A\). Translating back we find an \(A\)-algebra map \(A_i \to A\) which gives the same closed point as the original map \(A_i \to A^\wedge\) (as \(11 > 1\)). The base change \(U \to X\) of \(V \to X_{A_i}\) by this ring map will therefore be a finite étale morphism whose special fibre is isomorphic to \(U_0\).
Fully faithfulness when \((A, I\) is a henselian pair and \(A\) is a Noetherian G-ring. This can be deduced from essential surjectivity in exactly the same manner as was done in the case that \(A\) is complete Noetherian.
General case. Let \((A, I)\) be a henselian pair. Set \(S = \Spec(A)\) and denote \(S_0 = \Spec(A/I)\). By Limits, Lemma 0A0P we can write \(X \to \Spec(A)\) as a cofiltered limit of proper morphisms \(X_i \to S_i\) with \(S_i\) affine and of finite type over \(\mathbf{Z}\). Write \(S_i = \Spec(A_i)\) and denote \(I_i \subset A_i\) the inverse image of \(I\) by the map \(A_i \to A\). Set \(S_{i, 0} = \Spec(A_i/I_i)\). Since \(S = \lim S_i\) we have \(A = \colim A_i\). Thus we also have \(I = \colim I_i\) and \(A/I = \colim A_i/I_i\). The ring \(A_i\) is a Noetherian G-ring (More on Algebra, Proposition 07PX). Denote \((A_i^h, I_i^h)\) the henselization of the pair \((A_i, I_i)\). By More on Algebra, Lemma 0A04 we see that \(A = \colim A_i^h\). By More on Algebra, Lemma 0AH3 the rings \(A_i^h\) are G-rings. Thus we see that \(A = \colim A_i^h\) and \[X = \lim (X_i \times_{S_i} \Spec(A_i^h))\] as schemes. The category of schemes finite étale over \(X\) is the limit of the category of schemes finite étale over \(X_i \times_{S_i} \Spec(A_i^h)\) (by Limits, Lemmas 01ZM, 01ZO, and 07RP) The same thing is true for schemes finite étale over \(X_0 = \lim (X_i \times_{S_i} S_{i, 0})\). Thus we formally deduce the result for \(X / \Spec(A)\) from the result for the \((X_i \times_{S_i} \Spec(A_i^h)) / \Spec(A_i^h)\) which we dealt with above.
Lemma
Let \(k'/k\) be an extension of algebraically closed fields. Let \(X\) be a proper scheme over \(k\). Then the functor \[U \longmapsto U_{k'}\] is an equivalence of categories between schemes finite étale over \(X\) and schemes finite étale over \(X_{k'}\).
Proof
Let us prove the functor is essentially surjective. Let \(U' \to X_{k'}\) be a finite étale morphism. Write \(k' = \colim A_i\) as a filtered colimit of finite type \(k\)-algebras. By Limits, Lemma 01ZM there exists an \(i\) and a finitely presented morphism \(U_i \to X_{A_i}\) whose base change to \(X_{k'}\) is \(U'\). After increasing \(i\) we may assume that \(U_i \to X_{A_i}\) is finite and étale (Limits, Lemmas 01ZO and 07RP). Since \(k\) is algebraically closed we can find a \(k\)-valued point \(t\) in \(\Spec(A_i)\). Let \(U = (U_i)_t\) be the fibre of \(U_i\) over \(t\). Let \(A_i^h\) be the henselization of \((A_i)_{\mathfrak m}\) where \(\mathfrak m\) is the maximal ideal corresponding to the point \(t\). By Lemma 0A48 we see that \((U_i)_{A_i^h} = U \times \Spec(A_i^h)\) as schemes over \(X_{A_i^h}\). Now since \(A_i^h\) is algebraic over \(A_i\) (see for example discussion in Smoothing Ring Maps, Example 0A1W) and since \(k'\) is algebraically closed we can find a ring map \(A_i^h \to k'\) extending the given inclusion \(A_i \subset k'\). Hence we conclude that \(U'\) is isomorphic to the base change of \(U\). The proof of fully faithfulness is exactly the same.
Local connectedness
In this section we ask when \(\pi_1(U) \to \pi_1(X)\) is surjective for \(U\) a dense open of a scheme \(X\). We will see that this is the case (roughly) when \(U \cap B\) is connected for any small “ball” \(B\) around a point \(x \in X \setminus U\).
Lemma
Let \(f : X \to Y\) be a morphism of schemes. If \(f(X)\) is dense in \(Y\) then the base change functor \(\textit{F\'Et}_Y \to \textit{F\'Et}_X\) is faithful.
Proof
Since the category of finite étale coverings has an internal hom (Lemma 0BL7) it suffices to prove the following: Given \(W\) finite étale over \(Y\) and a morphism \(s : X \to W\) over \(X\) there is at most one section \(t : Y \to W\) such that \(s = t \circ f\). Consider two sections \(t_1, t_2 : Y \to W\) such that \(s = t_1 \circ f = t_2 \circ f\). Since the equalizer of \(t_1\) and \(t_2\) is closed in \(Y\) (Schemes, Lemma 01KM) and since \(f(X)\) is dense in \(Y\) we see that \(t_1\) and \(t_2\) agree on \(Y_{red}\). Then it follows that \(t_1\) and \(t_2\) have the same image which is an open and closed subscheme of \(W\) mapping isomorphically to \(Y\) (Étale Morphisms, Proposition 024T) hence they are equal.
The condition in the following lemma that the punctured spectrum of the strict henselization is connected follows for example from the assumption that the local ring is geometrically unibranch, see More on Algebra, Lemma 06DM. There is a partial converse in Properties, Lemma 0BQ4.
Lemma
Let \((A, \mathfrak m)\) be a local ring. Set \(X = \Spec(A)\) and let \(U = X \setminus \{\mathfrak m\}\). If the punctured spectrum of the strict henselization of \(A\) is connected, then \[\textit{F\'Et}_X \longrightarrow \textit{F\'Et}_U,\quad Y \longmapsto Y \times_X U\] is a fully faithful functor.
Proof
Assume \(A\) is strictly henselian. In this case any finite étale cover \(Y\) of \(X\) is isomorphic to a finite disjoint union of copies of \(X\). Thus it suffices to prove that any morphism \(U \to U \amalg \ldots \amalg U\) over \(U\), extends uniquely to a morphism \(X \to X \amalg \ldots \amalg X\) over \(X\). If \(U\) is connected (in particular nonempty), then this is true.
The general case. Since the category of finite étale coverings has an internal hom (Lemma 0BL7) it suffices to prove the following: Given \(Y\) finite étale over \(X\) any morphism \(s : U \to Y\) over \(X\) extends to a morphism \(t : X \to Y\) over \(X\). Let \(A^{sh}\) be the strict henselization of \(A\) and denote \(X^{sh} = \Spec(A^{sh})\), \(U^{sh} = U \times_X X^{sh}\), \(Y^{sh} = Y \times_X X^{sh}\). By the first paragraph and our assumption on \(A\), we can extend the base change \(s^{sh} : U^{sh} \to Y^{sh}\) of \(s\) to \(t^{sh} : X^{sh} \to Y^{sh}\). Set \(A' = A^{sh} \otimes_A A^{sh}\). Then the two pullbacks \(t'_1, t'_2\) of \(t^{sh}\) to \(X' = \Spec(A')\) are extensions of the pullback \(s'\) of \(s\) to \(U' = U \times_X X'\). As \(A \to A'\) is flat we see that \(U' \subset X'\) is (topologically) dense by going down for \(A \to A'\) (Algebra, Lemma 00HS). Thus \(t'_1 = t'_2\) by Lemma 0BQE. Hence \(t^{sh}\) descends to a morphism \(t : X \to Y\) for example by Descent, Lemma 023Q.
In view of Lemma 0BLQ it is interesting to know when the punctured spectrum of a ring (and of its strict henselization) is connected. There is a famous lemma due to Hartshorne which gives a sufficient condition, see Local Cohomology, Lemma 0BLR.
Lemma
Let \(X\) be a scheme. Let \(U \subset X\) be a dense open. Assume
the underlying topological space of \(X\) is Noetherian, and
for every \(x \in X \setminus U\) the punctured spectrum of the strict henselization of \(\mathcal{O}_{X, x}\) is connected.
Then \(\textit{F\'Et}_X \to \textit{F\'et}_U\) is fully faithful.
Proof
Let \(Y_1, Y_2\) be finite étale over \(X\) and let \(\varphi : (Y_1)_U \to (Y_2)_U\) be a morphism over \(U\). We have to show that \(\varphi\) lifts uniquely to a morphism \(Y_1 \to Y_2\) over \(X\). Uniqueness follows from Lemma 0BQE.
Let \(x \in X \setminus U\) be a generic point of an irreducible component of \(X \setminus U\). Set \(V = U \times_X \Spec(\mathcal{O}_{X, x})\). By our choice of \(x\) this is the punctured spectrum of \(\Spec(\mathcal{O}_{X, x})\). By Lemma 0BLQ we can extend the morphism \(\varphi_V : (Y_1)_V \to (Y_2)_V\) uniquely to a morphism \((Y_1)_{\Spec(\mathcal{O}_{X, x})} \to (Y_2)_{\Spec(\mathcal{O}_{X, x})}\). By Limits, Lemma 0BQ5 we find an open \(U \subset U'\) containing \(x\) and an extension \(\varphi' : (Y_1)_{U'} \to (Y_2)_{U'}\) of \(\varphi\). Since the underlying topological space of \(X\) is Noetherian this finishes the proof by Noetherian induction on the complement of the open over which \(\varphi\) is defined.
Lemma
Let \(X\) be a scheme. Let \(U \subset X\) be a dense open. Assume
\(U \to X\) is quasi-compact,
every point of \(X \setminus U\) is closed, and
for every \(x \in X \setminus U\) the punctured spectrum of the strict henselization of \(\mathcal{O}_{X, x}\) is connected.
Then \(\textit{F\'Et}_X \to \textit{F\'et}_U\) is fully faithful.
Proof
Let \(Y_1, Y_2\) be finite étale over \(X\) and let \(\varphi : (Y_1)_U \to (Y_2)_U\) be a morphism over \(U\). We have to show that \(\varphi\) lifts uniquely to a morphism \(Y_1 \to Y_2\) over \(X\). Uniqueness follows from Lemma 0BQE.
Let \(x \in X \setminus U\). Set \(V = U \times_X \Spec(\mathcal{O}_{X, x})\). Since every point of \(X \setminus U\) is closed \(V\) is the punctured spectrum of \(\Spec(\mathcal{O}_{X, x})\). By Lemma 0BLQ we can extend the morphism \(\varphi_V : (Y_1)_V \to (Y_2)_V\) uniquely to a morphism \((Y_1)_{\Spec(\mathcal{O}_{X, x})} \to (Y_2)_{\Spec(\mathcal{O}_{X, x})}\). By Limits, Lemma 0BQ5 (this uses that \(U\) is retrocompact in \(X\)) we find an open \(U \subset U'_x\) containing \(x\) and an extension \(\varphi'_x : (Y_1)_{U'_x} \to (Y_2)_{U'_x}\) of \(\varphi\). Note that given two points \(x, x' \in X \setminus U\) the morphisms \(\varphi'_x\) and \(\varphi'_{x'}\) agree over \(U'_x \cap U'_{x'}\) as \(U\) is dense in that open (Lemma 0BQE). Thus we can extend \(\varphi\) to \(\bigcup U'_x = X\) as desired.
Lemma
Let \(X\) be a scheme. Let \(U \subset X\) be a dense open. Assume
every quasi-compact open of \(X\) has finitely many irreducible components,
for every \(x \in X \setminus U\) the punctured spectrum of the strict henselization of \(\mathcal{O}_{X, x}\) is connected.
Then \(\textit{F\'Et}_X \to \textit{F\'et}_U\) is fully faithful.
Proof
Let \(Y_1, Y_2\) be finite étale over \(X\) and let \(\varphi : (Y_1)_U \to (Y_2)_U\) be a morphism over \(U\). We have to show that \(\varphi\) lifts uniquely to a morphism \(Y_1 \to Y_2\) over \(X\). Uniqueness follows from Lemma 0BQE. We will prove existence by showing that we can enlarge \(U\) if \(U \not = X\) and using Zorn’s lemma to finish the proof.
Let \(x \in X \setminus U\) be a generic point of an irreducible component of \(X \setminus U\). Set \(V = U \times_X \Spec(\mathcal{O}_{X, x})\). By our choice of \(x\) this is the punctured spectrum of \(\Spec(\mathcal{O}_{X, x})\). By Lemma 0BLQ we can extend the morphism \(\varphi_V : (Y_1)_V \to (Y_2)_V\) (uniquely) to a morphism \((Y_1)_{\Spec(\mathcal{O}_{X, x})} \to (Y_2)_{\Spec(\mathcal{O}_{X, x})}\). Choose an affine neighbourhood \(W \subset X\) of \(x\). Since \(U \cap W\) is dense in \(W\) it contains the generic points \(\eta_1, \ldots, \eta_n\) of \(W\). Choose an affine open \(W' \subset W \cap U\) containing \(\eta_1, \ldots, \eta_n\). Set \(V' = W' \times_X \Spec(\mathcal{O}_{X, x})\). By Limits, Lemma 0BQ5 applied to \(x \in W \supset W'\) we find an open \(W' \subset W'' \subset W\) with \(x \in W''\) and a morphism \(\varphi'' : (Y_1)_{W''} \to (Y_2)_{W''}\) agreeing with \(\varphi\) over \(W'\). Since \(W'\) is dense in \(W'' \cap U\), we see by Lemma 0BQE that \(\varphi\) and \(\varphi''\) agree over \(U \cap W'\). Thus \(\varphi\) and \(\varphi''\) glue to a morphism \(\varphi'\) over \(U' = U \cup W''\) agreeing with \(\varphi\) over \(U\). Observe that \(x \in U'\) so that we’ve extended \(\varphi\) to a strictly larger open.
Consider the set \(\mathcal{S}\) of pairs \((U', \varphi')\) where \(U \subset U'\) and \(\varphi'\) is an extension of \(\varphi\). We endow \(\mathcal{S}\) with a partial ordering in the obvious manner. If \((U'_i, \varphi'_i)\) is a totally ordered subset, then it has a maximum \((U', \varphi')\). Just take \(U' = \bigcup U'_i\) and let \(\varphi' : (Y_1)_{U'} \to (Y_2)_{U'}\) be the morphism agreeing with \(\varphi'_i\) over \(U'_i\). Thus Zorn’s lemma applies and \(\mathcal{S}\) has a maximal element. By the argument above we see that this maximal element is an extension of \(\varphi\) over all of \(X\).
Lemma
Let \((A, \mathfrak m)\) be a local ring. Set \(X = \Spec(A)\) and \(U = X \setminus \{\mathfrak m\}\). Let \(U^{sh}\) be the punctured spectrum of the strict henselization \(A^{sh}\) of \(A\). Assume \(U\) is quasi-compact and \(U^{sh}\) is connected. Then the sequence \[\pi_1(U^{sh}, \overline{u}) \to \pi_1(U, \overline{u}) \to \pi_1(X, \overline{u}) \to 1\] is exact in the sense of Lemma 0BS9 part (1).
Proof
The map \(\pi_1(U) \to \pi_1(X)\) is surjective by Lemmas 0BLQ and 0BN6.
Write \(X^{sh} = \Spec(A^{sh})\). 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 \(Y \times_X U^{sh}\). It follows that the composition \(\pi_1(U^{sh}) \to \pi_1(U) \to \pi_1(X)\) is trivial, see Lemma 0BS8.
To finish the proof, it suffices according to Lemma 0BS9 to show the following: Given a finite étale morphism \(V \to U\) such that \(V \times_U U^{sh}\) is a disjoint union of copies of \(U^{sh}\), we can find a finite étale morphism \(Y \to X\) with \(V \cong Y \times_X U\) over \(U\). The assumption implies that there exists a finite étale morphism \(Y^{sh} \to X^{sh}\) and an isomorphism \(V \times_U U^{sh} \cong Y^{sh} \times_{X^{sh}} U^{sh}\). Consider the following diagram \[\xymatrix{ U \ar[d] & U^{sh} \ar[d] \ar[l] & U^{sh} \times_U U^{sh} \ar[d] \ar@<1ex>[l] \ar@<-1ex>[l] & U^{sh} \times_U U^{sh} \times_U U^{sh} \ar[d] \ar@<1ex>[l] \ar[l] \ar@<-1ex>[l] \\ X & X^{sh} \ar[l] & X^{sh} \times_X X^{sh} \ar@<1ex>[l] \ar@<-1ex>[l] & X^{sh} \times_X X^{sh} \times_X X^{sh} \ar@<1ex>[l] \ar[l] \ar@<-1ex>[l] }\] Since \(U \subset X\) is quasi-compact by assumption, all the downward arrows are quasi-compact open immersions. Let \(\xi \in X^{sh} \times_X X^{sh}\) be a point not in \(U^{sh} \times_U U^{sh}\). Then \(\xi\) lies over the closed point \(x^{sh}\) of \(X^{sh}\). Consider the local ring homomorphism \[A^{sh} = \mathcal{O}_{X^{sh}, x^{sh}} \to \mathcal{O}_{X^{sh} \times_X X^{sh}, \xi}\] determined by the first projection \(X^{sh} \times_X X^{sh}\). This is a filtered colimit of local homomorphisms which are localizations étale ring maps. Since \(A^{sh}\) is strictly henselian, we conclude that it is an isomorphism. Since this holds for every \(\xi\) in the complement it follows there are no specializations among these points and hence every such \(\xi\) is a closed point (you can also prove this directly). As the local ring at \(\xi\) is isomorphic to \(A^{sh}\), it is strictly henselian and has connected punctured spectrum. Similarly for points \(\xi\) of \(X^{sh} \times_X X^{sh} \times_X X^{sh}\) not in \(U^{sh} \times_U U^{sh} \times_U U^{sh}\). It follows from Lemma 0BSA that pullback along the vertical arrows induce fully faithful functors on the categories of finite étale schemes. Thus the canonical descent datum on \(V \times_U U^{sh}\) relative to the fpqc covering \(\{U^{sh} \to U\}\) translates into a descent datum for \(Y^{sh}\) relative to the fpqc covering \(\{X^{sh} \to X\}\). Since \(Y^{sh} \to X^{sh}\) is finite hence affine, this descent datum is effective (Descent, Lemma 0245). Thus we get an affine morphism \(Y \to X\) and an isomorphism \(Y \times_X X^{sh} \to Y^{sh}\) compatible with descent data. By fully faithfulness of descent data (as in Descent, Lemma 02W0) we get an isomorphism \(V \to U \times_X Y\). Finally, \(Y \to X\) is finite étale as \(Y^{sh} \to X^{sh}\) is, see Descent, Lemmas 02VN and 02LA.
Let \(X\) be an irreducible scheme. Let \(\eta \in X\) be the generic point. The canonical morphism \(\eta \to X\) induces a canonical map [0BQH]\[\begin{equation} \text{Gal}(\kappa(\eta)^{sep}/\kappa(\eta)) = \pi_1(\eta, \overline{\eta}) \longrightarrow \pi_1(X, \overline{\eta}) \end{equation}\] The identification on the left hand side is Lemma 0BNE.
Lemma
Let \(X\) be an irreducible, geometrically unibranch scheme. For any nonempty open \(U \subset X\) the canonical map \[\pi_1(U, \overline{u}) \longrightarrow \pi_1(X, \overline{u})\] is surjective. The map (0BQH) \(\pi_1(\eta, \overline{\eta}) \to \pi_1(X, \overline{\eta})\) is surjective as well.
Proof
By Lemma 0BQB we may replace \(X\) by its reduction. Thus we may assume that \(X\) is an integral scheme. By Lemma 0BN6 the assertion of the lemma translates into the statement that the functors \(\textit{F\'Et}_X \to \textit{F\'Et}_U\) and \(\textit{F\'Et}_X \to \textit{F\'Et}_\eta\) are fully faithful.
The result for \(\textit{F\'Et}_X \to \textit{F\'Et}_U\) follows from Lemma 0BQG and the fact that for a local ring \(A\) which is geometrically unibranch its strict henselization has an irreducible spectrum. See More on Algebra, Lemma 06DM.
Observe that the residue field \(\kappa(\eta) = \mathcal{O}_{X, \eta}\) is the filtered colimit of \(\mathcal{O}_X(U)\) over \(U \subset X\) nonempty open affine. Hence \(\textit{F\'Et}_\eta\) is the colimit of the categories \(\textit{F\'Et}_U\) over such \(U\), see Limits, Lemmas 01ZM, 01ZO, and 07RP. A formal argument then shows that fully faithfulness for \(\textit{F\'Et}_X \to \textit{F\'Et}_\eta\) follows from the fully faithfulness of the functors \(\textit{F\'Et}_X \to \textit{F\'Et}_U\).
Lemma
Let \(X\) be a scheme. Let \(x_1, \ldots, x_n \in X\) be a finite number of closed points such that
\(U = X \setminus \{x_1, \ldots, x_n\}\) is connected and is a retrocompact open of \(X\), and
for each \(i\) the punctured spectrum \(U_i^{sh}\) of the strict henselization of \(\mathcal{O}_{X, x_i}\) is connected.
Then the map \(\pi_1(U) \to \pi_1(X)\) is surjective and the kernel is the smallest closed normal subgroup of \(\pi_1(U)\) containing the image of \(\pi_1(U_i^{sh}) \to \pi_1(U)\) for \(i = 1, \ldots, n\).
Proof
Surjectivity follows from Lemmas 0BSA and 0BN6. We can consider the sequence of maps \[\pi_1(U) \to \ldots \to \pi_1(X \setminus \{x_1, x_2\}) \to \pi_1(X \setminus \{x_1\}) \to \pi_1(X)\] A group theory argument then shows it suffices to prove the statement on the kernel in the case \(n = 1\) (details omitted). Write \(x = x_1\), \(U^{sh} = U_1^{sh}\), set \(A = \mathcal{O}_{X, x}\), and let \(A^{sh}\) be the strict henselization. Consider the diagram \[\xymatrix{ U \ar[d] & \Spec(A) \setminus \{\mathfrak m\} \ar[l] \ar[d] & U^{sh} \ar[d] \ar[l] \\ X & \Spec(A) \ar[l] & \Spec(A^{sh}) \ar[l] }\] By Lemma 0BS9 we have to show finite étale morphisms \(V \to U\) which pull back to trivial coverings of \(U^{sh}\) extend to finite étale schemes over \(X\). By Lemma 0BSB we know the corresponding statement for finite étale schemes over the punctured spectrum of \(A\). However, by Limits, Lemma 0BPA schemes of finite presentation over \(X\) are the same thing as schemes of finite presentation over \(U\) and \(A\) glued over the punctured spectrum of \(A\). This finishes the proof.
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 0BML). 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 \(L/K\) be a finite extension. Consider the normalization \(Y \to X\) of \(X\) in the morphism \(\Spec(L) \to X\) as defined in Morphisms, Section 0BAK. We will say (in this setting) that \(X\) is unramified in \(L\) if \(Y \to X\) is an unramified morphism of schemes. In Lemma 0BTF we will elucidate this condition. Observe that the scheme theoretic fibre of \(Y \to X\) over \(\Spec(K)\) is \(\Spec(L)\). Hence the field extension \(L/K\) is separable if \(X\) is unramified in \(L\), see Morphisms, Lemmas 02G7.
Lemma
In the situation above the following are equivalent
\(X\) is unramified in \(L\),
\(Y \to X\) is étale, and
\(Y \to X\) 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 01WJ. 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. Since a normal scheme is geometrically unibranch (Properties, Lemma 0BQ3), we see that the morphism \(Y \to X\) is étale by More on Morphisms, Lemma 0GS9. We also give a direct proof in the next paragraph.
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 04HJ. Shrinking we may assume \(U\) is quasi-compact. Then \(U\) has finitely many irreducible components (Descent, Lemma 0BAL). Since \(U\) is normal (Descent, Lemma 034F) the irreducible components of \(U\) are open and closed (Properties, Lemma 0357) 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 \(\Spec(L) \times_X U\) by More on Morphisms, Lemma 03GV. Every point of \(\Spec(L) \times_X U\) maps to \(\xi\). Thus every \(V_j\) contains a point mapping to \(\xi\) by Morphisms, Lemma 0AXP. Thus \(V_j \to U\) is an isomorphism as \(U = \overline{\{\xi\}}\). Thus \(Y \times_X U \to U\) is étale. By Descent, Lemma 02VN we conclude that \(Y \to X\) is étale over the image of \(U \to X\) (an open neighbourhood of \(x\)).
Lemma
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 034F. Since \(Y \to X\) is flat every generic point of \(Y\) maps to the generic point of \(X\) by Morphisms, Lemma 03HV. 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 0357. 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 035I 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 0AB1).
Proposition
Let \(X\) be a normal integral scheme with function field \(K\). Then the canonical map (0BQH) \[\text{Gal}(K^{sep}/K) = \pi_1(\eta, \overline{\eta}) \longrightarrow \pi_1(X, \overline{\eta})\] 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 0BQ3). Hence Lemma 0BQI 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 \[\xymatrix{ \textit{F\'Et}_X \ar[r] \ar[d] \ar[rd]_c & \textit{F\'Et}_\eta \ar[d] \\ \textit{Finite-}\pi_1(X, \overline{\eta})\textit{-sets} \ar[r] & \textit{Finite-}\text{Gal}(K^{sep}/K)\textit{-sets} }\] is fully faithful. The left vertical arrow is the equivalence of Theorem 0BND and the right vertical arrow is the equivalence of Lemma 0BNE. The lower horizontal arrow is induced by the map of the proposition. By Lemmas 0BQK and 0BQL we see that the essential image of \(c\) consists of \(\text{Gal}(K^{sep}/K)\textit{-Sets}\) isomorphic to sets of the form \[S = \Hom_K(\prod\nolimits_{i = 1, \ldots, n} L_i, K^{sep}) = \coprod\nolimits_{i = 1, \ldots, n} \Hom_K(L_i, K^{sep})\] 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 0BML and Lemma 0BMM). The proof is done.
Lemma
Let \((A, \mathfrak m)\) be a normal local ring. Set \(X = \Spec(A)\). 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 \[\pi_1(\Spec(K^{sh})) \to \pi_1(\Spec(K)) \to \pi_1(X) \to 1\] is exact in the sense of Lemma 0BS9 part (1).
Proof
Note that \(A^{sh}\) is a normal domain, see More on Algebra, Lemma 06DI. The map \(\pi_1(\Spec(K)) \to \pi_1(X)\) is surjective by Proposition 0BQM.
Write \(X^{sh} = \Spec(A^{sh})\). 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 \(Y \times_X \Spec(K^{sh})\). It follows that the composition \(\pi_1(\Spec(K^{sh})) \to \pi_1(X)\) is trivial, see Lemma 0BS8.
To finish the proof, it suffices according to Lemma 0BS9 to show the following: Given a finite étale morphism \(V \to \Spec(K)\) such that \(V \times_{\Spec(K)} \Spec(K^{sh})\) is a disjoint union of copies of \(\Spec(K^{sh})\), we can find a finite étale morphism \(Y \to X\) with \(V \cong Y \times_X \Spec(K)\) over \(\Spec(K)\). Write \(V = \Spec(L)\), 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 \(Y = \Spec(B)\) and the proof is complete. By Algebra, Lemma 03GG (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 02VN) as desired.
Group actions and integral closure
In this section we continue the discussion of More on Algebra, Section 0BRE. Recall that a normal local ring is a domain by definition.
Lemma
Let \(A\) be a normal domain whose fraction field \(K\) is separably algebraically closed. Let \(\mathfrak p \subset A\) be a nonzero prime ideal. Then the residue field \(\kappa(\mathfrak p)\) is algebraically closed.
Proof
Assume the lemma is not true to get a contradiction. Then there exists a monic irreducible polynomial \(P(T) \in \kappa(\mathfrak p)[T]\) of degree \(d > 1\). After replacing \(P\) by \(a^d P(a^{-1}T)\) for suitable \(a \in A\) (to clear denominators) we may assume that \(P\) is the image of a monic polynomial \(Q\) in \(A[T]\). Observe that \(Q\) is irreducible in \(K[T]\). Namely a factorization over \(K\) leads to a factorization over \(A\) by Algebra, Lemma 00H6 which we could reduce modulo \(\mathfrak p\) to get a factorization of \(P\). As \(K\) is separably closed, \(Q\) is not a separable polynomial (Fields, Definition 09H1). Then the characteristic of \(K\) is \(p > 0\) and \(Q\) has vanishing linear term (Fields, Definition 09H1). However, then we can replace \(Q\) by \(Q + a T\) where \(a \in \mathfrak p\) is nonzero to get a contradiction.
Lemma
A normal local ring with separably closed fraction field is strictly henselian.
Proof
Let \((A, \mathfrak m, \kappa)\) be normal local with separably closed fraction field \(K\). If \(A = K\), then we are done. If not, then the residue field \(\kappa\) is algebraically closed by Lemma 0BSP and it suffices to check that \(A\) is henselian. Let \(f \in A[T]\) be monic and let \(a_0 \in \kappa\) be a root of multiplicity \(1\) of the reduction \(\overline{f} \in \kappa[T]\). Let \(f = \prod f_i\) be the factorization in \(K[T]\). By Algebra, Lemma 00H6 we have \(f_i \in A[T]\). Thus \(a_0\) is a root of \(f_i\) for some \(i\). After replacing \(f\) by \(f_i\) we may assume \(f\) is irreducible. Then, since the derivative \(f'\) cannot be zero in \(A[T]\) as \(a_0\) is a single root, we conclude that \(f\) is linear due to the fact that \(K\) is separably algebraically closed. Thus \(A\) is henselian, see Algebra, Definition 04GF.
Lemma
Let \(G\) be a finite group acting on a ring \(R\). Let \(R^G \to A\) be a ring map. Let \(\mathfrak q' \subset A \otimes_{R^G} R\) be a prime lying over the prime \(\mathfrak q \subset R\). Then \[I_\mathfrak q = \{\sigma \in G \mid \sigma(\mathfrak q) = \mathfrak q\text{ and } \sigma \bmod \mathfrak q = \text{id}_{\kappa(\mathfrak q)}\}\] is equal to \[I_{\mathfrak q'} = \{\sigma \in G \mid \sigma(\mathfrak q') = \mathfrak q'\text{ and } \sigma \bmod \mathfrak q' = \text{id}_{\kappa(\mathfrak q')}\}\]
Proof
Since \(\mathfrak q\) is the inverse image of \(\mathfrak q'\) and since \(\kappa(\mathfrak q) \subset \kappa(\mathfrak q')\), we get \(I_{\mathfrak q'} \subset I_\mathfrak q\). Conversely, if \(\sigma \in I_\mathfrak q\), the \(\sigma\) acts trivially on the fibre ring \(A \otimes_{R^G} \kappa(\mathfrak q)\). Thus \(\sigma\) fixes all the primes lying over \(\mathfrak q\) and induces the identity on their residue fields.
Lemma
Let \(G\) be a finite group acting on a ring \(R\). Let \(\mathfrak q \subset R\) be a prime. Set \[I = \{\sigma \in G \mid \sigma(\mathfrak q) = \mathfrak q \text{ and } \sigma \bmod \mathfrak q = \text{id}_\mathfrak q\}\] Then \(R^G \to R^I\) is étale at \(R^I \cap \mathfrak q\).
Proof
The strategy of the proof is to use étale localization to reduce to the case where \(R \to R^I\) is a local isomorphism at \(R^I \cap \mathfrak p\). Let \(R^G \to A\) be an étale ring map. We claim that if the result holds for the action of \(G\) on \(A \otimes_{R^G} R\) and some prime \(\mathfrak q'\) of \(A \otimes_{R^G} R\) lying over \(\mathfrak q\), then the result is true.
To check this, note that since \(R^G \to A\) is flat we have \(A = (A \otimes_{R^G} R)^G\), see More on Algebra, Lemma 0BRH. By Lemma 0BSS the group \(I\) does not change. Then a second application of More on Algebra, Lemma 0BRH shows that \(A \otimes_{R^G} R^I = (A \otimes_{R^G} R)^I\) (because \(R^I \to A \otimes_{R^G} R^I\) is flat). Thus \[\xymatrix{ \Spec((A \otimes_{R^G} R)^I) \ar[d] \ar[r] & \Spec(R^I) \ar[d] \\ \Spec(A) \ar[r] & \Spec(R^G) }\] is cartesian and the horizontal arrows are étale. Thus if the left vertical arrow is étale in some open neighbourhood \(W\) of \((A \otimes_{R^G} R)^I \cap \mathfrak q'\), then the right vertical arrow is étale at the points of the (open) image of \(W\) in \(\Spec(R^I)\), see Descent, Lemma 05B5. In particular the morphism \(\Spec(R^I) \to \Spec(R^G)\) is étale at \(R^I \cap \mathfrak q\).
Let \(\mathfrak p = R^G \cap \mathfrak q\). By More on Algebra, Lemma 0BRI the fibre of \(\Spec(R) \to \Spec(R^G)\) over \(\mathfrak p\) is finite. Moreover the residue field extensions at these points are algebraic, normal, with finite automorphism groups by More on Algebra, Lemma 0BRJ. Thus we may apply More on Morphisms, Lemma 0BSR to the integral ring map \(R^G \to R\) and the prime \(\mathfrak p\). Combined with the claim above we reduce to the case where \(R = A_1 \times \ldots \times A_n\) with each \(A_i\) having a single prime \(\mathfrak q_i\) lying over \(\mathfrak p\) such that the residue field extensions \(\kappa(\mathfrak q_i)/\kappa(\mathfrak p)\) are purely inseparable. Of course \(\mathfrak q\) is one of these primes, say \(\mathfrak q = \mathfrak q_1\).
It may not be the case that \(G\) permutes the factors \(A_i\) (this would be true if the spectrum of \(A_i\) were connected, for example if \(R^G\) was local). This we can fix as follows; we suggest the reader think this through for themselves, perhaps using idempotents instead of topology. Recall that the product decomposition gives a corresponding disjoint union decomposition of \(\Spec(R)\) by open and closed subsets \(U_i\). Since \(G\) is finite, we can refine this covering by a finite disjoint union decomposition \(\Spec(R) = \coprod_{j \in J} W_j\) by open and closed subsets \(W_j\), such that for all \(j \in J\) there exists a \(j' \in J\) with \(\sigma(W_j) = W_{j'}\). The union of the \(W_j\) not meeting \(\{\mathfrak q_1, \ldots, \mathfrak q_n\}\) is a closed subset not meeting the fibre over \(\mathfrak p\) hence maps to a closed subset of \(\Spec(R^G)\) not meeting \(\mathfrak p\) as \(\Spec(R) \to \Spec(R^G)\) is closed. Hence after replacing \(R^G\) by a principal localization (permissible by the claim) we may assume each \(W_j\) meets one of the points \(\mathfrak q_i\). Then we set \(U_i = W_j\) if \(\mathfrak q_i \in W_j\). The corresponding product decomposition \(R = A_1 \times \ldots \times A_n\) is one where \(G\) permutes the factors \(A_i\).
Thus we may assume we have a product decomposition \(R = A_1 \times \ldots \times A_n\) compatible with \(G\)-action, where each \(A_i\) has a single prime \(\mathfrak q_i\) lying over \(\mathfrak p\) and the field extensions \(\kappa(\mathfrak q_i)/\kappa(\mathfrak p)\) are purely inseparable. Write \(A' = A_2 \times \ldots \times A_n\) so that \[R = A_1 \times A'\] Since \(\mathfrak q = \mathfrak q_1\) we find that every \(\sigma \in I\) preserves the product decomposition above. Hence \[R^I = (A_1)^I \times (A')^I\] Observe that \(I = D = \{\sigma \in G \mid \sigma(\mathfrak q) = \mathfrak q\}\) because \(\kappa(\mathfrak q)/\kappa(\mathfrak p)\) is purely inseparable. Since the action of \(G\) on primes over \(\mathfrak p\) is transitive (More on Algebra, Lemma 0BRI) we conclude that, the index of \(I\) in \(G\) is \(n\) and we can write \(G = eI \amalg \sigma_2I \amalg \ldots \amalg \sigma_nI\) so that \(A_i = \sigma_i(A_1)\) for \(i = 2, \ldots, n\). It follows that \[R^G = (A_1)^I.\] Thus the map \(R^G \to R^I\) is étale at \(R^I \cap \mathfrak q\) and the proof is complete.
The following lemma generalizes More on Algebra, Lemma 09EH.
Lemma
Let \(A\) be a normal domain with fraction field \(K\). Let \(L/K\) be a (possibly infinite) Galois extension. Let \(G = \text{Gal}(L/K)\) and let \(B\) be the integral closure of \(A\) in \(L\). Let \(\mathfrak q \subset B\). Set \[I = \{\sigma \in G \mid \sigma(\mathfrak q) = \mathfrak q \text{ and } \sigma \bmod \mathfrak q = \text{id}_{\kappa(\mathfrak q)}\}\] Then \((B^I)_{B^I \cap \mathfrak q}\) is a filtered colimit of étale \(A\)-algebras.
Proof
We can write \(L\) as the filtered colimit of finite Galois extensions of \(K\). Hence it suffices to prove this lemma in case \(L/K\) is a finite Galois extension, see Algebra, Lemma 0BSJ. Since \(A = B^G\) as \(A\) is integrally closed in \(K = L^G\) the result follows from Lemma 0BST.
Ramification theory
In this section we continue the discussion of More on Algebra, Section 09E3 and we relate it to our discussion of the fundamental groups of schemes.
Let \((A, \mathfrak m, \kappa)\) be a normal local ring with fraction field \(K\). Choose a separable algebraic closure \(K^{sep}\). Let \(A^{sep}\) be the integral closure of \(A\) in \(K^{sep}\). Choose maximal ideal \(\mathfrak m^{sep} \subset A^{sep}\). Let \(A \subset A^h \subset A^{sh}\) be the henselization and strict henselization. Observe that \(A^h\) and \(A^{sh}\) are normal rings as well (More on Algebra, Lemma 06DI). Denote \(K^h\) and \(K^{sh}\) their fraction fields. Since \((A^{sep})_{\mathfrak m^{sep}}\) is strictly henselian by Lemma 0BSQ we can choose an \(A\)-algebra map \(A^{sh} \to (A^{sep})_{\mathfrak m^{sep}}\). Namely, first choose a \(\kappa\)-embedding4 \(\kappa(\mathfrak m^{sh}) \to \kappa(\mathfrak m^{sep})\) and then extend (uniquely) to an \(A\)-algebra homomorphism by Algebra, Lemma 04GU. We get the following diagram \[\xymatrix{ K^{sep} & K^{sh} \ar[l] & K^h \ar[l] & K \ar[l] \\ (A^{sep})_{\mathfrak m^{sep}} \ar[u] & A^{sh} \ar[u] \ar[l] & A^h \ar[u] \ar[l] & A \ar[u] \ar[l] }\] We can take the fundamental groups of the spectra of these rings. Of course, since \(K^{sep}\), \((A^{sep})_{\mathfrak m^{sep}}\), and \(A^{sh}\) are strictly henselian, for them we obtain trivial groups. Thus the interesting part is the following [0BSV]\[\begin{equation} \vcenter{ \xymatrix{ \pi_1(U^{sh}) \ar[r] \ar[rd]_1 & \pi_1(U^h) \ar[d] \ar[r] & \pi_1(U) \ar[d] \\ & \pi_1(X^h) \ar[r] & \pi_1(X) } } \end{equation}\] Here \(X^h\) and \(X\) are the spectra of \(A^h\) and \(A\) and \(U^{sh}\), \(U^h\), \(U\) are the spectra of \(K^{sh}\), \(K^h\), and \(K\). The label \(1\) means that the map is trivial; this follows as it factors through the trivial group \(\pi_1(X^{sh})\). On the other hand, the profinite group \(G = \text{Gal}(K^{sep}/K)\) acts on \(A^{sep}\) and we can make the following definitions \[D = \{\sigma \in G \mid \sigma(\mathfrak m^{sep}) = \mathfrak m^{sep}\} \supset I = \{\sigma \in D \mid \sigma \bmod \mathfrak m^{sep} = \text{id}_{\kappa(\mathfrak m^{sep})}\}\] These groups are sometimes called the decomposition group and the inertia group especially when \(A\) is a discrete valuation ring.
Lemma
In the situation described above, via the isomorphism \(\pi_1(U) = \text{Gal}(K^{sep}/K)\) the diagram (0BSV) translates into the diagram \[\xymatrix{ I \ar[r] \ar[rd]_1 & D \ar[d] \ar[r] & \text{Gal}(K^{sep}/K) \ar[d] \\ & \text{Gal}(\kappa(\mathfrak m^{sh})/\kappa) \ar[r] & \text{Gal}(M/K) }\] where \(K^{sep}/M/K\) is the maximal subextension unramified with respect to \(A\). Moreover, the vertical arrows are surjective, the kernel of the left vertical arrow is \(I\) and the kernel of the right vertical arrow is the smallest closed normal subgroup of \(\text{Gal}(K^{sep}/K)\) containing \(I\).
Proof
By construction the group \(D\) acts on \((A^{sep})_{\mathfrak m^{sep}}\) over \(A\). By the uniqueness of \(A^{sh} \to (A^{sep})_{\mathfrak m^{sep}}\) given the map on residue fields (Algebra, Lemma 04GU) we see that the image of \(A^{sh} \to (A^{sep})_{\mathfrak m^{sep}}\) is contained in \(((A^{sep})_{\mathfrak m^{sep}})^I\). On the other hand, Lemma 0BSU shows that \(((A^{sep})_{\mathfrak m^{sep}})^I\) is a filtered colimit of étale extensions of \(A\). Since \(A^{sh}\) is the maximal such extension, we conclude that \(A^{sh} = ((A^{sep})_{\mathfrak m^{sep}})^I\). Hence \(K^{sh} = (K^{sep})^I\).
Recall that \(I\) is the kernel of a surjective map \(D \to \text{Aut}(\kappa(\mathfrak m^{sep})/\kappa)\), see More on Algebra, Lemma 0BRK. We have \(\text{Aut}(\kappa(\mathfrak m^{sep})/\kappa) = \text{Gal}(\kappa(\mathfrak m^{sh})/\kappa)\) as we have seen above that these fields are the algebraic and separable algebraic closures of \(\kappa\). On the other hand, any automorphism of \(A^{sh}\) over \(A\) is an automorphism of \(A^{sh}\) over \(A^h\) by the uniqueness in Algebra, Lemma 04GS. Furthermore, \(A^{sh}\) is the colimit of finite étale extensions \(A^h \subset A'\) which correspond \(1\)-to-\(1\) with finite separable extension \(\kappa'/\kappa\), see Algebra, Remark 0BSL. Thus \[\text{Aut}(A^{sh}/A) = \text{Aut}(A^{sh}/A^h) = \text{Gal}(\kappa(\mathfrak m^{sh})/\kappa)\] Let \(\kappa'/\kappa\) be a finite Galois extension with Galois group \(G\). Let \(A^h \subset A'\) be the finite étale extension corresponding to \(\kappa \subset \kappa'\) by Algebra, Lemma 04GK. Then it follows that \((A')^G = A^h\) by looking at fraction fields and degrees (small detail omitted). Taking the colimit we conclude that \((A^{sh})^{\text{Gal}(\kappa(\mathfrak m^{sh})/\kappa)} = A^h\). Combining all of the above, we find \(A^h = ((A^{sep})_{\mathfrak m^{sep}})^D\). Hence \(K^h = (K^{sep})^D\).
Since \(U\), \(U^h\), \(U^{sh}\) are the spectra of the fields \(K\), \(K^h\), \(K^{sh}\) we see that the top lines of the diagrams correspond via Lemma 0BNE. By Lemma 09ZS we have \(\pi_1(X^h) = \text{Gal}(\kappa(\mathfrak m^{sh})/\kappa)\). The exactness of the sequence \(1 \to I \to D \to \text{Gal}(\kappa(\mathfrak m^{sh})/\kappa) \to 1\) was pointed out above. By Proposition 0BQM we see that \(\pi_1(X) = \text{Gal}(M/K)\). Finally, the statement on the kernel of \(\text{Gal}(K^{sep}/K) \to \text{Gal}(M/K) = \pi_1(X)\) follows from Lemma 0BSM. This finishes the proof.
Let \(X\) be a normal integral scheme with function field \(K\). Let \(K^{sep}\) be a separable algebraic closure of \(K\). Let \(X^{sep} \to X\) be the normalization of \(X\) in \(K^{sep}\). Since \(G = \text{Gal}(K^{sep}/K)\) acts on \(K^{sep}\) we obtain a right action of \(G\) on \(X^{sep}\). For \(y \in X^{sep}\) define \[D_y = \{\sigma \in G \mid \sigma(y) = y\} \supset I_y = \{\sigma \in D \mid \sigma \bmod \mathfrak m_y = \text{id}_{\kappa(y)} \}\] similarly to the above. On the other hand, for \(x \in X\) let \(\mathcal{O}_{X, x}^{sh}\) be a strict henselization, let \(K_x^{sh}\) be the fraction field of \(\mathcal{O}_{X, x}^{sh}\) and choose a \(K\)-embedding \(K_x^{sh} \to K^{sep}\).
Lemma
Let \(X\) be a normal integral scheme with function field \(K\). With notation as above, the following three subgroups of \(\text{Gal}(K^{sep}/K) = \pi_1(\Spec(K))\) are equal
the kernel of the surjection \(\text{Gal}(K^{sep}/K) \longrightarrow \pi_1(X)\),
the smallest normal closed subgroup containing \(I_y\) for all \(y \in X^{sep}\), and
the smallest normal closed subgroup containing \(\text{Gal}(K^{sep}/K_x^{sh})\) for all \(x \in X\).
Proof
The equivalence of (2) and (3) follows from Lemma 0BSW which tells us that \(I_y\) is conjugate to \(\text{Gal}(K^{sep}/K_x^{sh})\) if \(y\) lies over \(x\). By Lemma 0BSM we see that \(\text{Gal}(K^{sep}/K_x^{sh})\) maps trivially to \(\pi_1(\Spec(\mathcal{O}_{X, x}))\) and therefore the subgroup \(N \subset G = \text{Gal}(K^{sep}/K)\) of (2) and (3) is contained in the kernel of \(G \longrightarrow \pi_1(X)\).
To prove the other inclusion, since \(N\) is normal, it suffices to prove: given \(N \subset U \subset G\) with \(U\) open normal, the quotient map \(G \to G/U\) factors through \(\pi_1(X)\). In other words, if \(L/K\) is the Galois extension corresponding to \(U\), then we have to show that \(X\) is unramified in \(L\) (Section 0BQJ, especially Proposition 0BQM). It suffices to do this when \(X\) is affine (we do this so we can refer to algebra results in the rest of the proof). Let \(Y \to X\) be the normalization of \(X\) in \(L\). The inclusion \(L \subset K^{sep}\) induces a morphism \(\pi : X^{sep} \to Y\). For \(y \in X^{sep}\) the inertia group of \(\pi(y)\) in \(\text{Gal}(L/K)\) is the image of \(I_y\) in \(\text{Gal}(L/K)\); this follows from More on Algebra, Lemma 0BSX. Since \(N \subset U\) all these inertia groups are trivial. We conclude that \(Y \to X\) is étale by applying Lemma 0BST. (Alternative: you can use Lemma 0BSM to see that the pullback of \(Y\) to \(\Spec(\mathcal{O}_{X, x})\) is étale for all \(x \in X\) and then conclude from there with a bit more work.)
Example
Let \(X\) be a normal integral Noetherian scheme with function field \(K\). Purity of branch locus (see below) tells us that if \(X\) is regular, then it suffices in Lemma 0BTD to consider the inertia groups \(I = \pi_1(\Spec(K_x^{sh}))\) for points \(x\) of codimension \(1\) in \(X\). In general this is not enough however. Namely, let \(Y = \mathbf{A}_k^n = \Spec(k[t_1, \ldots, t_n])\) where \(k\) is a field not of characteristic \(2\). Let \(G = \{\pm 1\}\) be the group of order \(2\) acting on \(Y\) by multiplication on the coordinates. Set \[X = \Spec(k[t_it_j, i, j \in \{1, \ldots, n\}])\] The embedding \(k[t_it_j] \subset k[t_1, \ldots, t_n]\) defines a degree \(2\) morphism \(Y \to X\) which is unramified everywhere except over the maximal ideal \(\mathfrak m = (t_it_j)\) which is a point of codimension \(n\) in \(X\).
Lemma
Let \(X\) be an integral normal scheme with function field \(K\). Let \(L/K\) be a finite extension. Let \(Y \to X\) be the normalization of \(X\) in \(L\). The following are equivalent
\(X\) is unramified in \(L\) as defined in Section 0BQJ,
\(Y \to X\) is an unramified morphism of schemes,
\(Y \to X\) is an étale morphism of schemes,
\(Y \to X\) is a finite étale morphism of schemes,
for \(x \in X\) the projection \(Y \times_X \Spec(\mathcal{O}_{X, x}) \to \Spec(\mathcal{O}_{X, x})\) is unramified,
same as in (5) but with \(\mathcal{O}_{X, x}^h\),
same as in (5) but with \(\mathcal{O}_{X, x}^{sh}\),
for \(x \in X\) the scheme theoretic fibre \(Y_x\) is étale over \(x\) of degree \(\geq [L : K]\).
If \(L/K\) is Galois with Galois group \(G\), then these are also equivalent to
for \(y \in Y\) the group \(I_y = \{g \in G \mid g(y) = y\text{ and } g \bmod \mathfrak m_y = \text{id}_{\kappa(y)}\}\) is trivial.
Proof
The equivalence of (1) and (2) is the definition of (1). The equivalence of (2), (3), and (4) is Lemma 0BQK. It is straightforward to prove that (4) \(\Rightarrow\) (5), (5) \(\Rightarrow\) (6), (6) \(\Rightarrow\) (7).
Assume (7). Observe that \(\mathcal{O}_{X, x}^{sh}\) is a normal local domain (More on Algebra, Lemma 06DI). Let \(L^{sh} = L \otimes_K K_x^{sh}\) where \(K_x^{sh}\) is the fraction field of \(\mathcal{O}_{X, x}^{sh}\). Then \(L^{sh} = \prod_{i = 1, \ldots, n} L_i\) with \(L_i/K_x^{sh}\) finite separable. By Algebra, Lemma 03GG (and a limit argument we omit) we see that \(Y \times_X \Spec(\mathcal{O}_{X, x}^{sh})\) is the integral closure of \(\Spec(\mathcal{O}_{X, x}^{sh})\) in \(L^{sh}\). Hence by Lemma 0BQK (applied to the factors \(L_i\) of \(L^{sh}\)) we see that \(Y \times_X \Spec(\mathcal{O}_{X, x}^{sh}) \to \Spec(\mathcal{O}_{X, x}^{sh})\) is finite étale. Looking at the generic point we see that the degree is equal to \([L : K]\) and hence we see that (8) is true.
Assume (8). Assume that \(x \in X\) and that the scheme theoretic fibre \(Y_x\) is étale over \(x\) of degree \(\geq [L : K]\). Observe that this means that \(Y\) has \(\geq [L : K]\) geometric points lying over \(x\). We will show that \(Y \to X\) is finite étale over a neighbourhood of \(x\). This will prove (1) holds. To prove this we may assume \(X = \Spec(R)\), the point \(x\) corresponds to the prime \(\mathfrak p \subset R\), and \(Y = \Spec(S)\). We apply More on Morphisms, Lemma 0BSR and we find an étale neighbourhood \((U, u) \to (X, x)\) such that \(Y \times_X U = V_1 \amalg \ldots \amalg V_m\) such that \(V_i\) has a unique point \(v_i\) lying over \(u\) with \(\kappa(v_i)/\kappa(u)\) purely inseparable. Shrinking \(U\) if necessary we may assume \(U\) is a normal integral scheme with generic point \(\xi\) (use Descent, Lemmas 0BAL and 034F and Properties, Lemma 0357). By our remark on geometric points we see that \(m \geq [L : K]\). On the other hand, by More on Morphisms, Lemma 03GV we see that \(\coprod V_i \to U\) is the normalization of \(U\) in \(\Spec(L) \times_X U\). As \(K \subset \kappa(\xi)\) is finite separable, we can write \(\Spec(L) \times_X U = \Spec(\prod_{i = 1, \ldots, n} L_i)\) with \(L_i/\kappa(\xi)\) finite and \([L : K] = \sum [L_i : \kappa(\xi)]\). Since \(V_j\) is nonempty for each \(j\) and \(m \geq [L : K]\) we conclude that \(m = n\) and \([L_i : \kappa(\xi)] = 1\) for all \(i\). Then \(V_j \to U\) is an isomorphism in particular étale, hence \(Y \times_X U \to U\) is étale. By Descent, Lemma 02VN we conclude that \(Y \to X\) is étale over the image of \(U \to X\) (an open neighbourhood of \(x\)).
Assume \(L/K\) is Galois and (9) holds. Then \(Y \to X\) is étale by Lemma 0BSU. We omit the proof that (1) implies (9).
In the case of infinite Galois extensions of discrete valuation rings we can say a tiny bit more. To do so we introduce the following notation. A subset \(S \subset \mathbf{N}\) of integers is multiplicativity directed if \(1 \in S\) and for \(n, m \in S\) there exists \(k \in S\) with \(n | k\) and \(m | k\). Define a partial ordering on \(S\) by the rule \(n \geq_S m\) if and only if \(m | n\). Given a field \(\kappa\) we obtain an inverse system of finite groups \(\{\mu_n(\kappa)\}_{n \in S}\) with transition maps \[\mu_n(\kappa) \longrightarrow \mu_m(\kappa),\quad \zeta \longmapsto \zeta^{n/m}\] for \(n \geq_S m\). Then we can form the profinite group \[\lim_{n \in S} \mu_n(\kappa)\] Observe that the limit is cofiltered (as \(S\) is directed). The construction is functorial in \(\kappa\). In particular \(\text{Aut}(\kappa)\) acts on this profinite group. For example, if \(S = \{1, n\}\), then this gives \(\mu_n(\kappa)\). If \(S = \{1, \ell, \ell^2, \ell^3, \ldots\}\) for some prime \(\ell\) different from the characteristic of \(\kappa\) this produces \(\lim_n \mu_{\ell^n}(\kappa)\) which is sometimes called the \(\ell\)-adic Tate module of the multiplicative group of \(\kappa\) (compare with More on Algebra, Example 0BKG).
Lemma
Let \(A\) be a discrete valuation ring with fraction field \(K\). Let \(L/K\) be a (possibly infinite) Galois extension. Let \(B\) be the integral closure of \(A\) in \(L\). Let \(\mathfrak m\) be a maximal ideal of \(B\). Let \(G = \text{Gal}(L/K)\), \(D = \{\sigma \in G \mid \sigma(\mathfrak m) = \mathfrak m\}\), and \(I = \{\sigma \in D \mid \sigma \bmod \mathfrak m = \text{id}_{\kappa(\mathfrak m)}\}\). The decomposition group \(D\) fits into a canonical exact sequence \[1 \to I \to D \to \text{Aut}(\kappa(\mathfrak m)/\kappa_A) \to 1\] The inertia group \(I\) fits into a canonical exact sequence \[1 \to P \to I \to I_t \to 1\] such that
\(P\) is a normal subgroup of \(D\),
\(P\) is a pro-\(p\)-group if the characteristic of \(\kappa_A\) is \(p > 1\) and \(P = \{1\}\) if the characteristic of \(\kappa_A\) is zero,
there is a multiplicatively directed \(S \subset \mathbf{N}\) such that \(\kappa(\mathfrak m)\) contains a primitive \(n\)th root of unity for each \(n \in S\) (elements of \(S\) are prime to \(p\)),
there exists a canonical surjective map \[\theta_{can} : I \to \lim_{n \in S} \mu_n(\kappa(\mathfrak m))\] whose kernel is \(P\), which satisfies \(\theta_{can}(\tau \sigma \tau^{-1}) = \tau(\theta_{can}(\sigma))\) for \(\tau \in D\), \(\sigma \in I\), and which induces an isomorphism \(I_t \to \lim_{n \in S} \mu_n(\kappa(\mathfrak m))\).
Proof
This is mostly a reformulation of the results on finite Galois extensions proved in More on Algebra, Section 09E3. The surjectivity of the map \(D \to \text{Aut}(\kappa(\mathfrak m)/\kappa)\) is More on Algebra, Lemma 0BRK. This gives the first exact sequence.
To construct the second short exact sequence let \(\Lambda\) be the set of finite Galois subextensions, i.e., \(\lambda \in \Lambda\) corresponds to \(L/L_\lambda/K\). Set \(G_\lambda = \text{Gal}(L_\lambda/K)\). Recall that \(G_\lambda\) is an inverse system of finite groups with surjective transition maps and that \(G = \lim_{\lambda \in \Lambda} G_\lambda\), see Fields, Lemma 0BU2. We let \(B_\lambda\) be the integral closure of \(A\) in \(L_\lambda\). Then we set \(\mathfrak m_\lambda = \mathfrak m \cap B_\lambda\) and we denote \(P_\lambda, I_\lambda, D_\lambda\) the wild inertia, inertia, and decomposition group of \(\mathfrak m_\lambda\), see More on Algebra, Lemma 09EE. For \(\lambda \geq \lambda'\) the restriction defines a commutative diagram \[\xymatrix{ P_\lambda \ar[d] \ar[r] & I_\lambda \ar[d] \ar[r] & D_\lambda \ar[d] \ar[r] & G_\lambda \ar[d] \\ P_{\lambda'} \ar[r] & I_{\lambda'} \ar[r] & D_{\lambda'} \ar[r] & G_{\lambda'} }\] with surjective vertical maps, see More on Algebra, Lemma 0BU7.
From the definitions it follows immediately that \(I = \lim I_\lambda\) and \(D = \lim D_\lambda\) under the isomorphism \(G = \lim G_\lambda\) above. Since \(L = \colim L_\lambda\) we have \(B = \colim B_\lambda\) and \(\kappa(\mathfrak m) = \colim \kappa(\mathfrak m_\lambda)\). Since the transition maps of the system \(D_\lambda\) are compatible with the maps \(D_\lambda \to \text{Aut}(\kappa(\mathfrak m_\lambda)/\kappa)\) (see More on Algebra, Lemma 0BU7) we see that the map \(D \to \text{Aut}(\kappa(\mathfrak m)/\kappa)\) is the limit of the maps \(D_\lambda \to \text{Aut}(\kappa(\mathfrak m_\lambda)/\kappa)\).
There exist canonical maps \[\theta_{\lambda, can} : I_\lambda \longrightarrow \mu_{n_\lambda}(\kappa(\mathfrak m_\lambda))\] where \(n_\lambda = |I_\lambda|/|P_\lambda|\), where \(\mu_{n_\lambda}(\kappa(\mathfrak m_\lambda))\) has order \(n_\lambda\), such that \(\theta_{\lambda, can}(\tau \sigma \tau^{-1}) = \tau(\theta_{\lambda, can}(\sigma))\) for \(\tau \in D_\lambda\) and \(\sigma \in I_\lambda\), and such that we get commutative diagrams \[\xymatrix{ I_\lambda \ar[r]_-{\theta_{\lambda, can}} \ar[d] & \mu_{n_\lambda}(\kappa(\mathfrak m_\lambda)) \ar[d]^{(-)^{n_\lambda/n_{\lambda'}}} \\ I_{\lambda'} \ar[r]^-{\theta_{\lambda', can}} & \mu_{n_{\lambda'}}(\kappa(\mathfrak m_{\lambda'})) }\] see More on Algebra, Remark 0BU8.
Let \(S \subset \mathbf{N}\) be the collection of integers \(n_\lambda\). Since \(\Lambda\) is directed, we see that \(S\) is multiplicatively directed. By the displayed commutative diagrams above we can take the limits of the maps \(\theta_{\lambda, can}\) to obtain \[\theta_{can} : I \to \lim_{n \in S} \mu_n(\kappa(\mathfrak m)).\] This map is continuous (small detail omitted). Since the transition maps of the system of \(I_\lambda\) are surjective and \(\Lambda\) is directed, the projections \(I \to I_\lambda\) are surjective. For every \(\lambda\) the diagram \[\xymatrix{ I \ar[d] \ar[r]_-{\theta_{can}} & \lim_{n \in S} \mu_n(\kappa(\mathfrak m)) \ar[d] \\ I_{\lambda} \ar[r]^-{\theta_{\lambda, can}} & \mu_{n_\lambda}(\kappa(\mathfrak m_\lambda)) }\] commutes. Hence the image of \(\theta_{can}\) surjects onto the finite group \(\mu_{n_\lambda}(\kappa(\mathfrak m)) = \mu_{n_\lambda}(\kappa(\mathfrak m_\lambda))\) of order \(n_\lambda\) (see above). It follows that the image of \(\theta_{can}\) is dense. On the other hand \(\theta_{can}\) is continuous and the source is a profinite group. Hence \(\theta_{can}\) is surjective by a topological argument.
The property \(\theta_{can}(\tau \sigma \tau^{-1}) = \tau(\theta_{can}(\sigma))\) for \(\tau \in D\), \(\sigma \in I\) follows from the corresponding properties of the maps \(\theta_{\lambda, can}\) and the compatibility of the map \(D \to \text{Aut}(\kappa(\mathfrak m))\) with the maps \(D_\lambda \to \text{Aut}(\kappa(\mathfrak m_\lambda))\). Setting \(P = \Ker(\theta_{can})\) this implies that \(P\) is a normal subgroup of \(D\). Setting \(I_t = I/P\) we obtain the isomorphism \(I_t \to \lim_{n \in S} \mu_n(\kappa(\mathfrak m))\) from the surjectivity of \(\theta_{can}\).
To finish the proof we show that \(P = \lim P_\lambda\) which proves that \(P\) is a pro-\(p\)-group. Recall that the tame inertia group \(I_{\lambda, t} = I_\lambda/P_\lambda\) has order \(n_\lambda\). Since the transition maps \(P_\lambda \to P_{\lambda'}\) are surjective and \(\Lambda\) is directed, we obtain a short exact sequence \[1 \to \lim P_\lambda \to I \to \lim I_{\lambda, t} \to 1\] (details omitted). Since for each \(\lambda\) the map \(\theta_{\lambda, can}\) induces an isomorphism \(I_{\lambda, t} \cong \mu_{n_\lambda}(\kappa(\mathfrak m))\) the desired result follows.
Lemma
Let \(A\) be a discrete valuation ring with fraction field \(K\). Let \(K^{sep}\) be a separable closure of \(K\). Let \(A^{sep}\) be the integral closure of \(A\) in \(K^{sep}\). Let \(\mathfrak m^{sep}\) be a maximal ideal of \(A^{sep}\). Let \(\mathfrak m = \mathfrak m^{sep} \cap A\), let \(\kappa = A/\mathfrak m\), and let \(\overline{\kappa} = A^{sep}/\mathfrak m^{sep}\). Then \(\overline{\kappa}\) is an algebraic closure of \(\kappa\). Let \(G = \text{Gal}(K^{sep}/K)\), \(D = \{\sigma \in G \mid \sigma(\mathfrak m^{sep}) = \mathfrak m^{sep}\}\), and \(I = \{\sigma \in D \mid \sigma \bmod \mathfrak m^{sep} = \text{id}_{\kappa(\mathfrak m^{sep})}\}\). The decomposition group \(D\) fits into a canonical exact sequence \[1 \to I \to D \to \text{Gal}(\kappa^{sep}/\kappa) \to 1\] where \(\kappa^{sep} \subset \overline{\kappa}\) is the separable closure of \(\kappa\). The inertia group \(I\) fits into a canonical exact sequence \[1 \to P \to I \to I_t \to 1\] such that
\(P\) is a normal subgroup of \(D\),
\(P\) is a pro-\(p\)-group if the characteristic of \(\kappa_A\) is \(p > 1\) and \(P = \{1\}\) if the characteristic of \(\kappa_A\) is zero,
there exists a canonical surjective map \[\theta_{can} : I \to \lim_{n\text{ prime to }p} \mu_n(\kappa^{sep})\] whose kernel is \(P\), which satisfies \(\theta_{can}(\tau \sigma \tau^{-1}) = \tau(\theta_{can}(\sigma))\) for \(\tau \in D\), \(\sigma \in I\), and which induces an isomorphism \(I_t \to \lim_{n\text{ prime to }p} \mu_n(\kappa^{sep})\).
Proof
The field \(\overline{\kappa}\) is the algebraic closure of \(\kappa\) by Lemma 0BSP. Most of the statements immediately follow from the corresponding parts of Lemma 0BUA. For example because \(\text{Aut}(\overline{\kappa}/\kappa) = \text{Gal}(\kappa^{sep}/\kappa)\) we obtain the first sequence. Then the only other assertion that needs a proof is the fact that with \(S\) as in Lemma 0BUA the limit \(\lim_{n \in S} \mu_n(\overline{\kappa})\) is equal to \(\lim_{n\text{ prime to }p} \mu_n(\kappa^{sep})\). To see this it suffices to show that every integer \(n\) prime to \(p\) divides an element of \(S\). Let \(\pi \in A\) be a uniformizer and consider the splitting field \(L\) of the polynomial \(X^n - \pi\). Since the polynomial is separable we see that \(L\) is a finite Galois extension of \(K\). Choose an embedding \(L \to K^{sep}\). Observe that if \(B\) is the integral closure of \(A\) in \(L\), then the ramification index of \(A \to B_{\mathfrak m^{sep} \cap B}\) is divisible by \(n\) (because \(\pi\) has an \(n\)th root in \(B\); in fact the ramification index equals \(n\) but we do not need this). Then it follows from the construction of the \(S\) in the proof of Lemma 0BUA that \(n\) divides an element of \(S\).
Geometric and arithmetic fundamental groups
In this section we work out what happens when comparing the fundamental group of a scheme \(X\) over a field \(k\) with the fundamental group of \(X_{\overline{k}}\) where \(\overline{k}\) is the algebraic closure of \(k\).
Lemma
Let \(I\) be a directed set. Let \(X_i\) be an inverse system of quasi-compact and quasi-separated schemes over \(I\) with affine transition morphisms. Let \(X = \lim X_i\) as in Limits, Section 01YV. Then there is an equivalence of categories \[\colim \textit{F\'Et}_{X_i} = \textit{F\'Et}_X\] If \(X_i\) is connected for all sufficiently large \(i\) and \(\overline{x}\) is a geometric point of \(X\), then \[\pi_1(X, \overline{x}) = \lim \pi_1(X_i, \overline{x})\]
Proof
The equivalence of categories follows from Limits, Lemmas 01ZM, 01ZO, and 07RP. The second statement is formal given the statement on categories.
Lemma
Let \(k\) be a field with perfection \(k^{perf}\). Let \(X\) be a connected scheme over \(k\). Then \(X_{k^{perf}}\) is connected and \(\pi_1(X_{k^{perf}}) \to \pi_1(X)\) is an isomorphism.
Proof
Special case of topological invariance of the fundamental group. See Proposition 0BQN. To see that \(\Spec(k^{perf}) \to \Spec(k)\) is a universal homeomorphism you can use Algebra, Lemma 0BRD.
Lemma
Let \(k\) be a field with algebraic closure \(\overline{k}\). Let \(X\) be a quasi-compact and quasi-separated scheme over \(k\). If the base change \(X_{\overline{k}}\) is connected, then there is a short exact sequence \[1 \to \pi_1(X_{\overline{k}}) \to \pi_1(X) \to \pi_1(\Spec(k)) \to 1\] of profinite topological groups.
Proof
Connected objects of \(\textit{F\'Et}_{\Spec(k)}\) are of the form \(\Spec(k') \to \Spec(k)\) with \(k'/k\) a finite separable extension. Then \(X_{\Spec{k'}}\) is connected, as the morphism \(X_{\overline{k}} \to X_{\Spec(k')}\) is surjective and \(X_{\overline{k}}\) is connected by assumption. Thus \(\pi_1(X) \to \pi_1(\Spec(k))\) is surjective by Lemma 0BN6.
Before we go on, note that we may assume that \(k\) is a perfect field. Namely, we have \(\pi_1(X_{k^{perf}}) = \pi_1(X)\) and \(\pi_1(\Spec(k^{perf})) = \pi_1(\Spec(k))\) by Lemma 0BTW.
It is clear that the composition of the functors \(\textit{F\'Et}_{\Spec(k)} \to \textit{F\'Et}_X \to \textit{F\'Et}_{X_{\overline{k}}}\) sends objects to disjoint unions of copies of \(X_{\Spec(\overline{k})}\). Therefore the composition \(\pi_1(X_{\overline{k}}) \to \pi_1(X) \to \pi_1(\Spec(k))\) is the trivial homomorphism by Lemma 0BS8.
Let \(U \to X\) be a finite étale morphism with \(U\) connected. Observe that \(U \times_X X_{\overline{k}} = U_{\overline{k}}\). Suppose that \(U_{\overline{k}} \to X_{\overline{k}}\) has a section \(s : X_{\overline{k}} \to U_{\overline{k}}\). Then \(s(X_{\overline{k}})\) is an open connected component of \(U_{\overline{k}}\). For \(\sigma \in \text{Gal}(\overline{k}/k)\) denote \(s^\sigma\) the base change of \(s\) by \(\Spec(\sigma)\). Since \(U_{\overline{k}} \to X_{\overline{k}}\) is finite étale it has only a finite number of sections. Thus \[\overline{T} = \bigcup s^\sigma(X_{\overline{k}})\] is a finite union and we see that \(\overline{T}\) is a \(\text{Gal}(\overline{k}/k)\)-stable open and closed subset. By Varieties, Lemma 038B we see that \(\overline{T}\) is the inverse image of a closed subset \(T \subset U\). Since \(U_{\overline{k}} \to U\) is open (Morphisms, Lemma 0383) we conclude that \(T\) is open as well. As \(U\) is connected we see that \(T = U\). Hence \(U_{\overline{k}}\) is a (finite) disjoint union of copies of \(X_{\overline{k}}\). By Lemma 0BTS we conclude that the image of \(\pi_1(X_{\overline{k}}) \to \pi_1(X)\) is normal.
Let \(V \to X_{\overline{k}}\) be a finite étale cover. Recall that \(\overline{k}\) is the union of finite separable extensions of \(k\). By Lemma 0BTV we find a finite separable extension \(k'/k\) and a finite étale morphism \(U \to X_{k'}\) such that \(V = X_{\overline{k}} \times_{X_{k'}} U = U \times_{\Spec(k')} \Spec(\overline{k})\). Then the composition \(U \to X_{k'} \to X\) is finite étale and \(U \times_{\Spec(k)} \Spec(\overline{k})\) contains \(V = U \times_{\Spec(k')} \Spec(\overline{k})\) as an open and closed subscheme. (Because \(\Spec(\overline{k})\) is an open and closed subscheme of \(\Spec(k') \times_{\Spec(k)} \Spec(\overline{k})\) via the multiplication map \(k' \otimes_k \overline{k} \to \overline{k}\).) By Lemma 0BN7 we conclude that \(\pi_1(X_{\overline{k}}) \to \pi_1(X)\) is injective.
Finally, we have to show that for any finite étale morphism \(U \to X\) such that \(U_{\overline{k}}\) is a disjoint union of copies of \(X_{\overline{k}}\) there is a finite étale morphism \(V \to \Spec(k)\) and a surjection \(V \times_{\Spec(k)} X \to U\). See Lemma 0BS9. Arguing as above using Lemma 0BTV we find a finite separable extension \(k'/k\) such that there is an isomorphism \(U_{k'} \cong \coprod_{i = 1, \ldots, n} X_{k'}\). Thus setting \(V = \coprod_{i = 1, \ldots, n} \Spec(k')\) we conclude.
Lemma
Let \(k\) be a field with algebraic closure \(\overline{k}\). Let \(X\) be a connected scheme of finite type over \(k\) and let \(\overline{x} \in X(\overline{k})\). Set \(G_k = \pi_1(\Spec(k), \Spec(\overline{k}))\). Let \(C\) be the connected component of \(X_{\overline{k}}\) containing \(\overline{x}\) and let \(H \subset G_k\) be the stabilizer of \(C\). Then the image of \[\pi_1(X, \overline{x}) \longrightarrow G_k\] is \(H\), and there is a short exact sequence \[1 \longrightarrow \pi_1(C, \overline{x}) \longrightarrow \pi_1(X, \overline{x}) \longrightarrow H \longrightarrow 1.\]
Proof
The connected components of \(X_{\overline{k}}\) are open and closed and there are only finitely many of them. The group \(G_k\) acts transitively on this finite set: a union of orbits would be a \(G_k\)-stable open and closed subset and hence would descend to an open and closed subset of \(X\) by Varieties, Lemma 038B, applied also to its complement. This would contradict the connectedness of \(X\). In particular, \(H\) is open in \(G_k\).
Let \(k' \subset \overline{k}\) be the finite separable extension with \(H = \text{Gal}(\overline{k}/k')\). Applying the same descent lemma over \(k'\) to \(C\) and its complement gives an open and closed subscheme \(X' \subset X_{k'}\) whose base change to \(\overline{k}\) is \(C\). View \(X' \to X\) as a morphism of \(k\)-schemes. After base change to \(\overline{k}\) its source is the disjoint union of the conjugates of \(C\), one for every coset of \(H\) in \(G_k\), and the morphism identifies these with all the connected components of \(X_{\overline{k}}\). It is therefore an isomorphism after this faithfully flat base change, hence an isomorphism by Descent, Lemma 02L4.
Thus \(X\), as a scheme, has the \(k'\)-structure \(X'\), and its base change to \(\overline{k}\) over \(k'\) is the connected scheme \(C\). Applying Lemma 0BTX to \(X'/k'\) gives the displayed exact sequence and identifies the image in \(G_k\) with \(\text{Gal}(\overline{k}/k') = H\).
Homotopy exact sequence
In this section we discuss the following result. Let \(f : X \to S\) be a flat proper morphism of finite presentation whose geometric fibres are connected and reduced. Assume \(S\) is connected and let \(\overline{s}\) be a geometric point of \(S\). Then there is an exact sequence \[\pi_1(X_{\overline{s}}) \to \pi_1(X) \to \pi_1(S) \to 1\] of fundamental groups. See Proposition 0C0J.
Lemma
Let \(f : X \to S\) be a quasi-compact and quasi-separated morphism of connected schemes. If \[\mathcal{O}_S \longrightarrow f_*\mathcal{O}_X\] is an isomorphism, then for every geometric point \(\overline{x}\) of \(X\), with image \(\overline{s}\) in \(S\), the map \[\pi_1(X, \overline{x}) \longrightarrow \pi_1(S, \overline{s})\] is surjective. In particular, this applies when \(f\) is proper and the displayed map is an isomorphism.
Proof
Let \(S' \to S\) be a connected finite étale morphism and set \(X' = X \times_S S'\). Since \(S' \to S\) is flat, flat base change gives \[\mathcal{O}_{S'} \longrightarrow f'_*\mathcal{O}_{X'}\] as an isomorphism, where \(f' : X' \to S'\) is the projection; see Cohomology of Schemes, Lemma 02KH. Hence \[\Gamma(S', \mathcal{O}_{S'}) = \Gamma(X', \mathcal{O}_{X'}).\] If \(X'\) were disconnected, the right hand side would contain a nontrivial idempotent, contrary to the connectedness of \(S'\). Thus pullback by \(f\) sends every connected finite étale cover of \(S\) to a connected finite étale cover of \(X\). The result follows from Lemma 0BN6.
Lemma
Let \(f : X \to S\) be a proper morphism such that \(\mathcal{O}_S \to f_*\mathcal{O}_X\) is an isomorphism. Let \(p : T \to S\) be a finite unramified morphism, let \(t \in T\), and let \(s = p(t)\). If the base change \[X \times_S T \longrightarrow X\] is étale at every point over \(t\), then \(p\) is étale at \(t\). Consequently, if the base change is étale, then \(p\) is étale.
Proof
First, \(f\) is surjective. Indeed, its image is closed, and on the complement the equality \(\mathcal{O}_S = f_*\mathcal{O}_X\) would identify a nonzero structure sheaf with zero.
The assertion is étale local around \(s\) and \(t\). After an étale base change the equality of structure sheaves is preserved by Cohomology of Schemes, Lemma 02KH. Thus, by Étale Morphisms, Lemma 04HJ, we may assume that \(S\) is affine and \[T = \coprod T_i\] with every \(T_i \to S\) a closed immersion. Let \(T_1\) be the factor containing \(t\). The non-étale locus of \(X_{T_1} \to X\) has closed image in \(T_1\) by properness, and this image does not contain \(t\). After shrinking \(S\) around \(s\), we may therefore assume that \(X_{T_1} \to X\) is étale.
The morphism \(X_{T_1} \to X\) is both a closed immersion and étale, and hence is an open immersion by Étale Morphisms, Theorem 025G. Thus \(X_{T_1}\) is open and closed in \(X\). It is defined by an idempotent in \[\Gamma(X, \mathcal{O}_X) = \Gamma(S, \mathcal{O}_S).\] Let \(S_1 \subset S\) be the corresponding open and closed subscheme. Then \(X_{T_1} = X_{S_1}\) as subschemes of \(X\).
Let \(\mathcal{I}_1\) be the ideal defining \(T_1\). Over \(S_1\) its pullback to \(X\) is zero; the equality \(\mathcal{O}_S = f_*\mathcal{O}_X\) therefore gives \(\mathcal{I}_1|_{S_1} = 0\). Over \(S \setminus S_1\) the scheme \(X_{T_1}\) is empty. Surjectivity of \(f\) then implies that \(T_1\) is empty there. Hence \(T_1 = S_1\). Thus \(T_1 \to S\) is an open immersion as well as a closed immersion, proving that \(p\) is étale at \(t\).
Lemma
Let \(S\) be a locally Noetherian scheme. Let \(f : X \to S\) be a proper morphism such that \(\mathcal{O}_S \to f_*\mathcal{O}_X\) is an isomorphism. Let \(h : X' \to X\) be a finite morphism and let \[X' \longrightarrow T \longrightarrow S\] be the Stein factorization of \(f \circ h\). Let \(s \in S\), let \(t \in T\) lie over \(s\), and let \(C \subset X'_s\) be the open and closed connected component corresponding to \(t\). Assume
\(h\) is étale at every point of \(C\), and
the canonical morphism \[C \longrightarrow X_s \times_{\Spec(\kappa(s))} \Spec(\kappa(t))\] is an isomorphism.
Then there is an open neighbourhood \(V \subset T\) of \(t\) such that \(V \to S\) is étale and the canonical morphism \[X' \longrightarrow X \times_S T\] is an isomorphism over \(V\).
Proof
Write \(g : X' \to T\) for the first morphism in the Stein factorization and consider \[\varphi = (h, g) : X' \longrightarrow X \times_S T.\] By More on Morphisms, Theorem 03H2, the scheme \(T\) is the relative spectrum of \((f \circ h)_*\mathcal{O}_{X'}\). This algebra is coherent by Cohomology of Schemes, Proposition 02O5; hence \(T\) is finite over \(S\). It follows that \(\varphi\) is finite. Its fibre over \(t\) is the canonical morphism in assumption (2), and therefore is an isomorphism. The isomorphism locus of \(\varphi\) is open in \(X \times_S T\). Its complement has closed image in \(T\), because \(X \times_S T \to T\) is proper, and this image does not contain \(t\). Let \(W \subset T\) be the complement of this image. Thus \(W\) is an open neighbourhood of \(t\) and \(\varphi\) is an isomorphism over \(W\).
Choose \(c \in C\). Via the isomorphism over \(W\), the morphism \(h\) is the base change of \(T \to S\) by \(X \to S\) in a neighbourhood of \(c\). Consequently \[\Omega_{T/S, t} \otimes_{\mathcal{O}_{T, t}} \kappa(c) = \Omega_{X'/X, c} \otimes_{\mathcal{O}_{X', c}} \kappa(c) = 0\] by assumption (1). Since \(\kappa(c)\) is a field extension of \(\kappa(t)\), faithful flatness and Nakayama’s lemma give \(\Omega_{T/S, t} = 0\). The morphism \(T \to S\) is finite and hence locally of finite type, so Morphisms, Lemma 02G5 shows that it is unramified at \(t\). The isomorphism over \(W\) and assumption (1) also show that \(X \times_S T \to X\) is étale at every point over \(t\). Lemma pione-lemma-finite-unramified-etale-after-pullback now shows that \(T \to S\) is étale at \(t\). Intersecting \(W\) with the étale locus gives the required open neighbourhood \(V\).
Lemma
Let \(S\) be a locally Noetherian scheme. Let \(f : X \to S\) be a proper morphism such that \(\mathcal{O}_S \to f_*\mathcal{O}_X\) is an isomorphism. Let \(h : X' \to X\) be a finite étale morphism, let \[X' \longrightarrow T \longrightarrow S\] be the Stein factorization of \(f \circ h\), and let \(\varphi : X' \to X \times_S T\) be the canonical morphism. The following are equivalent:
\(T \to S\) is finite étale and \(\varphi\) is an isomorphism;
for every \(s \in S\), every \(t \in T\) over \(s\), and the connected component \(C \subset X'_s\) corresponding to \(t\), the canonical morphism \[C \longrightarrow X_s \times_{\Spec(\kappa(s))} \Spec(\kappa(t))\] is an isomorphism.
In particular, a finite étale cover of \(X\) is pulled back from a finite étale cover of \(S\) if and only if it is geometrically trivial on every fibre of \(f\); the cover of \(S\) is then unique up to unique isomorphism.
Proof
The fibres of \(f\) are geometrically connected by the structure-sheaf identity in the statement and More on Morphisms, Theorem 03H2. Thus if (1) holds, then (2) follows by taking fibres. Conversely, assume (2). The morphism \(h\) is étale along every \(C\), so Lemma pione-lemma-local-Stein-trivial-component, applied at every point of \(T\), shows that \(T \to S\) is étale and that \(\varphi\) is an isomorphism. The morphism \(T \to S\) is finite by the proof of Lemma pione-lemma-local-Stein-trivial-component.
Finally, if \(X'\) is pulled back from a finite étale cover \(T' \to S\), then flat base change and \(f_*\mathcal{O}_X = \mathcal{O}_S\) give \[(f \circ h)_*\mathcal{O}_{X'} = (T' \to S)_*\mathcal{O}_{T'}.\] Hence \(T'\) is the middle term of the Stein factorization, proving uniqueness.
Lemma
In the situation of Lemma pione-lemma-local-Stein-trivial-component, assume in addition that \(S\), \(X\), and \(X'\) are integral. Let \(K\), \(L\), and \(L'\) be their function fields. Then there is a finite separable extension \(K'/K\) contained in \(L'\) such that the multiplication map \[L \otimes_K K' \longrightarrow L'\] is an isomorphism. In particular, \(L\) and \(K'\) are linearly disjoint over \(K\) and \(L'\) is their compositum.
Proof
Let \(X' \to T \to S\) be the Stein factorization. The scheme \(T\) is integral: it is the relative spectrum of \((f \circ h)_*\mathcal{O}_{X'}\), and the ring of sections over every nonempty affine open of \(T\) is the ring of sections of a nonempty open of the integral scheme \(X'\), hence is a domain. Let \(K'\) be the function field of \(T\). The morphism \(T \to S\) is finite by the proof of Lemma pione-lemma-local-Stein-trivial-component.
The same lemma gives a nonempty open \(V \subset T\) which is étale over \(S\) and over which \[X' \longrightarrow X \times_S T\] is an isomorphism. Since \(T\) and \(S\) are integral, their generic points belong to \(V\) and its image, respectively. It follows that \(K'/K\) is finite separable and that the generic fibre product is integral. Thus \(L \otimes_K K'\) is a domain. It is a finite-dimensional \(L\)-algebra, so it is a field, and the generic form of the displayed isomorphism identifies it with \(L'\). This proves all the assertions.
Lemma
Let \(f : X \to S\) be a proper morphism of schemes. Let \(X \to S' \to S\) be the Stein factorization of \(f\), see More on Morphisms, Theorem 03H2. If \(f\) is of finite presentation, flat, with geometrically reduced fibres, then \(S' \to S\) is finite étale.
Proof
This follows from Derived Categories of Schemes, Lemma 0G7Y and the information contained in More on Morphisms, Theorem 03H2.
Proposition
Let \(f : X \to S\) be a flat proper morphism of finite presentation whose geometric fibres are connected and reduced. Assume \(S\) is connected and let \(\overline{s}\) be a geometric point of \(S\). Then there is an exact sequence \[\pi_1(X_{\overline{s}}) \to \pi_1(X) \to \pi_1(S) \to 1\] of fundamental groups.
Proof
Let \(Y \to X\) be a finite étale morphism. Consider the Stein factorization \[\xymatrix{ Y \ar[d] \ar[r] & X \ar[d] \\ T \ar[r] & S }\] of \(Y \to S\). By Lemma 0BUN the morphism \(T \to S\) is finite étale. In this way we obtain a functor \(\textit{F\'Et}_X \to \textit{F\'Et}_S\). For any finite étale morphism \(U \to S\) a morphism \(Y \to U \times_S X\) over \(X\) is the same thing as a morphism \(Y \to U\) over \(S\) and such a morphism factors uniquely through the Stein factorization, i.e., corresponds to a unique morphism \(T \to U\) (by the construction of the Stein factorization as a relative normalization in More on Morphisms, Lemma 03GY and factorization by Morphisms, Lemma 035I). Thus we see that the functors \(\textit{F\'Et}_X \to \textit{F\'Et}_S\) and \(\textit{F\'Et}_S \to \textit{F\'Et}_X\) are adjoints. Note that the Stein factorization of \(U \times_S X \to S\) is \(U\), because the fibres of \(U \times_S X \to U\) are geometrically connected.
By the discussion above and Categories, Lemma 07RB we conclude that \(\textit{F\'Et}_S \to \textit{F\'Et}_X\) is fully faithful, i.e., \(\pi_1(X) \to \pi_1(S)\) is surjective (Lemma 0BN6).
It is immediate that the composition \(\textit{F\'Et}_S \to \textit{F\'Et}_X \to \textit{F\'Et}_{X_{\overline{s}}}\) sends any \(U\) to a disjoint union of copies of \(X_{\overline{s}}\). Hence \(\pi_1(X_{\overline{s}}) \to \pi_1(X) \to \pi_1(S)\) is trivial by Lemma 0BS8.
Let \(Y \to X\) be a finite étale morphism with \(Y\) connected such that \(Y \times_X X_{\overline{s}}\) contains a connected component \(Z\) isomorphic to \(X_{\overline{s}}\). Consider the Stein factorization \(T\) as above. Let \(\overline{t} \in T_{\overline{s}}\) be the point corresponding to the fibre \(Z\). Observe that \(T\) is connected (as the image of a connected scheme) and by the surjectivity above \(T \times_S X\) is connected. Now consider the factorization \[\pi : Y \longrightarrow T \times_S X\] Let \(\overline{x} \in X_{\overline{s}}\) be any closed point. Note that \(\kappa(\overline{t}) = \kappa(\overline{s}) = \kappa(\overline{x})\) is an algebraically closed field. Then the fibre of \(\pi\) over \((\overline{t}, \overline{x})\) consists of a unique point, namely the unique point \(\overline{z} \in Z\) corresponding to \(\overline{x} \in X_{\overline{s}}\) via the isomorphism \(Z \to X_{\overline{s}}\). We conclude that the finite étale morphism \(\pi\) has degree \(1\) in a neighbourhood of \((\overline{t}, \overline{x})\). Since \(T \times_S X\) is connected it has degree \(1\) everywhere and we find that \(Y \cong T \times_S X\). Thus \(Y \times_X X_{\overline{s}}\) splits completely. Combining all of the above we see that Lemmas 0BS9 and 0BTS both apply and the proof is complete.
Lemma
Let \(f : X \to S\) be a flat proper morphism of finite presentation whose geometric fibres are connected and reduced. Assume \(S\) is connected and let \(\overline{s}\) be a geometric point of \(S\). For a finite étale morphism \(Y \to X\), the following are equivalent:
there is a finite étale morphism \(T \to S\) and an isomorphism \(Y \cong T \times_S X\) over \(X\), and
the finite étale cover \(Y \times_X X_{\overline{s}} \to X_{\overline{s}}\) is isomorphic to a finite disjoint union of copies of \(X_{\overline{s}}\).
When these conditions hold, \(T\) and the displayed isomorphism are unique up to unique isomorphism.
Proof
The finite étale cover \(Y \to X\) corresponds to a finite set with a continuous action of \(\pi_1(X)\). It is pulled back from \(S\) if and only if this action factors through \(\pi_1(X) \to \pi_1(S)\). By Proposition 0C0J, this map is surjective and its kernel is the image of \(\pi_1(X_{\overline{s}})\). Thus the action factors through \(\pi_1(S)\) if and only if its restriction to \(\pi_1(X_{\overline{s}})\) is trivial. By the Galois category description, the latter condition is exactly (2). Uniqueness follows from the full faithfulness of \(\textit{F\'Et}_S \to \textit{F\'Et}_X\), proved in Proposition 0C0J.
Lemma
Let \(k\) be a field. Let \(X\) and \(Y\) be connected schemes of finite type over \(k\). Assume that \(X\) or \(Y\) is proper over \(k\). Choose an algebraic closure \(\overline{k}\) and points \[\overline{x} \in X(\overline{k}), \qquad \overline{y} \in Y(\overline{k}), \qquad \overline{z} = (\overline{x}, \overline{y}).\] Set \(G_k = \pi_1(\Spec(k), \Spec(\overline{k}))\). Then the natural map \[\pi_1(X \times_k Y, \overline{z}) \longrightarrow \pi_1(X, \overline{x}) \times_{G_k} \pi_1(Y, \overline{y})\] is an isomorphism. Consequently, the natural map \[\pi_1(X \times_k Y, \overline{z}) \longrightarrow \pi_1(X, \overline{x}) \times \pi_1(Y, \overline{y})\] is injective. If \(k\) is algebraically closed, then it is an isomorphism.
Proof
We first prove the result when \(k\) is algebraically closed. By Lemma 0BQB we may replace \(X\) and \(Y\) by their reductions. The induced closed immersion between the products is a thickening as all schemes in question are Noetherian, so this replacement does not change the fundamental group of the product either. The resulting schemes are geometrically reduced by Varieties, Lemma 020I. After interchanging \(X\) and \(Y\) if necessary, we may assume that \(X\) is proper. The projection \[p : X \times_k Y \longrightarrow Y\] is flat, proper, and of finite presentation, and its geometric fibres are connected and reduced. The assertion is unaffected by changing base points, so we may choose them to be \(k\)-rational. Proposition 0C0J therefore gives an exact sequence \[\pi_1(X) \longrightarrow \pi_1(X \times_k Y) \longrightarrow \pi_1(Y) \longrightarrow 1.\] The inclusions of the two factors through these points and the two projections show that the natural map from the middle group to \(\pi_1(X) \times \pi_1(Y)\) is surjective. If an element maps to \((1, 1)\), exactness shows that it comes from \(\pi_1(X)\), and the first projection then shows that this element of \(\pi_1(X)\) is trivial. Thus the natural map is also injective, proving the algebraically closed case.
For a general field, let \(C_X\) and \(C_Y\) be the connected components of \(X_{\overline{k}}\) and \(Y_{\overline{k}}\) containing \(\overline{x}\) and \(\overline{y}\). Let \(H_X, H_Y \subset G_k\) be their stabilizers. The product \(C_X \times_{\overline{k}} C_Y\) is connected: its projection to \(C_Y\) is open and surjective with connected fibres. Thus it is the connected component of \((X \times_k Y)_{\overline{k}}\) containing \(\overline{z}\). Its stabilizer is \[H = H_X \cap H_Y.\] Let \(W\) be the connected component of \(X \times_k Y\) containing \(\overline{z}\); its fundamental group is the fundamental group of \(X \times_k Y\) at \(\overline{z}\). Set \[P = \pi_1(X, \overline{x}) \times_{G_k} \pi_1(Y, \overline{y}).\] Lemma pione-lemma-ses-field-component, applied to \(X\), \(Y\), and \(W\), gives a commutative diagram with exact rows \[\xymatrix{ 1 \ar[r] & \pi_1(C_X \times_{\overline{k}} C_Y) \ar[r] \ar[d]^{\cong} & \pi_1(W, \overline{z}) \ar[r] \ar[d] & H \ar[r] \ar@{=}[d] & 1 \\ 1 \ar[r] & \pi_1(C_X, \overline{x}) \times \pi_1(C_Y, \overline{y}) \ar[r] & P \ar[r] & H \ar[r] & 1. }\] The bottom row is exact because \(\pi_1(X, \overline{x}) \to H_X\) and \(\pi_1(Y, \overline{y}) \to H_Y\) are surjective. The left vertical arrow is an isomorphism by the algebraically closed case; if one of \(X\) and \(Y\) is proper, then the corresponding geometric component is proper. The middle vertical arrow is therefore an isomorphism by a diagram chase.
Specialization maps
In this section we construct specialization maps. Let \(f : X \to S\) be a proper morphism of schemes with geometrically connected fibres. Let \(s' \leadsto s\) be a specialization of points in \(S\). Let \(\overline{s}\) and \(\overline{s}'\) be geometric points lying over \(s\) and \(s'\). Then there is a specialization map \[sp : \pi_1(X_{\overline{s}'}) \longrightarrow \pi_1(X_{\overline{s}})\] The construction of this map is as follows. Let \(A\) be the strict henselization of \(\mathcal{O}_{S, s}\) with respect to \(\kappa(s) \subset \kappa(s)^{sep} \subset \kappa(\overline{s})\), see Algebra, Definition 04GQ. Since \(s' \leadsto s\) the point \(s'\) corresponds to a point of \(\Spec(\mathcal{O}_{S, s})\) and hence there is at least one point (and potentially many points) of \(\Spec(A)\) over \(s'\) whose residue field is a separable algebraic extension of \(\kappa(s')\). Since \(\kappa(\overline{s}')\) is algebraically closed we can choose a morphism \(\varphi : \overline{s}' \to \Spec(A)\) giving rise to a commutative diagram \[\xymatrix{ \overline{s}' \ar[r]_-\varphi \ar[rd] & \Spec(A) \ar[d] & \overline{s} \ar[l] \ar[ld] \\ & S }\] The specialization map is the composition \[\pi_1(X_{\overline{s}'}) \longrightarrow \pi_1(X_A) = \pi_1(X_{\kappa(s)^{sep}}) = \pi_1(X_{\overline{s}})\] where the first equality is Lemma 0A48 and the second follows from Lemmas 0BTW and 0A49. By construction the specialization map fits into a commutative diagram \[\xymatrix{ \pi_1(X_{\overline{s}'}) \ar[rr]_{sp} \ar[rd] & & \pi_1(X_{\overline{s}}) \ar[ld] \\ & \pi_1(X) }\] provided that \(X\) is connected. The specialization map depends on the choice of \(\varphi : \overline{s}' \to \Spec(A)\) above and we will write \(sp_\varphi\) if we want to indicate this.
Lemma
Consider a commutative diagram \[\xymatrix{ Y \ar[d]_g \ar[r] & X \ar[d]^f \\ T \ar[r] & S }\] of schemes where \(f\) and \(g\) are proper with geometrically connected fibres. Let \(t' \leadsto t\) be a specialization of points in \(T\) and consider a specialization map \(sp : \pi_1(Y_{\overline{t}'}) \to \pi_1(Y_{\overline{t}})\) as above. Then there is a commutative diagram \[\xymatrix{ \pi_1(Y_{\overline{t}'}) \ar[r]_{sp} \ar[d] & \pi_1(Y_{\overline{t}}) \ar[d] \\ \pi_1(X_{\overline{s}'}) \ar[r]^{sp} & \pi_1(X_{\overline{s}}) }\] of specialization maps where \(\overline{s}\) and \(\overline{s}'\) are the images of \(\overline{t}\) and \(\overline{t}'\).
Proof
Let \(B\) be the strict henselization of \(\mathcal{O}_{T, t}\) with respect to \(\kappa(t) \subset \kappa(t)^{sep} \subset \kappa(\overline{t})\). Pick \(\psi : \overline{t}' \to \Spec(B)\) lifting \(\overline{t}' \to T\) as in the construction of the specialization map. Let \(s\) and \(s'\) denote the images of \(t\) and \(t'\) in \(S\). Let \(A\) be the strict henselization of \(\mathcal{O}_{S, s}\) with respect to \(\kappa(s) \subset \kappa(s)^{sep} \subset \kappa(\overline{s})\). Since \(\kappa(\overline{s}) = \kappa(\overline{t})\), by the functoriality of strict henselization (Algebra, Lemma 04GU) we obtain a ring map \(A \to B\) fitting into the commutative diagram \[\xymatrix{ \overline{t}' \ar[r]_-\psi \ar[d] & \Spec(B) \ar[d] \ar[r] & T \ar[d] \\ \overline{s}' \ar[r]^-\varphi & \Spec(A) \ar[r] & S }\] Here the morphism \(\varphi : \overline{s}' \to \Spec(A)\) is simply taken to be the composition \(\overline{t}' \to \Spec(B) \to \Spec(A)\). Applying base change we obtain a commutative diagram \[\xymatrix{ Y_{\overline{t}'} \ar[r] \ar[d] & Y_B \ar[d] \\ X_{\overline{s}'} \ar[r] & X_A }\] and from the construction of the specialization map the commutativity of this diagram implies the commutativity of the diagram of the lemma.
Lemma
Let \(f : X \to S\) be a proper morphism with geometrically connected fibres. Let \(s'' \leadsto s' \leadsto s\) be specializations of points of \(S\). A composition of specialization maps \(\pi_1(X_{\overline{s}''}) \to \pi_1(X_{\overline{s}'}) \to \pi_1(X_{\overline{s}})\) is a specialization map \(\pi_1(X_{\overline{s}''}) \to \pi_1(X_{\overline{s}})\).
Proof
Let \(\mathcal{O}_{S, s} \to A\) be the strict henselization constructed using \(\kappa(s) \to \kappa(\overline{s})\). Let \(A \to \kappa(\overline{s}')\) be the map used to construct the first specialization map. Let \(\mathcal{O}_{S, s'} \to A'\) be the strict henselization constructed using \(\kappa(s') \subset \kappa(\overline{s}')\). By functoriality of strict henselization, there is a map \(A \to A'\) such that the composition with \(A' \to \kappa(\overline{s}')\) is the given map (Algebra, Lemma 08HR). Next, let \(A' \to \kappa(\overline{s}'')\) be the map used to construct the second specialization map. Then it is clear that the composition of the first and second specialization maps is the specialization map \(\pi_1(X_{\overline{s}''}) \to \pi_1(X_{\overline{s}})\) constructed using \(A \to A' \to \kappa(\overline{s}'')\).
Let \(X \to S\) be a proper morphism with geometrically connected fibres. Let \(R\) be a strictly henselian valuation ring with algebraically closed fraction field and let \(\Spec(R) \to S\) be a morphism. Let \(\eta, s \in \Spec(R)\) be the generic and closed point. Then we can consider the specialization map \[sp_R : \pi_1(X_\eta) \to \pi_1(X_s)\] for the base change \(X_R/\Spec(R)\). Note that this makes sense as both \(\eta\) and \(s\) have algebraically closed residue fields.
Lemma
Let \(f : X \to S\) be a proper morphism with geometrically connected fibres. Let \(s' \leadsto s\) be a specialization of points of \(S\) and let \(sp : \pi_1(X_{\overline{s}'}) \to \pi_1(X_{\overline{s}})\) be a specialization map. Then there exists a strictly henselian valuation ring \(R\) over \(S\) with algebraically closed fraction field such that \(sp\) is isomorphic to \(sp_R\) defined above.
Proof
Let \(\mathcal{O}_{S, s} \to A\) be the strict henselization constructed using \(\kappa(s) \to \kappa(\overline{s})\). Let \(A \to \kappa(\overline{s}')\) be the map used to construct \(sp\). Let \(R \subset \kappa(\overline{s}')\) be a valuation ring with fraction field \(\kappa(\overline{s}')\) dominating the image of \(A\). See Algebra, Lemma 00IA. Observe that \(R\) is strictly henselian for example by Lemma 0BSQ and Algebra, Lemma 00IC. Then the lemma is clear.
Let \(X \to S\) be a proper morphism with geometrically connected fibres. Let \(R\) be a strictly henselian discrete valuation ring and let \(\Spec(R) \to S\) be a morphism. Let \(\eta, s \in \Spec(R)\) be the generic and closed point. Then we can consider the specialization map \[sp_R : \pi_1(X_{\overline{\eta}}) \to \pi_1(X_s)\] for the base change \(X_R/\Spec(R)\). Note that this makes sense as \(s\) has algebraically closed residue field.
Lemma
Let \(f : X \to S\) be a proper morphism with geometrically connected fibres. Let \(s' \leadsto s\) be a specialization of points of \(S\) and let \(sp : \pi_1(X_{\overline{s}'}) \to \pi_1(X_{\overline{s}})\) be a specialization map. If \(S\) is Noetherian, then there exists a strictly henselian discrete valuation ring \(R\) over \(S\) such that \(sp\) is isomorphic to \(sp_R\) defined above.
Proof
Let \(\mathcal{O}_{S, s} \to A\) be the strict henselization constructed using \(\kappa(s) \to \kappa(\overline{s})\). Let \(A \to \kappa(\overline{s}')\) be the map used to construct \(sp\). Let \(R \subset \kappa(\overline{s}')\) be a discrete valuation ring dominating the image of \(A\), see Algebra, Lemma 00PH. Choose a diagram of fields \[\xymatrix{ \kappa(\overline{s}) \ar[r] & k \\ A/\mathfrak m_A \ar[r] \ar[u] & R/\mathfrak m_R \ar[u] }\] with \(k\) algebraically closed. Let \(R^{sh}\) be the strict henselization of \(R\) constructed using \(R \to k\). We obtain a map \(A \to R^{sh}\) from Algebra, Lemma 04GU. The ring \(R^{sh}\) is a discrete valuation ring by More on Algebra, Lemma 0AP3. Denote \(\eta, o\) the generic and closed point of \(\Spec(R^{sh})\). Since the diagram of schemes \[\xymatrix{ \overline{\eta} \ar[d] \ar[r] & \Spec(R^{sh}) \ar[d] & \Spec(k) \ar[d] \ar[l] \\ \overline{s}' \ar[r] & \Spec(A) & \overline{s} \ar[l] }\] commutes, we obtain a commutative diagram \[\xymatrix{ \pi_1(X_{\overline{\eta}}) \ar[d] \ar[r]_{sp_{R^{sh}}} & \pi_1(X_o) \ar[d] \\ \pi_1(X_{\overline{s}'}) \ar[r]^{sp} & X_{\overline{s}} }\] of specialization maps by the construction of these maps. Since the vertical arrows are isomorphisms (Lemma 0A49), this proves the lemma.
Restriction to a closed subscheme
In this section we prove some results about the restriction functor \[\textit{F\'Et}_X \longrightarrow \textit{F\'Et}_Y,\quad U \longmapsto V = U \times_X Y\] where \(X\) is a scheme and \(Y\) is a closed subscheme. Using the topological invariance of the fundamental group, we can relate the study of this functor to the completion functor on finite locally free modules.
In the following lemmas we use the concept of coherent formal modules defined in Cohomology of Schemes, Section 0EHN. Given a Noetherian scheme and a quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_X\) we will say an object \((\mathcal{F}_n)\) of \(\textit{Coh}(X, \mathcal{I})\) is finite locally free if each \(\mathcal{F}_n\) is a finite locally free \(\mathcal{O}_X/\mathcal{I}^n\)-module.
Lemma
Let \(X\) be a Noetherian scheme and let \(Y \subset X\) be a closed subscheme with ideal sheaf \(\mathcal{I} \subset \mathcal{O}_X\). Assume the completion functor \[\textit{Coh}(\mathcal{O}_X) \longrightarrow \textit{Coh}(X, \mathcal{I}),\quad \mathcal{F} \longmapsto \mathcal{F}^\wedge\] is fully faithful on the full subcategory of finite locally free objects (see above). Then the restriction functor \(\textit{F\'Et}_X \to \textit{F\'Et}_Y\) is fully faithful.
Proof
Since the category of finite étale coverings has an internal hom (Lemma 0BL7) it suffices to prove the following: Given \(U\) finite étale over \(X\) and a morphism \(t : Y \to U\) over \(X\) there exists a unique section \(s : X \to U\) such that \(t = s|_Y\). Picture \[\xymatrix{ & U \ar[d]^f \\ Y \ar[r] \ar[ru] & X \ar@{..>}@/^1em/[u] }\] Finding the dotted arrow \(s\) is the same thing as finding an \(\mathcal{O}_X\)-algebra map \[s^\sharp : f_*\mathcal{O}_U \longrightarrow \mathcal{O}_X\] which reduces modulo the ideal sheaf of \(Y\) to the given algebra map \(t^\sharp : f_*\mathcal{O}_U \to \mathcal{O}_Y\). By Lemma 0BQB we can lift \(t\) uniquely to a compatible system of maps \(t_n : Y_n \to U\) and hence a map \[\lim t_n^\sharp : f_*\mathcal{O}_U \longrightarrow \lim \mathcal{O}_{Y_n}\] of sheaves of algebras on \(X\). Since \(f_*\mathcal{O}_U\) is a finite locally free \(\mathcal{O}_X\)-module, we conclude that we get a unique \(\mathcal{O}_X\)-module map \(\sigma : f_*\mathcal{O}_U \to \mathcal{O}_X\) whose completion is \(\lim t_n^\sharp\). To see that \(\sigma\) is an algebra homomorphism, we need to check that the diagram \[\xymatrix{ f_*\mathcal{O}_U \otimes_{\mathcal{O}_X} f_*\mathcal{O}_U \ar[r] \ar[d]_{\sigma \otimes \sigma} & f_*\mathcal{O}_U \ar[d]^\sigma \\ \mathcal{O}_X \otimes_{\mathcal{O}_X} \mathcal{O}_X \ar[r] & \mathcal{O}_X }\] commutes. For every \(n\) we know this diagram commutes after restricting to \(Y_n\), i.e., the diagram commutes after applying the completion functor. Hence by faithfulness of the completion functor we conclude.
Lemma
Let \(X\) be a Noetherian scheme and let \(Y \subset X\) be a closed subscheme with ideal sheaf \(\mathcal{I} \subset \mathcal{O}_X\). Assume the completion functor \[\textit{Coh}(\mathcal{O}_X) \longrightarrow \textit{Coh}(X, \mathcal{I}),\quad \mathcal{F} \longmapsto \mathcal{F}^\wedge\] is an equivalence on full subcategories of finite locally free objects (see above). Then the restriction functor \(\textit{F\'Et}_X \to \textit{F\'Et}_Y\) is an equivalence.
Proof
The restriction functor is fully faithful by Lemma 0EL8.
Let \(U_1 \to Y\) be a finite étale morphism. To finish the proof we will show that \(U_1\) is in the essential image of the restriction functor.
For \(n \geq 1\) let \(Y_n\) be the \(n\)th infinitesimal neighbourhood of \(Y\). By Lemma 0BQB there is a unique finite étale morphism \(\pi_n : U_n \to Y_n\) whose base change to \(Y = Y_1\) recovers \(U_1 \to Y_1\). Consider the sheaves \(\mathcal{F}_n = \pi_{n, *}\mathcal{O}_{U_n}\). We may and do view \(\mathcal{F}_n\) as an \(\mathcal{O}_X\)-module on \(X\) which is locally isomorphic to \((\mathcal{O}_X/f^{n + 1}\mathcal{O}_X)^{\oplus r}\). This \((\mathcal{F}_n)\) is a finite locally free object of \(\textit{Coh}(X, \mathcal{I})\). By assumption there exists a finite locally free \(\mathcal{O}_X\)-module \(\mathcal{F}\) and a compatible system of isomorphisms \[\mathcal{F}/\mathcal{I}^n\mathcal{F} \to \mathcal{F}_n\] of \(\mathcal{O}_X\)-modules.
To construct an algebra structure on \(\mathcal{F}\) consider the multiplication maps \(\mathcal{F}_n \otimes_{\mathcal{O}_X} \mathcal{F}_n \to \mathcal{F}_n\) coming from the fact that \(\mathcal{F}_n = \pi_{n, *}\mathcal{O}_{U_n}\) are sheaves of algebras. These define a map \[(\mathcal{F}\otimes_{\mathcal{O}_X} \mathcal{F})^\wedge \longrightarrow \mathcal{F}^\wedge\] in the category \(\textit{Coh}(X, \mathcal{I})\). Hence by assumption we may assume there is a map \(\mu : \mathcal{F}\otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{F}\) whose restriction to \(Y_n\) gives the multiplication maps above. By faithfulness of the functor in the statement of the lemma, we conclude that \(\mu\) defines a commutative \(\mathcal{O}_X\)-algebra structure on \(\mathcal{F}\) compatible with the given algebra structures on \(\mathcal{F}_n\). Setting \[U = \underline{\Spec}_X((\mathcal{F}, \mu))\] we obtain a finite locally free scheme \(\pi : U \to X\) whose restriction to \(Y\) is isomorphic to \(U_1\). The discriminant of \(\pi\) is the zero set of the section \[\det(Q_\pi) : \mathcal{O}_X \longrightarrow \wedge^{top}(\pi_*\mathcal{O}_U)^{\otimes -2}\] constructed in Discriminants, Section 0BVH. Since the restriction of this to \(Y_n\) is an isomorphism for all \(n\) by Discriminants, Lemma 0BJF we conclude that it is an isomorphism. Thus \(\pi\) is étale by Discriminants, Lemma 0BJF.
Lemma
Let \(X\) be a Noetherian scheme and let \(Y \subset X\) be a closed subscheme with ideal sheaf \(\mathcal{I} \subset \mathcal{O}_X\). Let \(\mathcal{V}\) be the set of open subschemes \(V \subset X\) containing \(Y\) ordered by reverse inclusion. Assume the completion functor \[\colim_\mathcal{V} \textit{Coh}(\mathcal{O}_V) \longrightarrow \textit{Coh}(X, \mathcal{I}), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge\] defines is fully faithful on the full subcategory of finite locally free objects (see above). Then the restriction functor \(\colim_\mathcal{V} \textit{F\'Et}_V \to \textit{F\'Et}_Y\) is fully faithful.
Proof
Observe that \(\mathcal{V}\) is a directed set, so the colimits are as in Categories, Section 04AX. The rest of the argument is almost exactly the same as the argument in the proof of Lemma 0EL8; we urge the reader to skip it.
Since the category of finite étale coverings has an internal hom (Lemma 0BL7) it suffices to prove the following: Given \(U\) finite étale over \(V \in \mathcal{V}\) and a morphism \(t : Y \to U\) over \(V\) there exists a \(V' \geq V\) and a morphism \(s : V' \to U\) over \(V\) such that \(t = s|_Y\). Picture \[\xymatrix{ & & U \ar[d]^f \\ Y \ar[r] \ar[rru] & V' \ar@{..>}[ru] \ar[r] & V }\] Finding the dotted arrow \(s\) is the same thing as finding an \(\mathcal{O}_{V'}\)-algebra map \[s^\sharp : f_*\mathcal{O}_U|_{V'} \longrightarrow \mathcal{O}_{V'}\] which reduces modulo the ideal sheaf of \(Y\) to the given algebra map \(t^\sharp : f_*\mathcal{O}_U \to \mathcal{O}_Y\). By Lemma 0BQB we can lift \(t\) uniquely to a compatible system of maps \(t_n : Y_n \to U\) and hence a map \[\lim t_n^\sharp : f_*\mathcal{O}_U \longrightarrow \lim \mathcal{O}_{Y_n}\] of sheaves of algebras on \(V\). Observe that \(f_*\mathcal{O}_U\) is a finite locally free \(\mathcal{O}_V\)-module. Hence we get a \(V' \geq V\) a map \(\sigma : f_*\mathcal{O}_U|_{V'} \to \mathcal{O}_{V'}\) whose completion is \(\lim t_n^\sharp\). To see that \(\sigma\) is an algebra homomorphism, we need to check that the diagram \[\xymatrix{ (f_*\mathcal{O}_U \otimes_{\mathcal{O}_V} f_*\mathcal{O}_U)|_{V'} \ar[r] \ar[d]_{\sigma \otimes \sigma} & f_*\mathcal{O}_U|_{V'} \ar[d]^\sigma \\ \mathcal{O}_{V'} \otimes_{\mathcal{O}_{V'}} \mathcal{O}_{V'} \ar[r] & \mathcal{O}_{V'} }\] commutes. For every \(n\) we know this diagram commutes after restricting to \(Y_n\), i.e., the diagram commutes after applying the completion functor. Hence by faithfulness of the completion functor we deduce that there exists a \(V'' \geq V'\) such that \(\sigma|_{V''}\) is an algebra homomorphism as desired.
Lemma
Let \(X\) be a Noetherian scheme and let \(Y \subset X\) be a closed subscheme with ideal sheaf \(\mathcal{I} \subset \mathcal{O}_X\). Let \(\mathcal{V}\) be the set of open subschemes \(V \subset X\) containing \(Y\) ordered by reverse inclusion. Assume the completion functor \[\colim_\mathcal{V} \textit{Coh}(\mathcal{O}_V) \longrightarrow \textit{Coh}(X, \mathcal{I}), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge\] defines an equivalence of the full subcategories of finite locally free objects (see explanation above). Then the restriction functor \[\colim_\mathcal{V} \textit{F\'Et}_V \to \textit{F\'Et}_Y\] is an equivalence.
Proof
Observe that \(\mathcal{V}\) is a directed set, so the colimits are as in Categories, Section 04AX. The rest of the argument is almost exactly the same as the argument in the proof of Lemma 0EL9; we urge the reader to skip it.
The restriction functor is fully faithful by Lemma 0ELA.
Let \(U_1 \to Y\) be a finite étale morphism. To finish the proof we will show that \(U_1\) is in the essential image of the restriction functor.
For \(n \geq 1\) let \(Y_n\) be the \(n\)th infinitesimal neighbourhood of \(Y\). By Lemma 0BQB there is a unique finite étale morphism \(\pi_n : U_n \to Y_n\) whose base change to \(Y = Y_1\) recovers \(U_1 \to Y_1\). Consider the sheaves \(\mathcal{F}_n = \pi_{n, *}\mathcal{O}_{U_n}\). We may and do view \(\mathcal{F}_n\) as an \(\mathcal{O}_X\)-module on \(X\) which is locally isomorphic to \((\mathcal{O}_X/f^{n + 1}\mathcal{O}_X)^{\oplus r}\). This \((\mathcal{F}_n)\) is a finite locally free object of \(\textit{Coh}(X, \mathcal{I})\). By assumption there exists a \(V \in \mathcal{V}\) and a finite locally free \(\mathcal{O}_V\)-module \(\mathcal{F}\) and a compatible system of isomorphisms \[\mathcal{F}/\mathcal{I}^n\mathcal{F} \to \mathcal{F}_n\] of \(\mathcal{O}_V\)-modules.
To construct an algebra structure on \(\mathcal{F}\) consider the multiplication maps \(\mathcal{F}_n \otimes_{\mathcal{O}_V} \mathcal{F}_n \to \mathcal{F}_n\) coming from the fact that \(\mathcal{F}_n = \pi_{n, *}\mathcal{O}_{U_n}\) are sheaves of algebras. These define a map \[(\mathcal{F}\otimes_{\mathcal{O}_V} \mathcal{F})^\wedge \longrightarrow \mathcal{F}^\wedge\] in the category \(\textit{Coh}(X, \mathcal{I})\). Hence by assumption after shrinking \(V\) we may assume there is a map \(\mu : \mathcal{F}\otimes_{\mathcal{O}_V} \mathcal{F} \to \mathcal{F}\) whose restriction to \(Y_n\) gives the multiplication maps above. After possibly shrinking further we may assume \(\mu\) defines a commutative \(\mathcal{O}_V\)-algebra structure on \(\mathcal{F}\) compatible with the given algebra structures on \(\mathcal{F}_n\). Setting \[U = \underline{\Spec}_V((\mathcal{F}, \mu))\] we obtain a finite locally free scheme over \(V\) whose restriction to \(Y\) is isomorphic to \(U_1\). It follows that \(U \to V\) is étale at all points lying over \(Y\), see More on Morphisms, Lemma 0A43. Thus after shrinking \(V\) once more we may assume \(U \to V\) is finite étale. This finishes the proof.
Lemma
Let \(X\) be a scheme and let \(Y \subset X\) be a closed subscheme. If every connected component of \(X\) meets \(Y\), then the restriction functor \(\textit{F\'Et}_X \to \textit{F\'Et}_Y\) is faithful.
Proof
Let \(a, b : U \to U'\) be two morphisms of schemes finite étale over \(X\) whose restriction to \(Y\) are the same. The image of a connected component of \(U\) is an connected component of \(X\); this follows from Topology, Lemma 07VB applied to the restriction of \(U \to X\) to a connected component of \(X\). Hence the image of every connected component of \(U\) meets \(Y\) by assumption. We conclude that \(a = b\) after restriction to each connected component of \(U\) by Étale Morphisms, Proposition 024V. Since the equalizer of \(a\) and \(b\) is an open subscheme of \(U\) (as the diagonal of \(U'\) over \(X\) is open) we conclude.
Lemma
Let \(X\) be a Noetherian scheme and let \(Y \subset X\) be a closed subscheme. Let \(Y_n \subset X\) be the \(n\)th infinitesimal neighbourhood of \(Y\) in \(X\). Assume one of the following holds
\(X\) is quasi-affine and \(\Gamma(X, \mathcal{O}_X) \to \lim \Gamma(Y_n, \mathcal{O}_{Y_n})\) is an isomorphism, or
\(X\) has an ample invertible module \(\mathcal{L}\) and \(\Gamma(X, \mathcal{L}^{\otimes m}) \to \lim \Gamma(Y_n, \mathcal{L}^{\otimes m}|_{Y_n})\) is an isomorphism for all \(m \gg 0\), or
for every finite locally free \(\mathcal{O}_X\)-module \(\mathcal{E}\) the map \(\Gamma(X, \mathcal{E}) \to \lim \Gamma(Y_n, \mathcal{E}|_{Y_n})\) is an isomorphism.
Then the restriction functor \(\textit{F\'Et}_X \to \textit{F\'Et}_Y\) is fully faithful.
Proof
This lemma follows formally from Lemma 0EL8 and Algebraic and Formal Geometry, Lemma 0EKP.
Lemma
Let \(X\) be a Noetherian scheme and let \(Y \subset X\) be a closed subscheme. Let \(Y_n \subset X\) be the \(n\)th infinitesimal neighbourhood of \(Y\) in \(X\). Let \(\mathcal{V}\) be the set of open subschemes \(V \subset X\) containing \(Y\) ordered by reverse inclusion. Assume one of the following holds
\(X\) is quasi-affine and \[\colim_\mathcal{V} \Gamma(V, \mathcal{O}_V) \longrightarrow \lim \Gamma(Y_n, \mathcal{O}_{Y_n})\] is an isomorphism, or
\(X\) has an ample invertible module \(\mathcal{L}\) and \[\colim_\mathcal{V} \Gamma(V, \mathcal{L}^{\otimes m}) \longrightarrow \lim \Gamma(Y_n, \mathcal{L}^{\otimes m}|_{Y_n})\] is an isomorphism for all \(m \gg 0\), or
for every \(V \in \mathcal{V}\) and every finite locally free \(\mathcal{O}_V\)-module \(\mathcal{E}\) the map \[\colim_{V' \geq V} \Gamma(V', \mathcal{E}|_{V'}) \longrightarrow \lim \Gamma(Y_n, \mathcal{E}|_{Y_n})\] is an isomorphism.
Then the functor \[\colim_\mathcal{V} \textit{F\'Et}_V \to \textit{F\'Et}_Y\] is fully faithful.
Proof
This lemma follows formally from Lemma 0ELA and Algebraic and Formal Geometry, Lemma 0EK2.
Pushouts and fundamental groups
Here is the main result.
Lemma
In More on Morphisms, Situation 0ECI, for example if \(Z \to Y\) and \(Z \to X\) are closed immersions of schemes, there is an equivalence of categories \[\textit{F\'Et}_{Y \amalg_Z X} \longrightarrow \textit{F\'Et}_Y \times_{\textit{F\'Et}_Z} \textit{F\'Et}_X\]
Proof
The pushout exists by More on Morphisms, Proposition 0E25. The functor is given by sending a scheme \(U\) finite étale over the pushout to the base changes \(Y' = U \times_{Y \amalg_Z X} Y\) and \(X' = U \times_{Y \amalg_Z X} X\) and the natural isomorphism \(Y' \times_Y Z \to X' \times_X Z\) over \(Z\). To prove this functor is an equivalence we use More on Morphisms, Lemma 0ECL to construct a quasi-inverse functor. The only thing left to prove is to show that given a morphism \(U \to Y \amalg_Z X\) which is separated, quasi-finite and étale such that \(X' \to X\) and \(Y' \to Y\) are finite, then \(U \to Y \amalg_Z X\) is finite. This can either be deduced from the corresponding algebra fact (More on Algebra, Lemma 0D2H) or it can be seen because \[X' \amalg Y' \to U\] is surjective and \(X'\) and \(Y'\) are proper over \(Y \amalg_Z X\) (this uses the description of the pushout in More on Morphisms, Proposition 0E25) and then we can apply Morphisms, Lemma 0AH6 to conclude that \(U\) is proper over \(Y \amalg_Z X\). Since a quasi-finite and proper morphism is finite (More on Morphisms, Lemma 02LS) we win.
Finite étale covers of punctured spectra, I
We first prove some results á la Lefschetz.
Situation
Let \((A, \mathfrak m)\) be a Noetherian local ring and \(f \in \mathfrak m\). We set \(X = \Spec(A)\) and \(X_0 = \Spec(A/fA)\) and we let \(U = X \setminus \{\mathfrak m\}\) and \(U_0 = X_0 \setminus \{\mathfrak m\}\) be the punctured spectrum of \(A\) and \(A/fA\).
Recall that for a scheme \(X\) the category of schemes finite étale over \(X\) is denoted \(\textit{F\'Et}_X\), see Section 0BL6. In Situation 0BLF we will study the base change functors \[\xymatrix{ \textit{F\'Et}_X \ar[d] \ar[r] & \textit{F\'Et}_U \ar[d] \\ \textit{F\'Et}_{X_0} \ar[r] & \textit{F\'Et}_{U_0} }\] In many case the right vertical arrow is faithful.
Lemma
In Situation 0BLF. Assume one of the following holds
\(\dim(A/\mathfrak p) \geq 2\) for every minimal prime \(\mathfrak p \subset A\) with \(f \not \in \mathfrak p\), or
every connected component of \(U\) meets \(U_0\).
Then \[\textit{F\'Et}_U \longrightarrow \textit{F\'Et}_{U_0},\quad V \longmapsto V_0 = V \times_U U_0\] is a faithful functor.
Proof
Case (2) is immediate from Lemma 0EJX. Assumption (1) implies every irreducible component of \(U\) meets \(U_0\), see Algebra, Lemma 00KW. Hence (1) follows from (2).
Before we prove something more interesting, we need a couple of lemmas.
Lemma
In Situation 0BLF. Let \(V \to U\) be a finite morphism. Let \(A^\wedge\) be the \(\mathfrak m\)-adic completion of \(A\), let \(X' = \Spec(A^\wedge)\) and let \(U'\) and \(V'\) be the base changes of \(U\) and \(V\) to \(X'\). If \(Y' \to X'\) is a finite morphism such that \(V' = Y' \times_{X'} U'\), then there exists a finite morphism \(Y \to X\) such that \(V = Y \times_X U\) and \(Y' = Y \times_X X'\).
Proof
This is a straightforward application of More on Algebra, Proposition 05ER. Namely, choose generators \(f_1, \ldots, f_t\) of \(\mathfrak m\). For each \(i\) write \(V \times_U D(f_i) = \Spec(B_i)\). For \(1 \leq i, j \leq n\) we obtain an isomorphism \(\alpha_{ij} : (B_i)_{f_j} \to (B_j)_{f_i}\) of \(A_{f_if_j}\)-algebras because the spectrum of both represent \(V \times_U D(f_if_j)\). Write \(Y' = \Spec(B')\). Since \(V \times_U U' = Y \times_{X'} U'\) we get isomorphisms \(\alpha_i : B'_{f_i} \to B_i \otimes_A A^\wedge\). A straightforward argument shows that \((B', B_i, \alpha_i, \alpha_{ij})\) is an object of \(\text{Glue}(A \to A^\wedge, f_1, \ldots, f_t)\), see More on Algebra, Remark 05EL. Applying the proposition cited above (and using More on Algebra, Remark 05EU to obtain the algebra structure) we find an \(A\)-algebra \(B\) such that \(\text{Can}(B)\) is isomorphic to \((B', B_i, \alpha_i, \alpha_{ij})\). Setting \(Y = \Spec(B)\) we see that \(Y \to X\) is a morphism which comes equipped with compatible isomorphisms \(V \cong Y \times_X U\) and \(Y' = Y \times_X X'\) as desired.
Lemma
In Situation 0BLF assume \(A\) is henselian or more generally that \((A, (f))\) is a henselian pair. Let \(A^\wedge\) be the \(\mathfrak m\)-adic completion of \(A\), let \(X' = \Spec(A^\wedge)\) and let \(U'\) and \(U'_0\) be the base changes of \(U\) and \(U_0\) to \(X'\). If \(\textit{F\'Et}_{U'} \to \textit{F\'Et}_{U'_0}\) is fully faithful, then \(\textit{F\'Et}_U \to \textit{F\'Et}_{U_0}\) is fully faithful.
Proof
Assume \(\textit{F\'Et}_{U'} \longrightarrow \textit{F\'Et}_{U'_0}\) is a fully faithful. Since \(X' \to X\) is faithfully flat, it is immediate that the functor \(V \to V_0 = V \times_U U_0\) is faithful. Since the category of finite étale coverings has an internal hom (Lemma 0BL7) it suffices to prove the following: Given \(V\) finite étale over \(U\) we have \[\Mor_U(U, V) = \Mor_{U_0}(U_0, V_0)\] The we assume we have a morphism \(s_0 : U_0 \to V_0\) over \(U_0\) and we will produce a morphism \(s : U \to V\) over \(U\).
By our assumption there does exist a morphism \(s' : U' \to V'\) whose restriction to \(V'_0\) is the base change \(s'_0\) of \(s_0\). Since \(V' \to U'\) is finite étale this means that \(V' = s'(U') \amalg W'\) for some \(W' \to U'\) finite and étale. Choose a finite morphism \(Z' \to X'\) such that \(W' = Z' \times_{X'} U'\). This is possible by Zariski’s main theorem in the form stated in More on Morphisms, Lemma 05K0 (small detail omitted). Then \[V' = s'(U') \amalg W' \longrightarrow X' \amalg Z' = Y'\] is an open immersion such that \(V' = Y' \times_{X'} U'\). By Lemma 0BLH we can find \(Y \to X\) finite such that \(V = Y \times_X U\) and \(Y' = Y \times_X X'\). Write \(Y = \Spec(B)\) so that \(Y' = \Spec(B \otimes_A A^\wedge)\). Then \(B \otimes_A A^\wedge\) has an idempotent \(e'\) corresponding to the open and closed subscheme \(X'\) of \(Y' = X' \amalg Z'\).
The case \(A\) is henselian (slightly easier). The image \(\overline{e}\) of \(e'\) in \(B \otimes_A \kappa(\mathfrak m) = B/\mathfrak mB\) lifts to an idempotent \(e\) of \(B\) as \(A\) is henselian (because \(B\) is a product of local rings by Algebra, Lemma 04GG). Then we see that \(e\) maps to \(e'\) by uniqueness of lifts of idempotents (using that \(B \otimes_A A^\wedge\) is a product of local rings). Let \(Y_1 \subset Y\) be the open and closed subscheme corresponding to \(e\). Then \(Y_1 \times_X X' = s'(X')\) which implies that \(Y_1 \to X\) is an isomorphism (by faithfully flat descent) and gives the desired section.
The case where \((A, (f))\) is a henselian pair. Here we use that \(s'\) is a lift of \(s'_0\). Namely, let \(Y_{0, 1} \subset Y_0 = Y \times_X X_0\) be the closure of \(s_0(U_0) \subset V_0 = Y_0 \times_{X_0} U_0\). As \(X' \to X\) is flat, the base change \(Y'_{0, 1} \subset Y'_0\) is the closure of \(s'_0(U'_0)\) which is equal to \(X'_0 \subset Y'_0\) (see Morphisms, Lemma 081I). Since \(Y'_0 \to Y_0\) is submersive (Morphisms, Lemma 02JY) we conclude that \(Y_{0, 1}\) is open and closed in \(Y_0\). Let \(e_0 \in B/fB\) be the corresponding idempotent. By More on Algebra, Lemma 09XI we can lift \(e_0\) to an idempotent \(e \in B\). Then we conclude as before.
In Situation 0BLF fully faithfulness of the restriction functor \(\textit{F\'Et}_U \longrightarrow \textit{F\'Et}_{U_0}\) holds under fairly mild assumptions. In particular, the assumptions often do not imply \(U\) is a connected scheme, but the conclusion guarantees that \(U\) and \(U_0\) have the same number of connected components.
Lemma
In Situation 0BLF. Assume
\(A\) has a dualizing complex,
the pair \((A, (f))\) is henselian,
one of the following is true
\(A_f\) is \((S_2)\) and every irreducible component of \(X\) not contained in \(X_0\) has dimension \(\geq 3\), or
for every prime \(\mathfrak p \subset A\), \(f \not \in \mathfrak p\) we have \(\text{depth}(A_\mathfrak p) + \dim(A/\mathfrak p) > 2\).
Then the restriction functor \(\textit{F\'Et}_U \longrightarrow \textit{F\'Et}_{U_0}\) is fully faithful.
Proof
Let \(A'\) be the \(\mathfrak m\)-adic completion of \(A\). We will show that the hypotheses remain true for \(A'\). This is clear for conditions (a) and (b). Condition (c)(ii) is preserved by Local Cohomology, Lemma 0EHW. Next, assume (c)(i) holds. Since \(A\) is universally catenary (Dualizing Complexes, Lemma 0A80) we see that every irreducible component of \(\Spec(A')\) not contained in \(V(f)\) has dimension \(\geq 3\), see More on Algebra, Proposition 0AW6. Since \(A \to A'\) is flat with Gorenstein fibres, the condition that \(A_f\) is \((S_2)\) implies that \(A'_f\) is \((S_2)\). References used: Dualizing Complexes, Section 0BJM, More on Algebra, Section 0BIR, and Algebra, Lemma 0339. Thus by Lemma 0BLI we may assume that \(A\) is a Noetherian complete local ring.
Assume \(A\) is a complete local ring in addition to the other assumptions. By Lemma 0EL8 the result follows from Algebraic and Formal Geometry, Lemma 0EKT.
Lemma
In Situation 0BLF. Assume
\(H^1_\mathfrak m(A)\) and \(H^2_\mathfrak m(A)\) are annihilated by a power of \(f\), and
\(A\) is henselian or more generally \((A, (f))\) is a henselian pair.
Then the restriction functor \(\textit{F\'Et}_U \longrightarrow \textit{F\'Et}_{U_0}\) is fully faithful.
Proof
By Lemma 0BLI we may assume that \(A\) is a Noetherian complete local ring. (The assumptions carry over; use Dualizing Complexes, Lemma 0ALZ.) By Lemma 0EL8 the result follows from Algebraic and Formal Geometry, Lemma 0EKS.
Lemma
In Situation 0BLF assume \(A\) has depth \(\geq 3\) and \(A\) is henselian or more generally \((A, (f))\) is a henselian pair. Then the restriction functor \(\textit{F\'Et}_U \to \textit{F\'Et}_{U_0}\) is fully faithful.
Proof
The assumption of depth forces \(H^1_\mathfrak m(A) = H^2_\mathfrak m(A) = 0\), see Dualizing Complexes, Lemma 0AVZ. Hence Lemma 0BM6 applies.
Purity in local case, I
Let \((A, \mathfrak m)\) be a Noetherian local ring. Set \(X = \Spec(A)\) and let \(U = X \setminus \{\mathfrak m\}\) be the punctured spectrum. We say purity holds for \((A, \mathfrak m)\) if the restriction functor \[\textit{F\'Et}_X \longrightarrow \textit{F\'Et}_U\] is essentially surjective. In this section we try to understand how the question changes when one passes from \(X\) to a hypersurface \(X_0\) in \(X\), in other words, we study a kind of local Lefschetz property for the fundamental groups of punctured spectra. These results will be useful to proceed by induction on dimension in the proofs of our main results on local purity, namely, Lemma 0BMA, Proposition 0BPD, and Proposition 0EYC.
Lemma
Let \((A, \mathfrak m)\) be a Noetherian local ring. Set \(X = \Spec(A)\) and let \(U = X \setminus \{\mathfrak m\}\). Let \(\pi : Y \to X\) be a finite morphism such that \(\text{depth}(\mathcal{O}_{Y, y}) \geq 2\) for all closed points \(y \in Y\). Then \(Y\) is the spectrum of \(B = \mathcal{O}_Y(\pi^{-1}(U))\).
Proof
Set \(V = \pi^{-1}(U)\) and denote \(\pi' : V \to U\) the restriction of \(\pi\). Consider the \(\mathcal{O}_X\)-module map \[\pi_*\mathcal{O}_Y \longrightarrow j_*\pi'_*\mathcal{O}_V\] where \(j : U \to X\) is the inclusion morphism. We claim Divisors, Lemma 0E9I applies to this map. If so, then \(B = \Gamma(Y, \mathcal{O}_Y)\) and we see that the lemma holds. Let \(x \in X\) be the closed point. It suffices to show that \(\text{depth}((\pi_*\mathcal{O}_Y)_x) \geq 2\). Let \(y_1, \ldots, y_n \in Y\) be the points mapping to \(x\). By Algebra, Lemma 0AUK it suffices to show that \(\text{depth}(\mathcal{O}_{Y, y_i}) \geq 2\) for \(i = 1, \ldots, n\). Since this is the assumption of the lemma the proof is complete.
Lemma
Let \((A, \mathfrak m)\) be a Noetherian local ring. Set \(X = \Spec(A)\) and let \(U = X \setminus \{\mathfrak m\}\). Let \(V\) be finite étale over \(U\). Assume \(A\) has depth \(\geq 2\). The following are equivalent
\(V = Y \times_X U\) for some \(Y \to X\) finite étale,
\(B = \Gamma(V, \mathcal{O}_V)\) is finite étale over \(A\).
Proof
Denote \(\pi : V \to U\) the given finite étale morphism. Assume \(Y\) as in (1) exists. Let \(x \in X\) be the point corresponding to \(\mathfrak m\). Let \(y \in Y\) be a point mapping to \(x\). We claim that \(\text{depth}(\mathcal{O}_{Y, y}) \geq 2\). This is true because \(Y \to X\) is étale and hence \(A = \mathcal{O}_{X, x}\) and \(\mathcal{O}_{Y, y}\) have the same depth (Algebra, Lemma 0337). Hence Lemma 0BM8 applies and \(Y = \Spec(B)\).
The implication (2) \(\Rightarrow\) (1) is easier and the details are omitted.
Lemma
Let \((A, \mathfrak m)\) be a Noetherian local ring. Set \(X = \Spec(A)\) and let \(U = X \setminus \{\mathfrak m\}\). Assume \(A\) is normal of dimension \(\geq 2\). The functor \[\textit{F\'Et}_U \longrightarrow \left\{ \begin{matrix} \text{finite normal }A\text{-algebras }B\text{ such} \\ \text{that }\Spec(B) \to X\text{ is \'etale over }U \end{matrix} \right\}, \quad V \longmapsto \Gamma(V, \mathcal{O}_V)\] is an equivalence. Moreover, \(V = Y \times_X U\) for some \(Y \to X\) finite étale if and only if \(B = \Gamma(V, \mathcal{O}_V)\) is finite étale over \(A\).
Proof
Observe that \(\text{depth}(A) \geq 2\) because \(A\) is normal (Serre’s criterion for normality, Algebra, Lemma 031S). Thus the final statement follows from Lemma 0BLK. Given \(\pi : V \to U\) finite étale, set \(B = \Gamma(V, \mathcal{O}_V)\). If we can show that \(B\) is normal and finite over \(A\), then we obtain the displayed functor. Since there is an obvious quasi-inverse functor, this is also all that we have to show.
Since \(A\) is normal, the scheme \(V\) is normal (Descent, Lemma 034F). Hence \(V\) is a finite disjoint union of integral schemes (Properties, Lemma 033M). Thus we may assume \(V\) is integral. In this case the function field \(L\) of \(V\) (Morphisms, Section 01RR) is a finite separable extension of the fraction field of \(A\) (because we get it by looking at the generic fibre of \(V \to U\) and using Morphisms, Lemma 02GL). By Algebra, Lemma 032L the integral closure \(B' \subset L\) of \(A\) in \(L\) is finite over \(A\). By More on Algebra, Lemma 0BM4 we see that \(B'\) is a reflexive \(A\)-module, which in turn implies that \(\text{depth}_A(B') \geq 2\) by More on Algebra, Lemma 0AVB.
Let \(f \in \mathfrak m\). Then \(B_f = \Gamma(V \times_U D(f), \mathcal{O}_V)\) (Properties, Lemma 01P7). Hence \(B'_f = B_f\) because \(B_f\) is normal (see above), finite over \(A_f\) with fraction field \(L\). It follows that \(V = \Spec(B') \times_X U\). Then we conclude that \(B = B'\) from Lemma 0BM8 applied to \(\Spec(B') \to X\). This lemma applies because the localizations \(B'_{\mathfrak m'}\) of \(B'\) at maximal ideals \(\mathfrak m' \subset B'\) lying over \(\mathfrak m\) have depth \(\geq 2\) by Algebra, Lemma 0AUK and the remark on depth in the preceding paragraph.
Lemma
Let \((A, \mathfrak m)\) be a Noetherian local ring. Set \(X = \Spec(A)\) and let \(U = X \setminus \{\mathfrak m\}\). Let \(V\) be finite étale over \(U\). Let \(A^\wedge\) be the \(\mathfrak m\)-adic completion of \(A\), let \(X' = \Spec(A^\wedge)\) and let \(U'\) and \(V'\) be the base changes of \(U\) and \(V\) to \(X'\). The following are equivalent
\(V = Y \times_X U\) for some \(Y \to X\) finite étale, and
\(V' = Y' \times_{X'} U'\) for some \(Y' \to X'\) finite étale.
Proof
The implication (1) \(\Rightarrow\) (2) follows from taking the base change of a solution \(Y \to X\). Let \(Y' \to X'\) be as in (2). By Lemma 0BLH we can find \(Y \to X\) finite such that \(V = Y \times_X U\) and \(Y' = Y \times_X X'\). By descent we see that \(Y \to X\) is finite étale (Algebra, Lemmas 03C4 and 00U2). This finishes the proof.
The point of the following two lemmas is that the assumptions do not force \(A\) to have depth \(\geq 3\). For example if \(A\) is a complete normal local domain of dimension \(\geq 3\) and \(f \in \mathfrak m\) is nonzero, then the assumptions are satisfied.
Lemma
In Situation 0BLF. Let \(V\) be finite étale over \(U\). Assume
\(A\) has a dualizing complex,
the pair \((A, (f))\) is henselian,
one of the following is true
\(A_f\) is \((S_2)\) and every irreducible component of \(X\) not contained in \(X_0\) has dimension \(\geq 3\), or
for every prime \(\mathfrak p \subset A\), \(f \not \in \mathfrak p\) we have \(\text{depth}(A_\mathfrak p) + \dim(A/\mathfrak p) > 2\).
\(V_0 = V \times_U U_0\) is equal to \(Y_0 \times_{X_0} U_0\) for some \(Y_0 \to X_0\) finite étale.
Then \(V = Y \times_X U\) for some \(Y \to X\) finite étale.
Proof
We reduce to the complete case using Lemma 0BLL. (The assumptions carry over; see proof of Lemma 0EK5.)
In the complete case we can lift \(Y_0 \to X_0\) to a finite étale morphism \(Y \to X\) by More on Algebra, Lemma 09ZL; observe that \((A, fA)\) is a henselian pair by More on Algebra, Lemma 0ALJ. Then we can use Lemma 0EK5 to see that \(V\) is isomorphic to \(Y \times_X U\) and the proof is complete.
Lemma
In Situation 0BLF. Let \(V\) be finite étale over \(U\). Assume
\(H^1_\mathfrak m(A)\) and \(H^2_\mathfrak m(A)\) are annihilated by a power of \(f\),
\(V_0 = V \times_U U_0\) is equal to \(Y_0 \times_{X_0} U_0\) for some \(Y_0 \to X_0\) finite étale.
Then \(V = Y \times_X U\) for some \(Y \to X\) finite étale.
Proof
We reduce to the complete case using Lemma 0BLL. (The assumptions carry over; use Dualizing Complexes, Lemma 0ALZ.)
In the complete case we can lift \(Y_0 \to X_0\) to a finite étale morphism \(Y \to X\) by More on Algebra, Lemma 09ZL; observe that \((A, fA)\) is a henselian pair by More on Algebra, Lemma 0ALJ. Then we can use Lemma 0BM6 to see that \(V\) is isomorphic to \(Y \times_X U\) and the proof is complete.
Lemma
In Situation 0BLF. Let \(V\) be finite étale over \(U\). Assume
\(A\) has depth \(\geq 3\),
\(V_0 = V \times_U U_0\) is equal to \(Y_0 \times_{X_0} U_0\) for some \(Y_0 \to X_0\) finite étale.
Then \(V = Y \times_X U\) for some \(Y \to X\) finite étale.
Proof
The assumption of depth forces \(H^1_\mathfrak m(A) = H^2_\mathfrak m(A) = 0\), see Dualizing Complexes, Lemma 0AVZ. Hence Lemma 0BLS applies.
Purity of branch locus
We will use the discriminant of a finite locally free morphism. See Discriminants, Section 0BVH.
Lemma
Let \((A, \mathfrak m)\) be a Noetherian local ring with \(\dim(A) \geq 1\). Let \(f \in \mathfrak m\). Then there exist a \(\mathfrak p \in V(f)\) with \(\dim(A_\mathfrak p) = 1\).
Proof
By induction on \(\dim(A)\). If \(\dim(A) = 1\), then \(\mathfrak p = \mathfrak m\) works. If \(\dim(A) > 1\), then let \(Z \subset \Spec(A)\) be an irreducible component of dimension \(> 1\). Then \(V(f) \cap Z\) has dimension \(> 0\) (Algebra, Lemma 00KW). Pick a prime \(\mathfrak q \in V(f) \cap Z\), \(\mathfrak q \not = \mathfrak m\) corresponding to a closed point of the punctured spectrum of \(A\); this is possible by Properties, Lemma 02IM. Then \(\mathfrak q\) is not the generic point of \(Z\). Hence \(0 < \dim(A_\mathfrak q) < \dim(A)\) and \(f \in \mathfrak q A_\mathfrak q\). By induction on the dimension we can find \(f \in \mathfrak p \subset A_\mathfrak q\) with \(\dim((A_\mathfrak q)_\mathfrak p) = 1\). Then \(\mathfrak p \cap A\) works.
Lemma
Let \(f : X \to Y\) be a morphism of locally Noetherian schemes. Let \(x \in X\). Assume
\(f\) is flat,
\(f\) is quasi-finite at \(x\),
\(x\) is not a generic point of an irreducible component of \(X\),
for specializations \(x' \leadsto x\) with \(\dim(\mathcal{O}_{X, x'}) = 1\) our \(f\) is unramified at \(x'\).
Then \(f\) is étale at \(x\).
Proof
Observe that the set of points where \(f\) is unramified is the same as the set of points where \(f\) is étale and that this set is open. See Morphisms, Definitions 02G4 and 02GI and Lemma 02GV. To check \(f\) is étale at \(x\) we may work étale locally on the base and on the target (Descent, Lemmas 02VN and 036W). Thus we can apply More on Morphisms, Lemma 02LK and assume that \(f : X \to Y\) is finite and that \(x\) is the unique point of \(X\) lying over \(y = f(x)\). Then it follows that \(f\) is finite locally free (Morphisms, Lemma 02KB).
Assume \(f\) is finite locally free and that \(x\) is the unique point of \(X\) lying over \(y = f(x)\). By Discriminants, Lemma 0BJF we find a locally principal closed subscheme \(D_\pi \subset Y\) such that \(y' \in D_\pi\) if and only if there exists an \(x' \in X\) with \(f(x') = y'\) and \(f\) ramified at \(x'\). Thus we have to prove that \(y \not \in D_\pi\). Assume \(y \in D_\pi\) to get a contradiction.
By condition (3) we have \(\dim(\mathcal{O}_{X, x}) \geq 1\). We have \(\dim(\mathcal{O}_{X, x}) = \dim(\mathcal{O}_{Y, y})\) by Algebra, Lemma 00ON. By Lemma 0BJG we can find \(y' \in D_\pi\) specializing to \(y\) with \(\dim(\mathcal{O}_{Y, y'}) = 1\). Choose \(x' \in X\) with \(f(x') = y'\) where \(f\) is ramified. Since \(f\) is finite it is closed, and hence \(x' \leadsto x\). We have \(\dim(\mathcal{O}_{X, x'}) = \dim(\mathcal{O}_{Y, y'}) = 1\) as before. This contradicts property (4).
Lemma
Let \((A, \mathfrak m)\) be a regular local ring of dimension \(d \geq 2\). Set \(X = \Spec(A)\) and \(U = X \setminus \{\mathfrak m\}\). Then
the functor \(\textit{F\'Et}_X \to \textit{F\'Et}_U\) is essentially surjective, i.e., purity holds for \(A\),
any finite \(A \to B\) with \(B\) normal which induces a finite étale morphism on punctured spectra is étale.
Proof
Recall that a regular local ring is normal by Algebra, Lemma 0567. Hence (1) and (2) are equivalent by Lemma 0BM9. We prove the lemma by induction on \(d\).
The case \(d = 2\). In this case \(A \to B\) is flat. Namely, we have going down for \(A \to B\) by Algebra, Proposition 00H8. Then \(\dim(B_{\mathfrak m'}) = 2\) for all maximal ideals \(\mathfrak m' \subset B\) by Algebra, Lemma 00ON. Then \(B_{\mathfrak m'}\) is Cohen-Macaulay by Algebra, Lemma 031S. Hence and this is the important step Algebra, Lemma 00R4 applies to show \(A \to B_{\mathfrak m'}\) is flat. Then Algebra, Lemma 00HT shows \(A \to B\) is flat. Thus we can apply Lemma 0BJH (or you can directly argue using the easier Discriminants, Lemma 0BJF) to see that \(A \to B\) is étale.
The case \(d \geq 3\). Let \(V \to U\) be finite étale. Let \(f \in \mathfrak m_A\), \(f \not \in \mathfrak m_A^2\). Then \(A/fA\) is a regular local ring of dimension \(d - 1 \geq 2\), see Algebra, Lemma 00NQ. Let \(U_0\) be the punctured spectrum of \(A/fA\) and let \(V_0 = V \times_U U_0\). By Lemma 0BLM it suffices to show that \(V_0\) is in the essential image of \(\textit{F\'Et}_{\Spec(A/fA)} \to \textit{F\'Et}_{U_0}\). This follows from the induction hypothesis.
Lemma
Let \(f : X \to Y\) be a morphism of locally Noetherian schemes. Let \(x \in X\) and set \(y = f(x)\). Assume
\(\mathcal{O}_{X, x}\) is normal,
\(\mathcal{O}_{Y, y}\) is regular,
\(f\) is quasi-finite at \(x\),
\(\dim(\mathcal{O}_{X, x}) = \dim(\mathcal{O}_{Y, y}) \geq 1\)
for specializations \(x' \leadsto x\) with \(\dim(\mathcal{O}_{X, x'}) = 1\) our \(f\) is unramified at \(x'\).
Then \(f\) is étale at \(x\).
Proof
We will prove the lemma by induction on \(d = \dim(\mathcal{O}_{X, x}) = \dim(\mathcal{O}_{Y, y})\).
An uninteresting case is when \(d = 1\). In that case we are assuming that \(f\) is unramified at \(x\) and that \(\mathcal{O}_{Y, y}\) is a discrete valuation ring (Algebra, Lemma 00PD). Then \(\mathcal{O}_{X, x}\) is flat over \(\mathcal{O}_{Y, y}\) (otherwise the map would not be quasi-finite at \(x\)) and we see that \(f\) is flat at \(x\). Since flat \(+\) unramified is étale we conclude (some details omitted).
The case \(d \geq 2\). We will use induction on \(d\) to reduce to the case discussed in Lemma 0BMA. To check \(f\) is étale at \(x\) we may work étale locally on the base and on the target (Descent, Lemmas 02VN and 036W). Thus we can apply More on Morphisms, Lemma 02LK and assume that \(f : X \to Y\) is finite and that \(x\) is the unique point of \(X\) lying over \(y\). Here we use that étale extensions of local rings do not change dimension, normality, and regularity, see More on Algebra, Section 0AGY and Étale Morphisms, Section 025L.
Next, we can base change by \(\Spec(\mathcal{O}_{Y, y})\) and assume that \(Y\) is the spectrum of a regular local ring. It follows that \(X = \Spec(\mathcal{O}_{X, x})\) as every point of \(X\) necessarily specializes to \(x\).
The ring map \(\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}\) is finite and necessarily injective (by equality of dimensions). We conclude we have going down for \(\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}\) by Algebra, Proposition 00H8 (and the fact that a regular ring is a normal ring by Algebra, Lemma 0567). Pick \(x' \in X\), \(x' \not = x\) with image \(y' = f(x')\). Then \(\mathcal{O}_{X, x'}\) is normal as a localization of a normal domain. Similarly, \(\mathcal{O}_{Y, y'}\) is regular (see Algebra, Lemma 0AFS). We have \(\dim(\mathcal{O}_{X, x'}) = \dim(\mathcal{O}_{Y, y'})\) by Algebra, Lemma 00ON (we checked going down above). Of course these dimensions are strictly less than \(d\) as \(x' \not = x\) and by induction on \(d\) we conclude that \(f\) is étale at \(x'\).
Thus we arrive at the following situation: We have a finite local homomorphism \(A \to B\) of Noetherian local rings of dimension \(d \geq 2\), with \(A\) regular, \(B\) normal, which induces a finite étale morphism \(V \to U\) on punctured spectra. Our goal is to show that \(A \to B\) is étale. This follows from Lemma 0BMA and the proof is complete.
The following lemma is sometimes useful to find the maximal open subset over which a finite étale morphism extends.
Lemma
Let \(j : U \to X\) be an open immersion of locally Noetherian schemes such that \(\text{depth}(\mathcal{O}_{X, x}) \geq 2\) for \(x \not \in U\). Let \(\pi : V \to U\) be finite étale. Then
\(\mathcal{B} = j_*\pi_*\mathcal{O}_V\) is a reflexive coherent \(\mathcal{O}_X\)-algebra, set \(Y = \underline{\Spec}_X(\mathcal{B})\),
\(Y \to X\) is the unique finite morphism such that \(V = Y \times_X U\) and \(\text{depth}(\mathcal{O}_{Y, y}) \geq 2\) for \(y \in Y \setminus V\),
\(Y \to X\) is étale at \(y\) if and only if \(Y \to X\) is flat at \(y\), and
\(Y \to X\) is étale if and only if \(\mathcal{B}\) is finite locally free as an \(\mathcal{O}_X\)-module.
Moreover, (a) the construction of \(\mathcal{B}\) and \(Y \to X\) commutes with base change by flat morphisms \(X' \to X\) of locally Noetherian schemes, and (b) if \(V' \to U'\) is a finite étale morphism with \(U \subset U' \subset X\) open which restricts to \(V \to U\) over \(U\), then there is a unique isomorphism \(Y' \times_X U' = V'\) over \(U'\).
Proof
Observe that \(\pi_*\mathcal{O}_V\) is a finite locally free \(\mathcal{O}_U\)-module, in particular reflexive. By Divisors, Lemma 0EBJ the module \(j_*\pi_*\mathcal{O}_V\) is the unique reflexive coherent module on \(X\) restricting to \(\pi_*\mathcal{O}_V\) over \(U\). This proves (1).
By construction \(Y \times_X U = V\). Since \(\mathcal{B}\) is coherent, we see that \(Y \to X\) is finite. We have \(\text{depth}(\mathcal{B}_x) \geq 2\) for \(x \in X \setminus U\) by Divisors, Lemma 0EBI. Hence \(\text{depth}(\mathcal{O}_{Y, y}) \geq 2\) for \(y \in Y \setminus V\) by Algebra, Lemma 0AUK. Conversely, suppose that \(\pi' : Y' \to X\) is a finite morphism such that \(V = Y' \times_X U\) and \(\text{depth}(\mathcal{O}_{Y', y'}) \geq 2\) for \(y' \in Y' \setminus V\). Then \(\pi'_*\mathcal{O}_{Y'}\) restricts to \(\pi_*\mathcal{O}_V\) over \(U\) and satisfies \(\text{depth}((\pi'_*\mathcal{O}_{Y'})_x) \geq 2\) for \(x \in X \setminus U\) by Algebra, Lemma 0AUK. Then \(\pi'_*\mathcal{O}_{Y'}\) is canonically isomorphic to \(j_*\pi_*\mathcal{O}_V\) for example by Divisors, Lemma 0E9I. This proves (2).
If \(Y \to X\) is étale at \(y\), then \(Y \to X\) is flat at \(y\). Conversely, suppose that \(Y \to X\) is flat at \(y\). If \(y \in V\), then \(Y \to X\) is étale at \(y\). If \(y \not \in V\), then we check (1), (2), (3), and (4) of Lemma 0BJH hold to see that \(Y \to X\) is étale at \(y\). Parts (1) and (2) are clear and so is (3) since \(\text{depth}(\mathcal{O}_{Y, y}) \geq 2\). If \(y' \leadsto y\) is a specialization and \(\dim(\mathcal{O}_{Y, y'}) = 1\), then \(y' \in V\) since otherwise the depth of this local ring would be \(2\) a contradiction by Algebra, Lemma 00LK. Hence \(Y \to X\) is étale at \(y'\) and we conclude (4) of Lemma 0BJH holds too. This finishes the proof of (3).
Part (4) follows from (3) and the fact that \(((Y \to X)_*\mathcal{O}_Y)_x\) is a flat \(\mathcal{O}_{X, x}\)-module if and only if \(\mathcal{O}_{Y, y}\) is a flat \(\mathcal{O}_{X, x}\)-module for all \(y \in Y\) mapping to \(x\), see Algebra, Lemma 00HT. Here we also use that a finite flat module over a Noetherian ring is finite locally free, see Algebra, Lemma 00NX (and Algebra, Lemma 00FP).
As to the final assertions of the lemma, part (a) follows from flat base change, see Cohomology of Schemes, Lemma 02KH and part (b) follows from the uniqueness in (2) applied to the restriction \(Y \times_X U'\).
Lemma
Let \(j : U \to X\) be an open immersion of Noetherian schemes such that purity holds for \(\mathcal{O}_{X, x}\) for all \(x \not \in U\). Then \[\textit{F\'Et}_X \longrightarrow \textit{F\'Et}_U\] is essentially surjective.
Proof
Let \(V \to U\) be a finite étale morphism. By Noetherian induction it suffices to extend \(V \to U\) to a finite étale morphism to a strictly larger open subset of \(X\). Let \(x \in X \setminus U\) be the generic point of an irreducible component of \(X \setminus U\). Then the inverse image \(U_x\) of \(U\) in \(\Spec(\mathcal{O}_{X, x})\) is the punctured spectrum of \(\mathcal{O}_{X, x}\). By assumption \(V_x = V \times_U U_x\) is the restriction of a finite étale morphism \(Y_x \to \Spec(\mathcal{O}_{X, x})\) to \(U_x\). By Limits, Lemma 0BQ5 we find an open subscheme \(U \subset U' \subset X\) containing \(x\) and a morphism \(V' \to U'\) of finite presentation whose restriction to \(U\) recovers \(V \to U\) and whose restriction to \(\Spec(\mathcal{O}_{X, x})\) recovering \(Y_x\). Finally, the morphism \(V' \to U'\) is finite étale after possible shrinking \(U'\) to a smaller open by Limits, Lemma 0EY3.
Finite étale covers of punctured spectra, II
In this section we prove some variants of the material discussed in Section 0BLE. Suppose we have a Noetherian local ring \((A, \mathfrak m)\) and \(f \in \mathfrak m\). We set \(X = \Spec(A)\) and \(X_0 = \Spec(A/fA)\) and we let \(U = X \setminus \{\mathfrak m\}\) and \(U_0 = X_0 \setminus \{\mathfrak m\}\) be the punctured spectrum of \(A\) and \(A/fA\). All of this is exactly as in Situation 0BLF. The difference is that we will consider the restriction functor \[\colim_{U_0 \subset U' \subset U\text{ open}} \textit{F\'Et}_{U'} \longrightarrow \textit{F\'Et}_{U_0}\] In other words, we will not try to lift finite étale coverings of \(U_0\) to all of \(U\), but just to some open neighbourhood \(U'\) of \(U_0\) in \(U\).
Lemma
In Situation 0BLF. Let \(U' \subset U\) be open and contain \(U_0\). Assume for \(\mathfrak p \subset A\) minimal with \(\mathfrak p \in U'\), \(\mathfrak p \not \in U_0\) we have \(\dim(A/\mathfrak p) \geq 2\). Then \[\textit{F\'Et}_{U'} \longrightarrow \textit{F\'Et}_{U_0},\quad V' \longmapsto V_0 = V' \times_{U'} U_0\] is a faithful functor. Moreover, there exists a \(U'\) satisfying the assumption and any smaller open \(U'' \subset U'\) containing \(U_0\) also satisfies this assumption. In particular, the restriction functor \[\colim_{U_0 \subset U' \subset U\text{ open}} \textit{F\'Et}_{U'} \longrightarrow \textit{F\'Et}_{U_0}\] is faithful.
Proof
By Algebra, Lemma 00KW we see that \(V(\mathfrak p)\) meets \(U_0\) for every prime \(\mathfrak p\) of \(A\) with \(\dim(A/\mathfrak p) \geq 2\). Thus the displayed functor is faithful for a \(U\) as in the statement by Lemma 0EJX. To see the existence of such a \(U'\) note that for \(\mathfrak p \subset A\) with \(\mathfrak p \in U\), \(\mathfrak p \not \in U_0\) with \(\dim(A/\mathfrak p) = 1\) then \(\mathfrak p\) corresponds to a closed point of \(U\) and hence \(V(\mathfrak p) \cap U_0 = \emptyset\). Thus we can take \(U'\) to be the complement of the irreducible components of \(X\) which do not meet \(U_0\) and have dimension \(1\).
Lemma
In Situation 0BLF assume
\(A\) has a dualizing complex and is \(f\)-adically complete,
every irreducible component of \(X\) not contained in \(X_0\) has dimension \(\geq 3\).
Then the restriction functor \[\colim_{U_0 \subset U' \subset U\text{ open}} \textit{F\'Et}_{U'} \longrightarrow \textit{F\'Et}_{U_0}\] is fully faithful.
Proof
To prove this we may replace \(A\) by its reduction by the topological invariance of the fundamental group, see Lemma 0BQB. Then the result follows from Lemma 0ELA and Algebraic and Formal Geometry, Lemma 0EKU.
Lemma
In Situation 0BLF assume
\(A\) is \(f\)-adically complete,
\(f\) is a nonzerodivisor.
\(H^1_\mathfrak m(A/fA)\) is a finite \(A\)-module.
Then the restriction functor \[\colim_{U_0 \subset U' \subset U\text{ open}} \textit{F\'Et}_{U'} \longrightarrow \textit{F\'Et}_{U_0}\] is fully faithful.
Proof
Follows from Lemma 0ELA and Algebraic and Formal Geometry, Lemma 0EKV.
Finite étale covers of punctured spectra, III
In this section we study when in Situation 0BLF. the restriction functor \[\colim_{U_0 \subset U' \subset U\text{ open}} \textit{F\'Et}_{U'} \longrightarrow \textit{F\'Et}_{U_0}\] is an equivalence of categories.
Lemma
In Situation 0BLF assume
\(A\) has a dualizing complex and is \(f\)-adically complete,
one of the following is true
\(A_f\) is \((S_2)\) and every irreducible component of \(X\) not contained in \(X_0\) has dimension \(\geq 4\), or
if \(\mathfrak p \not \in V(f)\) and \(V(\mathfrak p) \cap V(f) \not = \{\mathfrak m\}\), then \(\text{depth}(A_\mathfrak p) + \dim(A/\mathfrak p) > 3\).
Then the restriction functor \[\colim_{U_0 \subset U' \subset U\text{ open}} \textit{F\'Et}_{U'} \longrightarrow \textit{F\'Et}_{U_0}\] is an equivalence.
Proof
This follows from Lemma 0EK1 and Algebraic and Formal Geometry, Lemma 0EKY.
Lemma
In Situation 0BLF assume
\(A\) is \(f\)-adically complete,
\(f\) is a nonzerodivisor,
\(H^1_\mathfrak m(A/fA)\) and \(H^2_\mathfrak m(A/fA)\) are finite \(A\)-modules.
Then the restriction functor \[\colim_{U_0 \subset U' \subset U\text{ open}} \textit{F\'Et}_{U'} \longrightarrow \textit{F\'Et}_{U_0}\] is an equivalence.
Proof
This follows from Lemma 0EK1 and Algebraic and Formal Geometry, Lemma 0EKZ.
Remark
Let \((A, \mathfrak m)\) be a complete local Noetherian ring and \(f \in \mathfrak m\) nonzero. Suppose that \(A_f\) is \((S_2)\) and every irreducible component of \(\Spec(A)\) has dimension \(\geq 4\). Then Lemma 0DXY tells us that the category \[\colim\nolimits_{U' \subset U\text{ open, }U_0 \subset U} \text{ category of schemes finite \'etale over }U'\] is equivalent to the category of schemes finite étale over \(U_0\). For example this holds if \(A\) is a normal domain of dimension \(\geq 4\)!
Finite étale covers of punctured spectra, IV
Let \(X, X_0, U, U_0\) be as in Situation 0BLF. In this section we ask when the restriction functor \[\textit{F\'Et}_U \longrightarrow \textit{F\'Et}_{U_0}\] is essentially surjective. We will do this by taking results from Section 0EK7 and then filling in the gaps using purity. Recall that we say purity holds for a Noetherian local ring \((A, \mathfrak m)\) if the restriction functor \(\textit{F\'Et}_X \to \textit{F\'Et}_U\) is essentially surjective where \(X = \Spec(A)\) and \(U = X \setminus \{\mathfrak m\}\).
Lemma
In Situation 0BLF assume
\(A\) has a dualizing complex and is \(f\)-adically complete,
one of the following is true
\(A_f\) is \((S_2)\) and every irreducible component of \(X\) not contained in \(X_0\) has dimension \(\geq 4\), or
if \(\mathfrak p \not \in V(f)\) and \(V(\mathfrak p) \cap V(f) \not = \{\mathfrak m\}\), then \(\text{depth}(A_\mathfrak p) + \dim(A/\mathfrak p) > 3\).
for every maximal ideal \(\mathfrak p \subset A_f\) purity holds for \((A_f)_\mathfrak p\).
Then the restriction functor \(\textit{F\'Et}_U \to \textit{F\'Et}_{U_0}\) is essentially surjective.
Proof
Let \(V_0 \to U_0\) be a finite étale morphism. By Lemma 0DXY there exists an open \(U' \subset U\) containing \(U_0\) and a finite étale morphism \(V' \to U\) whose base change to \(U_0\) is isomorphic to \(V_0 \to U_0\). Since \(U' \supset U_0\) we see that \(U \setminus U'\) consists of points corresponding to prime ideals \(\mathfrak p_1, \ldots, \mathfrak p_n\) as in (3). By assumption we can find finite étale morphisms \(V'_i \to \Spec(A_{\mathfrak p_i})\) agreeing with \(V' \to U'\) over \(U' \times_U \Spec(A_{\mathfrak p_i})\). By Limits, Lemma 0BPA applied \(n\) times we see that \(V' \to U'\) extends to a finite étale morphism \(V \to U\).
Lemma
Let \((A, \mathfrak m)\) be a Noetherian local ring. Let \(f \in \mathfrak m\). Assume
\(A\) is \(f\)-adically complete,
\(f\) is a nonzerodivisor,
\(H^1_\mathfrak m(A/fA)\) and \(H^2_\mathfrak m(A/fA)\) are finite \(A\)-modules,
for every maximal ideal \(\mathfrak p \subset A_f\) purity holds for \((A_f)_\mathfrak p\).
Then the restriction functor \(\textit{F\'Et}_U \to \textit{F\'Et}_{U_0}\) is essentially surjective.
Proof
The proof is identical to the proof of Lemma 0EK9 using Lemma 0BLV in stead of Lemma 0DXY.
Purity in local case, II
This section is the continuation of Section 0BM7. Recall that we say purity holds for a Noetherian local ring \((A, \mathfrak m)\) if the restriction functor \(\textit{F\'Et}_X \to \textit{F\'Et}_U\) is essentially surjective where \(X = \Spec(A)\) and \(U = X \setminus \{\mathfrak m\}\).
Lemma
Let \((A, \mathfrak m)\) be a Noetherian local ring. Let \(f \in \mathfrak m\). Assume
\(A\) has a dualizing complex and is \(f\)-adically complete,
one of the following is true
\(A_f\) is \((S_2)\) and every irreducible component of \(X\) not contained in \(X_0\) has dimension \(\geq 4\), or
if \(\mathfrak p \not \in V(f)\) and \(V(\mathfrak p) \cap V(f) \not = \{\mathfrak m\}\), then \(\text{depth}(A_\mathfrak p) + \dim(A/\mathfrak p) > 3\).
for every maximal ideal \(\mathfrak p \subset A_f\) purity holds for \((A_f)_\mathfrak p\), and
purity holds for \(A\).
Then purity holds for \(A/fA\).
Proof
Denote \(X = \Spec(A)\) and \(U = X \setminus \{\mathfrak m\}\) the punctured spectrum. Similarly we have \(X_0 = \Spec(A/fA)\) and \(U_0 = X_0 \setminus \{\mathfrak m\}\). Let \(V_0 \to U_0\) be a finite étale morphism. By Lemma 0EK9 we find a finite étale morphism \(V \to U\) whose base change to \(U_0\) is isomorphic to \(V_0 \to U_0\). By assumption (5) we find that \(V \to U\) extends to a finite étale morphism \(Y \to X\). Then the restriction of \(Y\) to \(X_0\) is the desired extension of \(V_0 \to U_0\).
Lemma
Let \((A, \mathfrak m)\) be a Noetherian local ring. Let \(f \in \mathfrak m\). Assume
\(A\) is \(f\)-adically complete,
\(f\) is a nonzerodivisor,
\(H^1_\mathfrak m(A/fA)\) and \(H^2_\mathfrak m(A/fA)\) are finite \(A\)-modules,
for every maximal ideal \(\mathfrak p \subset A_f\) purity holds for \((A_f)_\mathfrak p\),
purity holds for \(A\).
Then purity holds for \(A/fA\).
Proof
The proof is identical to the proof of Lemma 0DXZ using Lemma 0EKA in stead of Lemma 0EK9.
Now we can bootstrap the earlier results to prove that purity holds for complete intersections of dimension \(\geq 3\). Recall that a Noetherian local ring is called a complete intersection if its completion is the quotient of a regular local ring by the ideal generated by a regular sequence. See the discussion in Divided Power Algebra, Section 09PY.
Proposition
Let \((A, \mathfrak m)\) be a Noetherian local ring. If \(A\) is a complete intersection of dimension \(\geq 3\), then purity holds for \(A\) in the sense that any finite étale cover of the punctured spectrum extends.
Proof
By Lemma 0BLL we may assume that \(A\) is a complete local ring. By assumption we can write \(A = B/(f_1, \ldots, f_r)\) where \(B\) is a complete regular local ring and \(f_1, \ldots, f_r\) is a regular sequence. We will finish the proof by induction on \(r\). The base case is \(r = 0\) which follows from Lemma 0BMA which applies to regular rings of dimension \(\geq 2\).
Assume that \(A = B/(f_1, \ldots, f_r)\) and that the proposition holds for \(r - 1\). Set \(A' = B/(f_1, \ldots, f_{r - 1})\) and apply Lemma 0BPC to \(f_r \in A'\). This is permissible: condition (1) holds as \(f_1, \ldots, f_r\) is a regular sequence, condition (2) holds as \(B\) and hence \(A'\) is complete, condition (3) holds as \(A = A'/f_r A'\) is Cohen-Macaulay of dimension \(\dim(A) \geq 3\), see Dualizing Complexes, Lemma 0AVZ, condition (4) holds by induction hypothesis as \(\dim((A'_{f_r})_\mathfrak p) \geq 3\) for a maximal prime \(\mathfrak p\) of \(A'_{f_r}\) and as \((A'_{f_r})_\mathfrak p = B_\mathfrak q/(f_1, \ldots, f_{r - 1})\) for some \(\mathfrak q \subset B\), condition (5) holds by induction hypothesis.
Purity in local case, III
In this section is a continuation of the discussion in Sections 0BM7 and 0BPB.
Lemma
Let \((A, \mathfrak m)\) be a Noetherian local ring of depth \(\geq 2\). Let \(B = A[[x_1, \ldots, x_d]]\) with \(d \geq 1\). Set \(Y = \Spec(B)\) and \(Y_0 = V(x_1, \ldots, x_d)\). For any open subscheme \(V \subset Y\) with \(V_0 = V \cap Y_0\) equal to \(Y_0 \setminus \{\mathfrak m_B\}\) the restriction functor \[\textit{F\'Et}_V \longrightarrow \textit{F\'Et}_{V_0}\] is fully faithful.
Proof
Set \(I = (x_1, \ldots, x_d)\). Set \(X = \Spec(A)\). If we use the map \(Y \to X\) to identify \(Y_0\) with \(X\), then \(V_0\) is identified with the punctured spectrum \(U\) of \(A\). Pushing forward modules by this affine morphism we get \[\begin{align*} \lim_n \Gamma(V_0, \mathcal{O}_V/I^n\mathcal{O}_V) & = \lim_n \Gamma(V_0, \mathcal{O}_Y/I^n\mathcal{O}_Y) \\ & = \lim_n \Gamma(U, \mathcal{O}_U[x_1, \ldots, x_d]/(x_1, \ldots, x_d)^n) \\ & = \lim_n A[x_1, \ldots, x_d]/(x_1, \ldots, x_d)^n \\ & = B \end{align*}\] Namely, as the depth of \(A\) is \(\geq 2\) we have \(\Gamma(U, \mathcal{O}_U) = A\), see Local Cohomology, Lemma 0BK0. Thus for any \(V \subset Y\) open as in the lemma we get \[B = \Gamma(Y, \mathcal{O}_Y) \to \Gamma(V, \mathcal{O}_V) \to \lim_n \Gamma(V_0, \mathcal{O}_Y/I^n\mathcal{O}_Y) = B\] which implies both arrows are isomorphisms (small detail omitted). By Algebraic and Formal Geometry, Lemma 0EKP we conclude that \(\textit{Coh}(\mathcal{O}_V) \to \textit{Coh}(V, I\mathcal{O}_V)\) is fully faithful on the full subcategory of finite locally free objects. Thus we conclude by Lemma 0EL8.
Lemma
Let \((A, \mathfrak m)\) be a Noetherian local ring of depth \(\geq 2\). Let \(B = A[[x_1, \ldots, x_d]]\) with \(d \geq 1\). For any open \(V \subset Y = \Spec(B)\) which contains
any prime \(\mathfrak q \subset B\) such that \(\mathfrak q \cap A \not = \mathfrak m\),
the prime \(\mathfrak m B\)
the functor \(\textit{F\'Et}_Y \to \textit{F\'Et}_V\) is an equivalence. In particular purity holds for \(B\).
Proof
A prime \(\mathfrak q \subset B\) which is not contained in \(V\) lies over \(\mathfrak m\). In this case \(A \to B_\mathfrak q\) is a flat local homomorphism and hence \(\text{depth}(B_\mathfrak q) \geq 2\) (Algebra, Lemma 0337). Thus the functor is fully faithful by Lemma 0BQF combined with Local Cohomology, Lemma 0BLR.
Let \(W \to V\) be a finite étale morphism. Let \(B \to C\) be the unique finite ring map such that \(\Spec(C) \to Y\) is the finite morphism extending \(W \to V\) constructed in Lemma 0EY6. Observe that \(C = \Gamma(W, \mathcal{O}_W)\).
Set \(Y_0 = V(x_1, \ldots, x_d)\) and \(V_0 = V \cap Y_0\). Set \(X = \Spec(A)\). If we use the map \(Y \to X\) to identify \(Y_0\) with \(X\), then \(V_0\) is identified with the punctured spectrum \(U\) of \(A\). Thus we may view \(W_0 = W \times_Y Y_0\) as a finite étale scheme over \(U\). Then \[W_0 \times_U (U \times_X Y) \quad\text{and}\quad W \times_V (U \times_X Y)\] are schemes finite étale over \(U \times_X Y\) which restrict to isomorphic finite étale schemes over \(V_0\). By Lemma 0EY9 applied to the open \(U \times_X Y\) we obtain an isomorphism \[W_0 \times_U (U \times_X Y) \longrightarrow W \times_V (U \times_X Y)\] over \(U \times_X Y\).
Observe that \(C_0 = \Gamma(W_0, \mathcal{O}_{W_0})\) is a finite \(A\)-algebra by Lemma 0EY6 applied to \(W_0 \to U \subset X\) (exactly as we did for \(B \to C\) above). Since the construction in Lemma 0EY6 is compatible with flat base change and with change of opens, the isomorphism above induces an isomorphism \[\Psi : C \longrightarrow C_0 \otimes_A B\] of finite \(B\)-algebras. However, we know that \(\Spec(C) \to Y\) is étale at all points above at least one point of \(Y\) lying over \(\mathfrak m \in X\). Since \(\Psi\) is an isomorphism, we conclude that \(\Spec(C_0) \to X\) is étale above \(\mathfrak m\) (small detail omitted). Of course this means that \(A \to C_0\) is finite étale and hence \(B \to C\) is finite étale.
Lemma
Let \(f : X \to S\) be a morphism of schemes. Let \(U \subset X\) be an open subscheme. Assume
\(f\) is smooth,
\(S\) is Noetherian,
for \(s \in S\) with \(\text{depth}(\mathcal{O}_{S, s}) \leq 1\) we have \(X_s = U_s\),
\(U_s \subset X_s\) is dense for all \(s \in S\).
Then \(\textit{F\'Et}_X \to \textit{F\'Et}_U\) is an equivalence.
Proof
The functor is fully faithful by Lemma 0BQF combined with Local Cohomology, Lemma 0BLR (plus an application of Algebra, Lemma 0337 to check the depth condition).
Let \(\pi : V \to U\) be a finite étale morphism. Let \(Y \to X\) be the finite morphism constructed in Lemma 0EY6. We have to show that \(Y \to X\) is finite étale. To show that this is true for all points \(x \in X\) mapping to a given point \(s \in S\) we may perform a base change by a flat morphism \(S' \to S\) of Noetherian schemes such that \(s\) is in the image. This follows from the compatibility of the construction in Lemma 0EY6 with flat base change.
After enlarging \(U\) we may assume \(U \subset X\) is the maximal open over which \(Y \to X\) is finite étale. Let \(Z \subset X\) be the complement of \(U\). To get a contradiction, assume \(Z \not = \emptyset\). Let \(s \in S\) be a point in the image of \(Z \to S\) such that no strict generalization of \(s\) is in the image. Then after base change to \(\Spec(\mathcal{O}_{S, s})\) we see that \(S = \Spec(A)\) with \((A, \mathfrak m, \kappa)\) a local Noetherian ring of depth \(\geq 2\) and \(Z\) contained in the closed fibre \(X_s\) and nowhere dense in \(X_s\). Choose a closed point \(z \in Z\). Then \(\kappa(z)/\kappa\) is finite (by the Hilbert Nullstellensatz, see Algebra, Theorem 00FV). Choose a finite flat morphism \((S', s') \to (S, s)\) of local schemes realizing the residue field extension \(\kappa(z)/\kappa\), see Algebra, Lemma 07NE. After doing a base change by \(S' \to S\) we reduce to the case where \(\kappa(z) = \kappa\).
By More on Morphisms, Lemma 057G there exists a locally closed subscheme \(S' \subset X\) passing through \(z\) such that \(S' \to S\) is étale at \(z\). After performing the base change by \(S' \to S\), we may assume there is a section \(\sigma : S \to X\) such that \(\sigma(s) = z\). Choose an affine neighbourhood \(\Spec(B) \subset X\) of \(s\). Then \(A \to B\) is a smooth ring map which has a section \(\sigma : B \to A\). Denote \(I = \Ker(\sigma)\) and denote \(B^\wedge\) the \(I\)-adic completion of \(B\). Then \(B^\wedge \cong A[[x_1, \ldots, x_d]]\) for some \(d \geq 0\), see Algebra, Lemma 05D5. Observe that \(d > 0\) since otherwise we see that \(X \to S\) is étale at \(z\) which would imply that \(z\) is a generic point of \(X_s\) and hence \(z \in U\) by assumption (4). Similarly, if \(d > 0\), then \(\mathfrak m B^\wedge\) maps into \(U\) via the morphism \(\Spec(B^\wedge) \to X\). It suffices prove \(Y \to X\) is finite étale after base change to \(\Spec(B^\wedge)\). Since \(B \to B^\wedge\) is flat (Algebra, Lemma 00MB) this follows from Lemma 0EYA and the uniqueness in the construction of \(Y \to X\).
Proposition
Let \(A \to B\) be a local homomorphism of local Noetherian rings. Assume \(A\) has depth \(\geq 2\), \(A \to B\) is formally smooth for the \(\mathfrak m_B\)-adic topology, and \(\dim(B) > \dim(A)\). For any open \(V \subset Y = \Spec(B)\) which contains
any prime \(\mathfrak q \subset B\) such that \(\mathfrak q \cap A \not = \mathfrak m_A\),
the prime \(\mathfrak m_A B\)
the functor \(\textit{F\'Et}_Y \to \textit{F\'Et}_V\) is an equivalence. In particular purity holds for \(B\).
Proof
A prime \(\mathfrak q \subset B\) which is not contained in \(V\) lies over \(\mathfrak m_A\). In this case \(A \to B_\mathfrak q\) is a flat local homomorphism and hence \(\text{depth}(B_\mathfrak q) \geq 2\) (Algebra, Lemma 0337). Thus the functor is fully faithful by Lemma 0BQF combined with Local Cohomology, Lemma 0BLR.
Denote \(A^\wedge\) and \(B^\wedge\) the completions of \(A\) and \(B\) with respect to their maximal ideals. Observe that the assumptions of the proposition hold for \(A^\wedge \to B^\wedge\), see More on Algebra, Lemmas 07NV, 07NW, and 07ED. By the uniqueness and compatibility with flat base change of the construction of Lemma 0EY6 it suffices to prove the essential surjectivity for \(A^\wedge \to B^\wedge\) and the inverse image of \(V\) (details omitted; compare with Lemma 0BLL for the case where \(V\) is the punctured spectrum). By More on Algebra, Proposition 07PM this means we may assume \(A \to B\) is regular.
Let \(W \to V\) be a finite étale morphism. By Popescu’s theorem (Smoothing Ring Maps, Theorem 07GC) we can write \(B = \colim B_i\) as a filtered colimit of smooth \(A\)-algebras. We can pick an \(i\) and an open \(V_i \subset \Spec(B_i)\) whose inverse image is \(V\) (Limits, Lemma 01Z4). After increasing \(i\) we may assume there is a finite étale morphism \(W_i \to V_i\) whose base change to \(V\) is \(W \to V\), see Limits, Lemmas 01ZM, 01ZO, and 07RP. We may assume the complement of \(V_i\) is contained in the closed fibre of \(\Spec(B_i) \to \Spec(A)\) as this is true for \(V\) (either choose \(V_i\) this way or use the lemma above to show this is true for \(i\) large enough). Let \(\eta\) be the generic point of the closed fibre of \(\Spec(B) \to \Spec(A)\). Since \(\eta \in V\), the image of \(\eta\) is in \(V_i\). Hence after replacing \(V_i\) by an affine open neighbourhood of the image of the closed point of \(\Spec(B)\), we may assume that the closed fibre of \(\Spec(B_i) \to \Spec(A)\) is irreducible and that its generic point is contained in \(V_i\) (details omitted; use that a scheme smooth over a field is a disjoint union of irreducible schemes). At this point we may apply Lemma 0EYB to see that \(W_i \to V_i\) extends to a finite étale morphism \(\Spec(C_i) \to \Spec(B_i)\) and pulling back to \(\Spec(B)\) we conclude that \(W\) is in the essential image of the functor \(\textit{F\'Et}_Y \to \textit{F\'Et}_V\) as desired.
Lefschetz for the fundamental group
Of course we have already proven a bunch of results of this type in the local case. In this section we discuss the projective case.
Proposition
Let \(k\) be a field. Let \(X\) be a proper scheme over \(k\). Let \(\mathcal{L}\) be an ample invertible \(\mathcal{O}_X\)-module. Let \(s \in \Gamma(X, \mathcal{L})\). Let \(Y = Z(s)\) be the zero scheme of \(s\). Assume that for all \(x \in X \setminus Y\) we have \[\text{depth}(\mathcal{O}_{X, x}) + \dim(\overline{\{x\}}) > 1\] Then the restriction functor \(\textit{F\'Et}_X \to \textit{F\'Et}_Y\) is fully faithful. In fact, for any open subscheme \(V \subset X\) containing \(Y\) the restriction functor \(\textit{F\'Et}_V \to \textit{F\'Et}_Y\) is fully faithful.
Proof
The first statement is a formal consequence of Lemma 0EJZ and Algebraic and Formal Geometry, Proposition 0EL1. The second statement follows from Lemma 0EJZ and Algebraic and Formal Geometry, Lemma 0EL2.
Proposition
Let \(k\) be a field. Let \(X\) be a proper scheme over \(k\). Let \(\mathcal{L}\) be an ample invertible \(\mathcal{O}_X\)-module. Let \(s \in \Gamma(X, \mathcal{L})\). Let \(Y = Z(s)\) be the zero scheme of \(s\). Let \(\mathcal{V}\) be the set of open subschemes of \(X\) containing \(Y\) ordered by reverse inclusion. Assume that for all \(x \in X \setminus Y\) we have \[\text{depth}(\mathcal{O}_{X, x}) + \dim(\overline{\{x\}}) > 2\] Then the restriction functor \[\colim_\mathcal{V} \textit{F\'Et}_V \to \textit{F\'Et}_Y\] is an equivalence.
Proof
This is a formal consequence of Lemma 0EK1 and Algebraic and Formal Geometry, Proposition 0EL7.
Proposition
Let \(k\) be a field. Let \(X\) be a proper scheme over \(k\). Let \(\mathcal{L}\) be an ample invertible \(\mathcal{O}_X\)-module. Let \(s \in \Gamma(X, \mathcal{L})\). Let \(Y = Z(s)\) be the zero scheme of \(s\). Assume that for all \(x \in X \setminus Y\) we have \[\text{depth}(\mathcal{O}_{X, x}) + \dim(\overline{\{x\}}) > 2\] and that for \(x \in X \setminus Y\) closed purity holds for \(\mathcal{O}_{X, x}\). Then the restriction functor \(\textit{F\'Et}_X \to \textit{F\'Et}_Y\) is an equivalence. If \(X\) or equivalently \(Y\) is connected, then \[\pi_1(Y, \overline{y}) \to \pi_1(X, \overline{y})\] is an isomorphism for any geometric point \(\overline{y}\) of \(Y\).
Proof
Fully faithfulness holds by Proposition 0ELC. By Proposition 0ELD any object of \(\textit{F\'Et}_Y\) is isomorphic to the fibre product \(U \times_V Y\) for some finite étale morphism \(U \to V\) where \(V \subset X\) is an open subscheme containing \(Y\). The complement \(T = X \setminus V\) is5 a finite set of closed points of \(X \setminus Y\). Say \(T = \{x_1, \ldots, x_n\}\). By assumption we can find finite étale morphisms \(V'_i \to \Spec(\mathcal{O}_{X, x_i})\) agreeing with \(U \to V\) over \(V \times_X \Spec(\mathcal{O}_{X, x_i})\). By Limits, Lemma 0BPA applied \(n\) times we see that \(U \to V\) extends to a finite étale morphism \(U' \to X\) as desired. See Lemma 0BQA for the final statement.
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
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 \(f : X \to \Spec(A)\). Let \(\omega_X\) be as in Resolution of Surfaces, Remark 0B4R. 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 0AWG. By Grauert-Riemenschneider (Resolution of Surfaces, Proposition 0AXD) we have \(R^1f_*\omega_X = 0\). Thus the trace map is a map \(f_*\omega_X \to \omega_A\). Then we can consider \[\mathcal{O}_{\Spec(A)} = f_*\mathcal{O}_X \to f_*\omega_X \to \omega_A\] 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 0AVM 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 031S and Dualizing Complexes, Lemma 0AWE). Hence \(f_*\omega_X \to \omega_A\) is surjective whence an isomorphism. Thus \(Rf_*\omega_X = \omega_A\) which by duality implies \(Rf_*\mathcal{O}_X = \mathcal{O}_{\Spec(A)}\). Whence \(H^1(X, \mathcal{O}_X) = 0\) which implies that \(A\) defines a rational singularity (see discussion in Resolution of Surfaces, Section 0AXE in particular Lemmas 0B4Q and 0AXF). 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 0B64 and \(\omega_A \cong A\) as we’ve seen above.
Lemma
Let \(f : X \to \Spec(A)\) be a finite type morphism. Let \(x \in X\) be a point. Assume
\(A\) is an excellent regular local ring,
\(\mathcal{O}_{X, x}\) is normal of dimension \(2\),
\(f\) 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 07QU). Thus we can choose a minimal resolution of singularities \(W \to \Spec(\mathcal{O}_{X, x})\), see Resolution of Surfaces, Theorem 0BGP. After possibly replacing \(X\) by an affine open neighbourhood of \(x\) we can find a proper morphism \(b : X' \to X\) such that \(X' \times_X \Spec(\mathcal{O}_{X, x}) = W\), see Limits, Lemma 0BPA. 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 \(W \to \Spec(\mathcal{O}_{X, x})\) 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 \(W \to \Spec(\mathcal{O}_{X, x})\) 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 \(f' = f \circ b : X' \to \Spec(A)\).
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 0AA3). The Cohen-Macaulay locus of \(X\) is open by Duality for Schemes, Lemma 0AWT (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 0EA0 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 0E9V we find this map is an isomorphism by Divisors, Lemma 0AVM. 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 0E9K. By Duality for Schemes, Lemma 0E9Z 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 \(W \to \Spec(\mathcal{O}_{X, x})\) we see that \(\omega_W = \mathcal{O}_W(E|_W)\). By Lemma 0EA2 we find \(E|_W = 0\). This means that \(f' : X' \to Y\) is étale by (the already used) Duality for Schemes, Lemma 0E9Z. This immediately finishes the proof, as étaleness of \(f'\) forces \(b\) to be an isomorphism.
Lemma
Let \(f : X \to Y\) be a morphism of locally Noetherian schemes. Let \(x \in X\) and set \(y = f(x)\). Assume
\(\mathcal{O}_{X, x}\) is normal of dimension \(\geq 1\),
\(\mathcal{O}_{Y, y}\) is regular,
\(f\) is locally of finite type, and
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 \(\Spec(\mathcal{O}_{Y, y}) \to Y\) without affecting the conclusion of the lemma, see Morphisms, Lemma 0476. Thus we may assume \(Y = \Spec(A)\) 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 \(y \in Y = \Spec(A)\).
Let \(X' = X \times_{\Spec(A)} \Spec(A^\wedge)\). Let \(x' \in X'\) be the unique point lying over \(x\). By the above we see that \(X'\) is étale over \(\Spec(A^\wedge)\) 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 0BG9. Then if we can show \(X' \to \Spec(A^\wedge)\) is étale at \(x'\), then \(f\) is étale at \(x\) (by the aforementioned Morphisms, Lemma 0476). Thus we may and do assume \(A\) is a regular complete local ring.
The case \(d = 2\) now follows from Lemma 0EA3.
Assume \(d > 2\). Let \(t \in \mathfrak m\), \(t \not \in \mathfrak m^2\). Set \(Y_0 = \Spec(A/tA)\) 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 0BMB.
Affineness of complement of ramification locus
Let \(f : X \to Y\) be a finite type morphism of Noetherian schemes with \(X\) normal and \(Y\) regular. Let \(V \subset X\) be the maximal open subscheme where \(f\) is étale. The discussion in [EGA, Chapter IV, Section 21.12] suggests that \(V \to X\) might be an affine morphism. Observe that if \(V \to X\) is affine, then we deduce purity of ramification locus (Lemma 0EA4) by using Divisors, Lemma 0BCU. Thus affineness of \(V \to X\) is a “strong” form of purity for the ramification locus. In this section we prove \(V \to X\) is affine when \(X\) and \(Y\) are equicharacteristic and excellent, see Theorem 0ECD. It seems reasonable to guess the result remains true for \(X\) and \(Y\) of mixed characteristic (but still excellent).
Lemma
Let \((A, \mathfrak m)\) be a regular local ring which contains a field. Let \(f : V \to \Spec(A)\) be étale and quasi-compact. Assume that \(\mathfrak m \not \in f(V)\) and assume that \(g : V \to \Spec(A) \setminus \{\mathfrak m\}\) is affine. Then \(H^i(V, \mathcal{O}_V)\), \(i > 0\) is isomorphic to a direct sum of copies of the injective hull of the residue field of \(A\).
Proof
Denote \(U = \Spec(A) \setminus \{\mathfrak m\}\) the punctured spectrum. Thus \(g : V \to U\) is affine. We have \(H^i(V, \mathcal{O}_V) = H^i(U, g_*\mathcal{O}_V)\) by Cohomology of Schemes, Lemma 089W. The \(\mathcal{O}_U\)-module \(g_*\mathcal{O}_V\) is quasi-coherent by Schemes, Lemma 01LC. For any quasi-coherent \(\mathcal{O}_U\)-module \(\mathcal{F}\) the cohomology \(H^i(U, \mathcal{F})\), \(i > 0\) is \(\mathfrak m\)-power torsion, see for example Local Cohomology, Lemma 0DWR. In particular, the \(A\)-modules \(H^i(V, \mathcal{O}_V)\), \(i > 0\) are \(\mathfrak m\)-power torsion. For any flat ring map \(A \to A'\) we have \(H^i(V, \mathcal{O}_V) \otimes_A A' = H^i(V', \mathcal{O}_{V'})\) where \(V' = V \times_{\Spec(A)} \Spec(A')\) by flat base change Cohomology of Schemes, Lemma 02KH. If we take \(A'\) to be the completion of \(A\) (flat by More on Algebra, Section 07NU), then we see that \[H^i(V, \mathcal{O}_V) = H^i(V, \mathcal{O}_V) \otimes_A A' = H^i(V', \mathcal{O}_{V'}),\quad\text{for } i > 0\] The first equality by the torsion property we just proved and More on Algebra, Lemma 05EC. Moreover, the injective hull of the residue field \(k\) is the same for \(A\) and \(A'\), see Dualizing Complexes, Lemma 08Z5. In this way we reduce to the case \(A = k[[x_1, \ldots, x_d]]\), see Algebra, Section 0323.
Assume the characteristic of \(k\) is \(p > 0\). Since \(F : A \to A\), \(a \mapsto a^p\) is flat (Local Cohomology, Lemma 0EC0) and since \(V \times_{\Spec(A), \Spec(F)} \Spec(A) \cong V\) as schemes over \(\Spec(A)\) by Étale Morphisms, Lemma 0EBS the above gives \(H^i(V, \mathcal{O}_V) \otimes_{A, F} A \cong H^i(V, \mathcal{O}_V)\). Thus we get the result by Local Cohomology, Lemma 0EC3.
Assume the characteristic of \(k\) is \(0\). By Local Cohomology, Lemma 0EC7 there are additive operators \(D_j\), \(j = 1, \ldots, d\) on \(H^i(V, \mathcal{O}_V)\) satisfying the Leibniz rule with respect to \(\partial_j = \partial/\partial x_j\). Thus we get the result by Local Cohomology, Lemma 0EC2.
Lemma
In the situation of Lemma 0ECB assume that \(H^i(V, \mathcal{O}_V) = 0\) for \(i \geq \dim(A) - 1\). Then \(V\) is affine.
Proof
Let \(k = A/\mathfrak m\). Since \(V \times_{\Spec(A)} \Spec(k) = \emptyset\), by cohomology and base change we have \[R\Gamma(V, \mathcal{O}_V) \otimes_A^\mathbf{L} k = 0\] See Derived Categories of Schemes, Lemma 08IB. Thus there is a spectral sequence (More on Algebra, Example 0662) \[E_2^{p, q} = \text{Tor}_{-p}(k, H^q(V, \mathcal{O}_V)),\quad d_2^{p, q} : E_2^{p, q} \to E_2^{p + 2, q - 1}\] and \(d_r^{p, q} : E_r^{p, q} \to E_r^{p + r, q - r + 1}\) converging to zero. By Lemma 0ECB, Dualizing Complexes, Lemma 0EBT, and our assumption \(H^i(V, \mathcal{O}_V) = 0\) for \(i \geq \dim(A) - 1\) we conclude that there is no nonzero differential entering or leaving the \((p, q) = (0, 0)\) spot. Thus \(H^0(V, \mathcal{O}_V) \otimes_A k = 0\). This means that if \(\mathfrak m = (x_1, \ldots, x_d)\) then we have an open covering \(V = \bigcup V \times_{\Spec(A)} \Spec(A_{x_i})\) by affine open subschemes \(V \times_{\Spec(A)} \Spec(A_{x_i})\) (because \(V\) is affine over the punctured spectrum of \(A\)) such that \(x_1, \ldots, x_d\) generate the unit ideal in \(\Gamma(V, \mathcal{O}_V)\). This implies \(V\) is affine by Properties, Lemma 01QF.
Theorem
Let \(Y\) be an excellent regular scheme over a field. Let \(f : X \to Y\) be a finite type morphism of schemes with \(X\) normal. Let \(V \subset X\) be the maximal open subscheme where \(f\) is étale. Then the inclusion morphism \(V \to X\) is affine.
Proof
Let \(x \in X\) with image \(y \in Y\). It suffices to prove that \(V \cap W\) is affine for some affine open neighbourhood \(W\) of \(x\). Since \(\Spec(\mathcal{O}_{X, x})\) is the limit of the schemes \(W\), this holds if and only if \[V_x = V \times_X \Spec(\mathcal{O}_{X, x})\] is affine (Limits, Lemma 01Z6). Thus, if the theorem holds for the morphism \(X \times_Y \Spec(\mathcal{O}_{Y, y}) \to \Spec(\mathcal{O}_{Y, y})\), then the theorem holds. In particular, we may assume \(Y\) is regular of finite dimension, which allows us to do induction on the dimension \(d = \dim(Y)\). Combining this with the same argument again, we may assume that \(Y\) is local with closed point \(y\) and that \(V \cap (X \setminus f^{-1}(\{y\}) \to X \setminus f^{-1}(\{y\})\) is affine.
Let \(x \in X\) be a point lying over \(y\). If \(x \in V\), then there is nothing to prove. Observe that \(f^{-1}(\{y\}) \cap V\) is a finite set of closed points (the fibres of an étale morphism are discrete). Thus after replacing \(X\) by an affine open neighbourhood of \(x\) we may assume \(y \not \in f(V)\). We have to prove that \(V\) is affine.
Let \(e(V)\) be the maximum \(i\) with \(H^i(V, \mathcal{O}_V) \not = 0\). As \(X\) is affine the integer \(e(V)\) is the maximum of the numbers \(e(V_x)\) where \(x \in X \setminus V\), see Local Cohomology, Lemma 0DXB and the characterization of cohomological dimension in Local Cohomology, Lemma 0DX7. We have \(e(V_x) \leq \dim(\mathcal{O}_{X, x}) - 1\) by Local Cohomology, Lemma 0DXC. If \(\dim(\mathcal{O}_{X, x}) \geq 2\) then purity of ramification locus (Lemma 0EA4) shows that \(V_x\) is strictly smaller than the punctured spectrum of \(\mathcal{O}_{X, x}\). Since \(\mathcal{O}_{X, x}\) is normal and excellent, this implies \(e(V_x) \leq \dim(\mathcal{O}_{X, x}) - 2\) by Hartshorne-Lichtenbaum vanishing (Local Cohomology, Lemma 0EB7). On the other hand, since \(X \to Y\) is of finite type and \(V \subset X\) is dense (after possibly replacing \(X\) by the closure of \(V\)), we see that \(\dim(\mathcal{O}_{X, x}) \leq d\) by the dimension formula (Morphisms, Lemma 02JU). Whence \(e(V) \leq \max(0, d - 2)\). Thus \(V\) is affine by Lemma 0ECC if \(d \geq 2\). If \(d = 1\) or \(d = 0\), then the punctured spectrum of \(\mathcal{O}_{Y, y}\) is affine and hence \(V\) is affine.
Specialization maps in the smooth proper case
In this section we discuss the following result. Let \(f : X \to S\) be a proper smooth morphism of schemes. Let \(s \leadsto s'\) be a specialization of points in \(S\). Then the specialization map \[sp : \pi_1(X_{\overline{s}}) \longrightarrow \pi_1(X_{\overline{s}'})\] of Section 0BUP is surjective and
if the characteristic of \(\kappa(s')\) is zero, then it is an isomorphism, or
if the characteristic of \(\kappa(s')\) is \(p > 0\), then it induces an isomorphism on maximal prime-to-\(p\) quotients.
Lemma
Let \(f : X \to S\) be a flat proper morphism with geometrically connected fibres. Let \(s' \leadsto s\) be a specialization. If \(X_s\) is geometrically reduced, then the specialization map \(sp : \pi_1(X_{\overline{s}'}) \to \pi_1(X_{\overline{s}})\) is surjective.
Proof
Since \(X_s\) is geometrically reduced, we may assume all fibres are geometrically reduced after possibly shrinking \(S\), see More on Morphisms, Lemma 0C0E. Let \(\mathcal{O}_{S, s} \to A \to \kappa(\overline{s}')\) be as in the construction of the specialization map, see Section 0BUP. Thus it suffices to show that \[\pi_1(X_{\overline{s}'}) \to \pi_1(X_A)\] is surjective. This follows from Proposition 0C0J and \(\pi_1(\Spec(A)) = \{1\}\).
Proposition
Let \(f : X \to S\) be a smooth proper morphism with geometrically connected fibres. Let \(s' \leadsto s\) be a specialization. If the characteristic to \(\kappa(s)\) is zero, then the specialization map \[sp : \pi_1(X_{\overline{s}'}) \to \pi_1(X_{\overline{s}})\] is an isomorphism.
Proof
The map is surjective by Lemma 0C0P. Thus we have to show it is injective.
We may assume \(S\) is affine. Then \(S\) is a cofiltered limit of affine schemes of finite type over \(\mathbf{Z}\). Hence we can assume \(X \to S\) is the base change of \(X_0 \to S_0\) where \(S_0\) is the spectrum of a finite type \(\mathbf{Z}\)-algebra and \(X_0 \to S_0\) is smooth and proper. See Limits, Lemma 01ZM, 0C0C, and 081F. By Lemma 0C0K we reduce to the case where the base is Noetherian.
Applying Lemma 0C0N we reduce to the case where the base \(S\) is the spectrum of a strictly henselian discrete valuation ring \(A\) and we are looking at the specialization map over \(A\). Let \(K\) be the fraction field of \(A\). Choose an algebraic closure \(\overline{K}\) which corresponds to a geometric generic point \(\overline{\eta}\) of \(\Spec(A)\). For \(\overline{K}/L/K\) finite separable, let \(B \subset L\) be the integral closure of \(A\) in \(L\). This is a discrete valuation ring by More on Algebra, Remark 09E8.
Let \(X \to \Spec(A)\) be as in the previous paragraph. To show injectivity of the specialization map it suffices to prove that every finite étale cover \(V\) of \(X_{\overline{\eta}}\) is the base change of a finite étale cover \(Y \to X\). Namely, then \(\pi_1(X_{\overline{\eta}}) \to \pi_1(X) = \pi_1(X_s)\) is injective by Lemma 0BN7.
Given \(V\) we can first descend \(V\) to \(V' \to X_{K^{sep}}\) by Lemma 0BTW and then to \(V'' \to X_L\) by Lemma 0BTV. Let \(Z \to X_B\) be the normalization of \(X_B\) in \(V''\). Observe that \(Z\) is normal and that \(Z_L = V''\) as schemes over \(X_L\). Hence \(Z \to X_B\) is finite étale over the generic fibre. The problem is that we do not know that \(Z \to X_B\) is everywhere étale. Since \(X \to \Spec(A)\) has geometrically connected smooth fibres, we see that the special fibre \(X_s\) is geometrically irreducible. Hence the special fibre of \(X_B \to \Spec(B)\) is irreducible; let \(\xi_B\) be its generic point. Let \(\xi_1, \ldots, \xi_r\) be the points of \(Z\) mapping to \(\xi_B\). Our first (and it will turn out only) problem is now that the extensions \[\mathcal{O}_{X_B, \xi_B} \subset \mathcal{O}_{Z, \xi_i}\] of discrete valuation rings may be ramified. Let \(e_i\) be the ramification index of this extension. Note that since the characteristic of \(\kappa(s)\) is zero, the ramification is tame!
To get rid of the ramification we are going to choose a further finite separable extension \(K^{sep}/L'/L/K\) such that the ramification index \(e\) of the induced extensions \(B'/B\) is divisible by \(e_i\). Consider the normalized base change \(Z'\) of \(Z\) with respect to \(\Spec(B') \to \Spec(B)\), see discussion in More on Morphisms, Section 09IJ. Let \(\xi_{i, j}\) be the points of \(Z'\) mapping to \(\xi_{B'}\) and to \(\xi_i\) in \(Z\). Then the local rings \[\mathcal{O}_{Z', \xi_{i, j}}\] are localizations of the integral closure of \(\mathcal{O}_{Z, \xi_i}\) in \(L' \otimes_L F_i\) where \(F_i\) is the fraction field of \(\mathcal{O}_{Z, \xi_i}\); details omitted. Hence Abhyankar’s lemma (More on Algebra, Lemma 0BRM) tells us that \[\mathcal{O}_{X_{B'}, \xi_{B'}} \subset \mathcal{O}_{Z', \xi_{i, j}}\] is unramified. We conclude that the morphism \(Z' \to X_{B'}\) is étale away from codimension \(1\). Hence by purity of branch locus (Lemma 0BMB) we see that \(Z' \to X_{B'}\) is finite étale!
However, since the residue field extension induced by \(A \to B'\) is trivial (as the residue field of \(A\) is algebraically closed being separably closed of characteristic zero) we conclude that \(Z'\) is the base change of a finite étale cover \(Y \to X\) by applying Lemma 0A48 twice (first to get \(Y\) over \(A\), then to prove that the pullback to \(B\) is isomorphic to \(Z'\)). This finishes the proof.
Let \(G\) be a profinite group. Let \(p\) be a prime number. The maximal prime-to-\(p\) quotient is by definition \[G' = \lim_{U \subset G\text{ open, normal, index prime to }p} G/U\] If \(X\) is a connected scheme and \(p\) is given, then the maximal prime-to-\(p\) quotient of \(\pi_1(X)\) is denoted \(\pi'_1(X)\).
Theorem
Let \(f : X \to S\) be a smooth proper morphism with geometrically connected fibres. Let \(s' \leadsto s\) be a specialization. If the characteristic of \(\kappa(s)\) is \(p\), then the specialization map \[sp : \pi_1(X_{\overline{s}'}) \to \pi_1(X_{\overline{s}})\] is surjective and induces an isomorphism \[\pi'_1(X_{\overline{s}'}) \cong \pi'_1(X_{\overline{s}})\] of the maximal prime-to-p quotients
Proof
This is proved in exactly the same manner as Proposition 0C0Q with the following differences
Given \(X/A\) we no longer show that the functor \(\textit{F\'Et}_X \to \textit{F\'Et}_{X_{\overline{\eta}}}\) is essentially surjective. We show only that Galois objects whose Galois group has order prime to \(p\) are in the essential image. This will be enough to conclude the injectivity of \(\pi'_1(X_{\overline{s}'}) \to \pi'_1(X_{\overline{s}})\) by exactly the same argument.
The extensions \(\mathcal{O}_{X_B, \xi_B} \subset \mathcal{O}_{Z, \xi_i}\) are tamely ramified as the associated extension of fraction fields is Galois with group of order prime to \(p\). See More on Algebra, Lemma 09EB.
The extension \(\kappa_B/\kappa_A\) is no longer necessarily trivial, but it is purely inseparable. Hence the morphism \(X_{\kappa_B} \to X_{\kappa_A}\) is a universal homeomorphism and induces an isomorphism of fundamental groups by Proposition 0BQN.
Lemma
Let \(k\) be an algebraically closed field. Let \(X\) be a smooth proper connected curve over \(k\) of genus \(g\). There exist elements \[a_1, b_1, \ldots, a_g, b_g \in \pi_1(X)\] which topologically generate \(\pi_1(X)\) and satisfy \[[a_1, b_1] \cdots [a_g, b_g] = 1, \qquad [a, b] = aba^{-1}b^{-1}.\]
Proof
Assume first that \(k\) has characteristic zero. We can find a subfield \(k_0 \subset k\) finitely generated over \(\mathbf{Q}\) and a smooth proper geometrically connected curve \(X_0\) over \(k_0\) whose base change to \(k\) is \(X\). Let \(\overline{k}_0 \subset k\) be the algebraic closure of \(k_0\) in \(k\), and choose an embedding \(\overline{k}_0 \to \mathbf{C}\). Base change gives isomorphisms \[\pi_1(X) \cong \pi_1(X_{0, \overline{k}_0}) \cong \pi_1(X_{0, \mathbf{C}})\] by Lemma 0A49. The genus is unchanged by Algebraic Curves, Lemma 0BY9. By the Riemann existence theorem [SGA1, Exposé XII, Corollaire 5.2], the last group is the profinite completion of the usual fundamental group of a compact Riemann surface of genus \(g\). The latter has the presentation \[\langle a_1, b_1, \ldots, a_g, b_g \mid [a_1, b_1] \cdots [a_g, b_g] = 1 \rangle.\] This proves the result in characteristic zero.
Assume that \(k\) has characteristic \(p > 0\). Choose a Cohen ring \(A\) with residue field \(k\) by Algebra, Lemma 0328. By Deformation Theory, Lemma defos-lemma-smooth-proper-curve-lift-complete-local, there is a smooth projective morphism \(\mathcal{X} \to \Spec(A)\) whose special fibre is \(X\). Its geometric fibres are connected by More on Morphisms, Lemma 0E0N; indeed, the number of geometric connected components is locally constant and is one on the special fibre. If \(\overline{K}\) is an algebraic closure of the fraction field of \(A\), then \(\mathcal{X}_{\overline{K}}\) has genus \(g\) by Algebraic Curves, Lemma 0GKA. The characteristic-zero case supplies \(2g\) generators with the stated relation for \(\pi_1(\mathcal{X}_{\overline{K}})\). Their images have the same relation and topologically generate \(\pi_1(X)\) because the specialization map is surjective by Lemma 0C0P.
Proposition
Let \(k\) be an algebraically closed field. Let \(X\) be a smooth projective connected scheme over \(k\). Then \(\pi_1(X)\) is topologically finitely generated.
Proof
We argue by induction on \(d = \dim(X)\). If \(d = 0\), then \(X = \Spec(k)\), and if \(d = 1\), the result follows from Lemma pione-lemma-fundamental-group-smooth-proper-curve-generators. Assume \(d \geq 2\). Since \(X\) is smooth and connected, it is irreducible. Choose a projective embedding of \(X\). There is a hyperplane section \(Y \subset X\) which is smooth and irreducible by More on Morphisms, Remark 0G4G.
For \(x \in X \setminus Y\), regularity of \(X\) and the dimension formula give \[\text{depth}(\mathcal{O}_{X, x}) + \dim(\overline{\{x\}}) = d > 1.\] Hence the restriction functor \(\textit{F\'Et}_X \to \textit{F\'Et}_Y\) is fully faithful by Proposition 0ELC. Consequently the map \(\pi_1(Y) \to \pi_1(X)\) is surjective by Lemma 0BN6. The induction hypothesis shows that \(\pi_1(Y)\) is topologically finitely generated, and the same is therefore true of its quotient \(\pi_1(X)\).
Tame ramification
Let \(X \to Y\) be a finite étale morphism of schemes of finite type over \(\mathbf{Z}\). There are many ways to define what it means for \(f\) to be tamely ramified at \(\infty\). The article [Kerz-Schmidt] discusses to what extent these notions agree.
In this section we discuss a different more elementary question which precedes the notion of tameness at infinity. Please compare with the (slightly different) discussion in [Grothendieck-Murre]. Assume we are given
a locally Noetherian scheme \(X\),
a dense open \(U \subset X\),
a finite étale morphism \(f : Y \to U\)
such that for every prime divisor \(Z \subset X\) with \(Z \cap U = \emptyset\) the local ring \(\mathcal{O}_{X, \xi}\) of \(X\) at the generic point \(\xi\) of \(Z\) is a discrete valuation ring. Setting \(K_\xi\) equal to the fraction field of \(\mathcal{O}_{X, \xi}\) we obtain a cartesian square \[\xymatrix{ \Spec(K_\xi) \ar[r] \ar[d] & U \ar[d] \\ \Spec(\mathcal{O}_{X, \xi}) \ar[r] & X }\] of schemes. In particular, we see that \(Y \times_U \Spec(K_\xi)\) is the spectrum of a finite separable algebra \(L_\xi/K_\xi\). Then we say \(Y\) is unramified over \(X\) in codimension \(1\), resp. \(Y\) is tamely ramified over \(X\) in codimension \(1\) if \(L_\xi/K_\xi\) is unramified, resp. tamely ramified with respect to \(\mathcal{O}_{X, \xi}\) for every \((Z, \xi)\) as above, see More on Algebra, Definition 09E9. More precisely, we decompose \(L_\xi\) into a product of finite separable field extensions of \(K_\xi\) and we require each of these to be unramified, resp. tamely ramified with respect to \(\mathcal{O}_{X, \xi}\).
Definition
Let \(X\) be a locally Noetherian scheme and let \(U \subset X\) be a connected dense open subscheme. Assume that the local ring of \(X\) at every prime divisor contained in \(X \setminus U\) is a discrete valuation ring. Let \(\overline{u}\) be a geometric point of \(U\). Denote by \[\textit{F\'Et}^t_{U/X} \subset \textit{F\'Et}_U\] the full subcategory of finite étale covers which are tamely ramified over \(X\) in codimension \(1\). The tame fundamental group of \(U\) with respect to \(X\) is \[\pi_1^t(U/X, \overline{u}) = \text{Aut}(Y \mapsto Y_{\overline{u}}), \qquad Y \in \textit{F\'Et}^t_{U/X}.\]
The category in the definition is a Galois category. Namely, tame extensions are preserved on passing to subextensions and admit common finite tame overextensions by More on Algebra, Lemmas 0EXV and 0EXX. Thus the subcategory is stable under finite limits, finite colimits, subobjects, and quotients in \(\textit{F\'Et}_U\). Restriction of the fibre functor gives a canonical continuous surjection \[\pi_1(U, \overline{u}) \longrightarrow \pi_1^t(U/X, \overline{u}).\]
Lemma
Let \(X' \to X\) be a morphism of locally Noetherian schemes. Let \(U \subset X\) be a dense open. Assume
\(U' = f^{-1}(U)\) is dense open in \(X'\),
for every prime divisor \(Z \subset X\) with \(Z \cap U = \emptyset\) the local ring \(\mathcal{O}_{X, \xi}\) of \(X\) at the generic point \(\xi\) of \(Z\) is a discrete valuation ring,
for every prime divisor \(Z' \subset X'\) with \(Z' \cap U' = \emptyset\) the local ring \(\mathcal{O}_{X', \xi'}\) of \(X'\) at the generic point \(\xi'\) of \(Z'\) is a discrete valuation ring,
if \(\xi' \in X'\) is as in (3), then \(\xi = f(\xi')\) is as in (2).
Then if \(f : Y \to U\) is finite étale and \(Y\) is unramified, resp. tamely ramified over \(X\) in codimension \(1\), then \(Y' = Y \times_X X' \to U'\) is finite étale and \(Y'\) is unramified, resp. tamely ramified over \(X'\) in codimension \(1\).
Proof
The only interesting fact in this lemma is the commutative algebra result given in More on Algebra, Lemma 0EXY.
Using the terminology introduced above, we can reformulate our purity results obtained earlier in the following pleasing manner.
Lemma
Let \(X\) be a locally Noetherian scheme. Let \(U \subset X\) be open and dense. Let \(Y \to U\) be a finite étale morphism. Assume
\(Y\) is unramified over \(X\) in codimension \(1\), and
\(\mathcal{O}_{X, x}\) is regular for all \(x \in X \setminus U\).
Then there exists a finite étale morphism \(Y' \to X\) whose restriction to \(U\) is \(Y\).
Proof
Let \(\xi \in X \setminus U\) be a generic point of an irreducible component of \(X \setminus U\) of codimension \(1\). Then \(\mathcal{O}_{X, \xi}\) is a discrete valuation ring. As in the discussion above, write \(Y \times_U \Spec(K_\xi) = \Spec(L_\xi)\). Denote \(B_\xi\) the integral closure of \(\mathcal{O}_{X, \xi}\) in \(L_\xi\). Our assumption that \(Y\) is unramified over \(X\) in codimension \(1\) signifies that \(\mathcal{O}_{X, \xi} \to B_\xi\) is finite étale. Thus we get \(Y_\xi \to \Spec(\mathcal{O}_{X, \xi})\) finite étale and an isomorphism \[Y \times_U \Spec(K_\xi) \cong Y_\xi \times_{\Spec(\mathcal{O}_{X, \xi})} \Spec(K_\xi)\] over \(\Spec(K_\xi)\). By Limits, Lemma 0BQ5 we find an open subscheme \(U \subset U' \subset X\) containing \(\xi\) and a morphism \(Y' \to U'\) of finite presentation whose restriction to \(U\) recovers \(Y\) and whose restriction to \(\Spec(\mathcal{O}_{X, \xi})\) recovers \(Y_\xi\). Finally, the morphism \(Y' \to U'\) is finite étale after possible shrinking \(U'\) to a smaller open by Limits, Lemma 0EY3. Repeating the argument with the other generic points of \(X \setminus U\) of codimension \(1\) we may assume that we have a finite étale morphism \(Y' \to U'\) extending \(Y \to U\) to an open subscheme containing \(U' \subset X\) containing \(U\) and all codimension \(1\) points of \(X \setminus U\). We finish by applying Lemma 0EY7 to \(Y' \to U'\). Namely, all local rings \(\mathcal{O}_{X, x}\) for \(x \in X \setminus U'\) are regular and have \(\dim(\mathcal{O}_{X, x}) \geq 2\). Hence we have purity for \(\mathcal{O}_{X, x}\) by Lemma 0BMA.
Lemma
Let \(X\) be a locally Noetherian scheme. Let \(D \subset X\) be an effective Cartier divisor such that \(D\) is a regular scheme. Let \(Y \to X \setminus D\) be a finite étale morphism. If \(Y\) is unramified over \(X\) in codimension \(1\), then there exists a finite étale morphism \(Y' \to X\) whose restriction to \(X \setminus D\) is \(Y\).
Proof
This is a special case of Lemma 0H2W. First, \(D\) is nowhere dense in \(X\) (see discussion in Divisors, Section 01WQ) and hence \(X \setminus D\) is dense in \(X\). Second, the ring \(\mathcal{O}_{X, x}\) is a regular local ring for all \(x \in D\) by Algebra, Lemma 00NU and our assumption that \(\mathcal{O}_{D, x}\) is regular.
Example
Let \(A\) be a Noetherian ring. Let \(f \in A\) be a nonzerodivisor such that \(A/fA\) is reduced. This implies that \(A_\mathfrak p\) is a discrete valuation ring with uniformizer \(f\) for any minimal prime \(\mathfrak p\) over \(f\). Let \(e \geq 1\) be an integer which is invertible in \(A\). Set \[C = A[x]/(x^e - f)\] Then \(\Spec(C) \to \Spec(A)\) is a finite locally free morphism which is étale over the spectrum of \(A_f\). The finite étale morphism \[\Spec(C_f) \longrightarrow \Spec(A_f)\] is tamely ramified over \(\Spec(A)\) in codimension \(1\). The tameness follows immediately from the characterization of tamely ramified extensions in More on Algebra, Lemma 0EXW.
Here is a version of Abhyankar’s lemma for regular divisors.
Lemma
Let \(X\) be a locally Noetherian scheme. Let \(D \subset X\) be an effective Cartier divisor such that \(D\) is a regular scheme. Let \(Y \to X \setminus D\) be a finite étale morphism. If \(Y\) is tamely ramified over \(X\) in codimension \(1\), then étale locally on \(X\) the morphism \(Y \to X\) is as given as a finite disjoint union of standard tamely ramified morphisms as described in Example 0EYF.
Proof
For every \(x \in X\) we are going to find an étale neighourhood \((U, u) \to (X, x)\) with \(U = \Spec(A)\) such that the base change \(Y \times_X U \to U\) is a finite disjoint union of standard tamely ramified morphisms as in Example 0EYF. We will assume \(x \in D\); the case \(x \not \in D\) follows from Étale Morphisms, Lemma 04HN and taking \(e = 1\) and \(f = 1\) in Example 0EYF.
In this paragraph we reduce to the case where \(X\) is the spectrum of a strictly henselian local ring and \(x\) is the closed point. Namely, shrinking \(X\) we may assume \(X = \Spec(A)\) and \(D \subset X\) is given by a nonzerodivisor \(f \in A\). Then \(Y\) is affine as a finite étale covering of \(\Spec(A_f)\). Write \(Y = \Spec(B)\). Let \(A^{sh}\) be the strict henselization of \(\mathcal{O}_{X, x}\). Since \(A \to A^{sh}\) is flat and \(x \in D\), we see that \(f\) maps to a nonzerodivisor of the maximal ideal of \(A^{sh}\). Observe that \(A^{sh}/fA^{sh} = (A/fA)^{sh}\) (Algebra, Lemma 05WS) is a regular local ring as the strict henselization of \(\mathcal{O}_{D, x}\), see More on Algebra, Lemma 06LN. By Lemma 0EYD the base change of \(Y\) to \(\Spec(A^{sh})\) is tamely ramified in codimension \(1\). Assume we’ve proven the assertion for the base change of \(Y\) to \(\Spec(A^{sh})\). Since \(A^{sh}\) is strictly henselian any étale neighbourhood has a section and we conclude that we have an isomorphism \[A^{sh} \otimes_A B \cong \prod\nolimits_{i \in I} A^{sh}_f[x]/(x^{e_i} - f)\] where \(I\) is a finite set and each \(e_i\) is an integer invertible in \(A^{sh}\). Write \(A^{sh} = \colim A_\lambda\) as a filtered colimit of étale \(A\)-algebras \(A_\lambda\). The displayed isomorphism descends to an isomorphism \[A_\lambda \otimes_A B \cong \prod\nolimits_{i \in I} (A_\lambda)_f[x]/(x^{e_i} - f)\] for suitably large \(\lambda\), see for example Algebra, Lemma 05N9. After increasing \(\lambda\) a bit more we may assume \(e_i\) is invertible in \(A_\lambda\). Then \(\Spec(A_\lambda) \to \Spec(A)\) is the desired étale neighbourhood of \(x\).
Assume \(X = \Spec(A)\) where \(A\) is a strictly henselian local ring and \(x \in X\) corresponds to the maximal ideal \(\mathfrak m\) of \(A\). Let \(f \in \mathfrak m\) be the nonzerodivisor cutting out \(D\). Then \(A/fA\) is regular and by Algebra, Lemma 00NU we find that \(A\) is regular too. We will use some properties of regular local rings, for example that they are normal domains, see Algebra, Lemmas 00NP and 0567. In particular \(A\) is a domain. As above we see that \(Y = \Spec(B)\) is affine and \(A_f \to B\) is finite étale. Hence \(B\) is normal too (Algebra, Lemma 033C) and we conclude \(Y\) is a finite disjoint union of spectra of normal domains by Algebra, Lemma 030C. Thus we may assume \(B\) is a domain too. Since \(A\) and \(A/fA\) are domains, we conclude that \(f\) is a prime element of \(A\) generating a height \(1\) prime \(\mathfrak p = fA\). Observe that \(A_\mathfrak p\) is a discrete valuation ring with uniformizer \(f\).
Let \(K\) be the fraction field of \(A\) and let \(L\) be the fraction field of \(B\). The tame ramification assumption means that \(L\) is tamely ramified with respect to \(A_\mathfrak p\). Let \(e \geq 1\) be the product of the ramification indices for \(L\) over \(A_\mathfrak p\) as in More on Algebra, Remark 09E8. Then \(e\) is invertible in \(\kappa(\mathfrak p)\) but at this point we do not know that \(e\) is invertible in \(A/fA\) or in \(A\).
Consider the finite free \(A\)-algebra \[A' = A[x]/(x^e - f)\] This is a local finite extension of a strictly henselian local ring and hence strictly henselian. Observe that \(f' = x\) is a nonzerodivisor in \(A'\) and that \(A'/f'A' \cong A/fA\) is a regular ring. So as before \(A'\) is regular and a fortiori a domain. Set \(B' = B \otimes_A A' = B \otimes_{A_f} A'_{f'}\). By Abhyankar’s lemma (More on Algebra, Lemma 0BRM) we see that \(\Spec(B')\) is unramified over \(\Spec(A')\) in codimension \(1\). Namely, by Lemma 0EYD we see that \(\Spec(B')\) is still at least tamely ramified over \(\Spec(A')\) in codimension \(1\). But Abhyankar’s lemma tells us that the ramification indices have all become equal to \(1\). By Lemma 0EYE we conclude that \(\Spec(B') \to \Spec(A'_{f'})\) extends to a finite étale morphism \(\Spec(C) \to \Spec(A')\). However, since \(A'\) is strictly henselian, we conclude that \(\Spec(C)\) is a finite disjoint union of copies of \(\Spec(A')\). Conclusion: there exists at least one morphism \(\Spec(A'_{f'}) \to \Spec(B')\). We conclude that there exists an inclusion \(B \subset A'_{f'}\) of \(A_f\)-algebras.
It follows that we have inclusions \(K \subset L \subset K[f^{1/e}]\). By More on Algebra, Lemma 09EV we conclude that \(L\) is equal to \(K[f^{1/n}]\) for some divisor \(n\) of \(e\). The normality of \(B\) then implies that \(B \cong A_f[y]/(y^n - f)\); details omitted. However, the ramification locus of the ring map \(A_f \to A_f[y]/(y^n - f)\) is cut out by \(ny^{n - 1}\) which must imply that \(n\) is a unit in \(A_f\). Since \(f\) is a prime element, this means that \(n = u f^s\) for some unit \(u\) of \(A\) and \(s \in \mathbf{Z}\). Since \(n\) maps to a unit in \(\kappa(\mathfrak p)\) we find \(s = 0\) and the proof is complete.
Lemma
In the situation of Lemma 0EYG the normalization of \(X\) in \(Y\) is a finite locally free morphism \(\pi : Y' \to X\) such that
the restriction of \(Y'\) to \(X \setminus D\) is isomorphic to \(Y\),
\(D' = \pi^{-1}(D)_{red}\) is an effective Cartier divisor on \(Y'\), and
\(D'\) is a regular scheme.
Moreover, étale locally on \(X\) the morphism \(Y' \to X\) is a finite disjoint union of morphisms \[\Spec(A[x]/(x^e - f)) \to \Spec(A)\] where \(A\) is a Noetherian ring, \(f \in A\) is a nonzerodivisor with \(A/fA\) regular, and \(e \geq 1\) is invertible in \(A\).
Proof
This is just an addendum to Lemma 0EYG and in fact the truth of this lemma follows almost immediately if you’ve read the proof of that lemma. But we can also deduce the lemma from the result of Lemma 0EYG. Namely, taking the normalization of \(X\) in \(Y\) commutes with étale base change, see More on Morphisms, Lemma 03GV. Hence we see that we may prove the statements on the local structure of \(Y' \to X\) étale locally on \(X\). Thus, by Lemma 0EYG we may assume that \(X = \Spec(A)\) where \(A\) is a Noetherian ring, that we have a nonzerodivisor \(f\in A\) such that \(A/fA\) is regular, and that \(Y\) is a finite disjoint union of spectra of rings \(A_f[x]/(x^e - f)\) where \(e\) is invertible in \(A\). We omit the verification that the integral closure of \(A\) in \(A_f[x]/(x^e - f)\) is equal to \(A' = A[x]/(x^e - f)\). (To see this argue that the localizations of \(A'\) at primes lying over \((f)\) are regular.) We omit the details.
Lemma
In the situation of Lemma 0EYG let \(Y' \to X\) be as in Lemma 0EYH. Let \(R\) be a discrete valuation ring with fraction field \(K\). Let \[t : \Spec(R) \to X\] be a morphism such that the scheme theoretic inverse image \(t^{-1}D\) is the reduced closed point of \(\Spec(R)\).
If \(t|_{\Spec(K)}\) lifts to a point of \(Y\), then we get a lift \(t' : \Spec(R) \to Y'\) such that \(Y' \to X\) is étale along \(t'(\Spec(R))\).
If \(\Spec(K) \times_X Y\) is isomorphic to a disjoint union of copies of \(\Spec(K)\), then \(Y' \to X\) is finite étale over an open neighbourhood of \(t(\Spec(R))\).
Proof
By the valuative criterion of properness applied to the finite morphism \(Y' \to X\) we see that \(\Spec(K)\)-valued points of \(Y\) matching \(t|_{\Spec(K)}\) as maps into \(X\) lift uniquely to morphisms \(t' : \Spec(R) \to Y'\). Thus statement (1) make sense.
Choose an étale neighbourhood \((U, u) \to (X, t(\mathfrak m_R))\) such that \(U = \Spec(A)\) and such that \(Y' \times_X U \to U\) has a description as in Lemma 0EYH for some \(f \in A\). Then \(\Spec(R) \times_X U \to \Spec(R)\) is étale and surjective. If \(R'\) denotes the local ring of \(\Spec(R) \times_X U\) lying over the closed point of \(\Spec(R)\), then \(R'\) is a discrete valuation ring and \(R \subset R'\) is an unramified extension of discrete valuation rings (More on Algebra, Lemma 0AP2). The assumption on \(t\) signifies that the map \(A \to R'\) corresponding to \[\Spec(R') \to \Spec(R) \times_X U \to U\] maps \(f\) to a uniformizer \(\pi \in R'\). Now suppose that \[Y' \times_X U = \coprod\nolimits_{i \in I} \Spec(A[x]/(x^{e_i} - f))\] for some \(e_i \geq 1\). Then we see that \[\Spec(R') \times_U (Y' \times_X U) = \coprod\nolimits_{i \in I} \Spec(R'[x]/(x^{e_i} - \pi))\] The rings \(R'[x]/(x^{e_i} - f)\) are discrete valuation rings (More on Algebra, Lemma 09EV) and hence have no map into the fraction field of \(R'\) unless \(e_i = 1\).
Proof of (1). In this case the map \(t' : \Spec(R) \to Y'\) base changes to determine a corresponding map \(t'' : \Spec(R') \to Y' \times_X U\) which must map into a summand corresponding to \(i \in I\) with \(e_i = 1\) by the discussion above. Thus clearly we see that \(Y' \times_X U \to U\) is étale along the image of \(t''\). Since being étale is a property one can check after étale base chamge, this proves (1).
Proof of (2). In this case the assumption implies that \(e_i = 1\) for all \(i \in I\). Thus \(Y' \times_X U \to U\) is finite étale and we conclude as before.
Lemma
Let \(S\) be an integral normal Noetherian scheme with generic point \(\eta\). Let \(f : X \to S\) be a smooth morphism with geometrically connected fibres. Let \(\sigma : S \to X\) be a section of \(f\). Let \(Z \to X_\eta\) be a finite étale Galois cover (Section 03SF) with group \(G\) of order invertible on \(S\) such that \(Z\) has a \(\kappa(\eta)\)-rational point mapping to \(\sigma(\eta)\). Then there exists a finite étale Galois cover \(Y \to X\) with group \(G\) whose restriction to \(X_\eta\) is \(Z\).
Proof
First assume \(S = \Spec(R)\) is the spectrum of a discrete valuation ring \(R\) with closed point \(s \in S\). Then \(X_s\) is an effective Cartier divisor in \(X\) and \(X_s\) is regular as a scheme smooth over a field. Moreover the generic fibre \(X_\eta\) is the open subscheme \(X \setminus X_s\). It follows from More on Algebra, Lemma 09EB and the assumption on \(G\) that \(Z\) is tamely ramified over \(X\) in codimension \(1\). Let \(Z' \to X\) be as in Lemma 0EYH. Observe that the action of \(G\) on \(Z\) extends to an action of \(G\) on \(Z'\). By Lemma 0EYI we see that \(Z' \to X\) is finite étale over an open neighbourhood of \(\sigma(y)\). Since \(X_s\) is irreducible, this implies \(Z \to X_\eta\) is unramified over \(X\) in codimension \(1\). Then we get a finite étale morphism \(Y \to X\) whose restriction to \(X_\eta\) is \(Z\) by Lemma 0EYE. Of course \(Y \cong Z'\) (details omitted; hint: compute étale locally) and hence \(Y\) is a Galois cover with group \(G\).
General case. Let \(U \subset S\) be a maximal open subscheme such that there exists a finite étale Galois cover \(Y \to X \times_S U\) with group \(G\) whose restriction to \(X_\eta\) is isomorphic to \(Z\). Assume \(U \not = S\) to get a contradiction. Let \(s \in S \setminus U\) be a generic point of an irreducible component of \(S \setminus U\). Then the inverse image \(U_s\) of \(U\) in \(\Spec(\mathcal{O}_{S, s})\) is the punctured spectrum of \(\mathcal{O}_{S, s}\). We claim \(Y \times_S U_s \to X \times_S U_s\) is the restriction of a finite étale Galois cover \(Y'_s \to X \times_S \Spec(\mathcal{O}_{S, s})\) with group \(G\).
Let us first prove the claim produces the desired contradiction. By Limits, Lemma 0BQ5 we find an open subscheme \(U \subset U' \subset S\) containing \(s\) and a morphism \(Y'' \to U'\) of finite presentation whose restriction to \(U\) recovers \(Y' \to U\) and whose restriction to \(\Spec(\mathcal{O}_{S, s})\) recovers \(Y'_s\). Moreover, by the equivalence of categories given in the lemma, we may assume after shrinking \(U'\) there is a morphism \(Y'' \to U' \times_S X\) and there is an action of \(G\) on \(Y''\) over \(U' \times_S X\) compatible with the given morphisms and actions after base change to \(U\) and \(\Spec(\mathcal{O}_{S, s})\). After shrinking \(U'\) further if necessary, we may assume \(Y'' \to U \times_S X\) is finite étale, see Limits, Lemma 0EY3. This means we have found a strictly larger open of \(S\) over which \(Y\) extends to a finite étale Galois cover with group \(G\) which gives the contradiction we were looking for.
Proof of the claim. We may and do replace \(S\) by \(\Spec(\mathcal{O}_{S, s})\). Then \(S = \Spec(A)\) where \((A, \mathfrak m)\) is a local normal domain. Also \(U \subset S\) is the punctured spectrum and we have a finite étale Galois cover \(Y \to X \times_S U\) with group \(G\). If \(\dim(A) = 1\), then we can construct the extension of \(Y\) to a Galois covering of \(X\) by the first paragraph of the proof. Thus we may assume \(\dim(A) \geq 2\) and hence \(\text{depth}(A) \geq 2\) as \(S\) is normal, see Algebra, Lemma 031S. Since \(X \to S\) is flat, we conclude that \(\text{depth}(\mathcal{O}_{X, x}) \geq 2\) for every point \(x \in X\) mapping to \(s\), see Algebra, Lemma 0337. Let \[Y' \longrightarrow X\] be the finite morphism constructed in Lemma 0EY6 using \(Y \to X \times_S U\). Observe that we obtain a canonical \(G\)-action on \(Y\). Thus all that remains is to show that \(Y'\) is étale over \(X\). In fact, by Lemma 0EYB (for example) it even suffices to show that \(Y' \to X\) is étale over the (unique) generic point of the fibre \(X_s\). This we do by a local calculation in a (formal) neighbourhood of \(\sigma(s)\).
Choose an affine open \(\Spec(B) \subset X\) containing \(\sigma(s)\). Then \(A \to B\) is a smooth ring map which has a section \(\sigma : B \to A\). Denote \(I = \Ker(\sigma)\) and denote \(B^\wedge\) the \(I\)-adic completion of \(B\). Then \(B^\wedge \cong A[[x_1, \ldots, x_d]]\) for some \(d \geq 0\), see Algebra, Lemma 05D5. Of course \(B \to B^\wedge\) is flat (Algebra, Lemma 00MB) and the image of \(\Spec(B^\wedge) \to X\) contains the generic point of \(X_s\). Let \(V \subset \Spec(B^\wedge)\) be the inverse image of \(U\). Consider the finite étale morphism \[W = Y \times_{(X \times_S U)} V \longrightarrow V\] By the compatibility of the construction of \(Y'\) with flat base change in Lemma 0EY6 we find that the base chang \(Y' \times_X \Spec(B^\wedge) \to \Spec(B^\wedge)\) is constructed from \(W \to V\) over \(\Spec(B^\wedge)\) by the procedure in Lemma 0EY6. Set \(V_0 = V \cap V(x_1, \ldots, x_d) \subset V\) and \(W_0 = W \times_V V_0\). This is a normal integral scheme which maps into \(\sigma(S)\) by the morphism \(\Spec(B^\wedge) \to X\) and in fact is identified with \(\sigma(U)\). Hence we know that \(W_0 \to V_0 = U\) completely decomposes as this is true for its generic fibre by our assumption on \(Z \to X_\eta\) having a \(\kappa(\eta)\)-rational point lying over \(\sigma(\eta)\) (and of course the \(G\)-action then implies the whole fibre \(Z_{\sigma(\eta)}\) is a disjoint union of copies of the scheme \(\eta = \Spec(\kappa(\eta))\)). Finally, by Lemma 0EY9 we have \[W_0 \times_U V \cong W\] This shows that \(W\) is a disjoint union of copies of \(V\) and hence \(Y' \times_X \Spec(B^\wedge)\) is a disjoint union of copies of \(\Spec(B^\wedge)\) and the proof is complete.
Lemma
Let \(S\) be a quasi-compact and quasi-separated integral normal scheme with generic point \(\eta\). Let \(f : X \to S\) be a quasi-compact and quasi-separated smooth morphism with geometrically connected fibres. Let \(\sigma : S \to X\) be a section of \(f\). Let \(Z \to X_\eta\) be a finite étale Galois cover (Section 03SF) with group \(G\) of order invertible on \(S\) such that \(Z\) has a \(\kappa(\eta)\)-rational point mapping to \(\sigma(\eta)\). Then there exists a finite étale Galois cover \(Y \to X\) with group \(G\) whose restriction to \(X_\eta\) is \(Z\).
Proof
If \(S\) is Noetherian, then this is the result of Lemma 0EYJ. The general case follows from this by a standard limit argument. We strongly urge the reader to skip the proof.
We can write \(S = \lim S_i\) as a directed limit of a system of schemes with affine transition morphisms and with \(S_i\) of finite type over \(\mathbf{Z}\), see Limits, Proposition 01ZA. For each \(i\) let \(S \to S'_i \to S_i\) be the normalization of \(S_i\) in \(S\), see Morphisms, Section 0BAK. Combining Algebra, Proposition 0335 Morphisms, Lemmas 03GR and 035L we conclude that \(S'_i\) is of finite type over \(\mathbf{Z}\), finite over \(S_i\), and that \(S'_i\) is an integral normal scheme such that \(S \to S'_i\) is dominant. By Morphisms, Lemma 035J we obtain transition morphisms \(S'_{i'} \to S'_i\) compatible with the transition morphisms \(S_{i'} \to S_i\) and with the morphisms with source \(S\). We claim that \(S = \lim S'_i\). Proof of claim omitted (hint: look on affine opens over a chosen affine open in \(S_i\) for some \(i\) to translate this into a straightforward algebra problem). We conclude that we may write \(S = \lim S_i\) as a directed limit of a system of normal integral schemes \(S_i\) with affine transition morphisms and with \(S_i\) of finite type over \(\mathbf{Z}\).
For some \(i\) we can find a smooth morphism \(X_i \to S_i\) of finite presentation whose base change to \(S\) is \(X \to S\). See Limits, Lemmas 01ZM and 0C0C. After increasing \(i\) we may assume the section \(\sigma\) lifts to a section \(\sigma_i : S_i \to X_i\) (by the equivalence of categories in Limits, Lemma 01ZM). We may replace \(X_i\) by the open subscheme \(X_i^0\) of it studied in More on Morphisms, Section 055K since the image of \(X \to X_i\) clearly maps into it (openness by More on Morphisms, Lemma 055R). Thus we may assume the fibres of \(X_i \to S_i\) are geometrically connected. After increasing \(i\) we may assume \(|G|\) is invertible on \(S_i\). Let \(\eta_i \in S_i\) be the generic point. Since \(X_\eta\) is the limit of the schemes \(X_{i, \eta_i}\) we can use the exact same arguments to descent \(Z \to X_\eta\) to some finite étale Galois cover \(Z_i \to X_{i, \eta_i}\) after possibly increasing \(i\). See Lemma 0BTV. After possibly increasing \(i\) once more we may assume \(Z_i\) has a \(\kappa(\eta_i)\)-rational point mapping to \(\sigma_i(\eta_i)\). Then we apply the lemma in the Noetherian case and we pullback to \(X\) to conclude.
Theorem
Let \(S\) be a locally Noetherian scheme. Let \(f : X \to S\) be a smooth proper morphism with geometrically connected fibres. Let \(D \subset X\) be an effective Cartier divisor which is smooth over \(S\), and set \(U = X \setminus D\). Given a specialization \(\overline{s}_1 \leadsto \overline{s}_0\) of geometric points of \(S\), there is a specialization homomorphism, well defined up to inner automorphism, \[sp^t : \pi_1^t(U_{\overline{s}_1}/X_{\overline{s}_1}) \longrightarrow \pi_1^t(U_{\overline{s}_0}/X_{\overline{s}_0}).\] It is surjective. If the characteristic of \(\kappa(\overline{s}_0)\) is zero, then it is an isomorphism. If that characteristic is \(p > 0\), then it induces an isomorphism on maximal prime-to-\(p\) quotients.
Proof
The smooth divisor \(D\) is a relative normal crossings divisor with one branch. The cited base change theorem for tame covers says that finite tame covers on the two geometric fibres are related by a specialization functor. Its degree-zero assertion makes this functor fully faithful. Hence the associated homomorphism of Galois groups is surjective by Lemma 0BN6.
If a finite group \(G\) has order invertible at \(\overline{s}_0\), the degree-one assertion of the same base change theorem identifies the \(G\)-torsors on the two fibres. Every connected Galois cover with group of order prime to the positive residue characteristic is such a torsor. Thus specialization gives an equivalence on the finite Galois objects of prime-to-\(p\) order when the residue characteristic is \(p > 0\), which is exactly the asserted isomorphism on maximal prime-to-\(p\) quotients. In residue characteristic zero this applies to every finite group and proves that \(sp^t\) itself is an isomorphism.
Proposition
Let \(k\) be an algebraically closed field. Let \(X\) be a smooth proper connected curve over \(k\) of genus \(g\), let \(s_1, \ldots, s_n\) be distinct closed points, and set \(U = X \setminus \{s_1, \ldots, s_n\}\). There are topological generators \[x_1, y_1, \ldots, x_g, y_g, \sigma_1, \ldots, \sigma_n \quad\text{of}\quad \pi_1^t(U/X)\] such that \(\sigma_j\) generates a tame inertia subgroup at \(s_j\) and \[\left(\prod_{i = 1}^g [x_i, y_i]\right) \sigma_1 \cdots \sigma_n = 1.\] If \(k\) has characteristic \(p > 0\), the maximal prime-to-\(p\) quotient of \(\pi_1^t(U/X)\) is the pro-prime-to-\(p\) completion of the group with these generators and this single relation. In characteristic zero, \(\pi_1^t(U/X)\) is the profinite completion of that group.
Equivalently, let \(G\) be any finite group if \(k\) has characteristic zero, and let \(G\) have order prime to \(p\) if \(k\) has characteristic \(p > 0\). Given generators \(x_i, y_i, \sigma_j\) of \(G\) satisfying the displayed relation, there exists a connected finite étale Galois cover of \(U\) with group \(G\) for which the inertia group above \(s_j\) is generated by \(\sigma_j\).
Proof
In characteristic zero every finite cover is tame along the boundary. After spreading out the pointed curve and using Theorem pione-theorem-tame-specialization in characteristic zero, we may embed the field of definition into \(\mathbf{C}\). The Riemann existence theorem [SGA1, Exposé XII, Corollaire 5.2] identifies the fundamental group with the profinite completion of the topological fundamental group of a genus \(g\) surface punctured at \(n\) points. This gives the generators, the relation, and the inertia description.
Suppose \(k\) has characteristic \(p > 0\). Choose a Cohen ring \(A\) with residue field \(k\). By Deformation Theory, Lemma defos-lemma-smooth-proper-curve-lift-complete-local, the curve \(X\) lifts to a smooth projective curve \(\mathcal{X}\) over \(A\). Each \(s_j\) lifts to a section of \(\mathcal{X} \to \Spec(A)\) by More on Algebra, Lemma 0H74; the sections are disjoint because their special-fibre points are distinct. Their union is a smooth relative effective Cartier divisor \(\mathcal{D}\). The geometric generic fibre has genus \(g\) as in the proof of Lemma pione-lemma-fundamental-group-smooth-proper-curve-generators.
Apply Theorem pione-theorem-tame-specialization to \((\mathcal{X}, \mathcal{D})\). The images of the characteristic-zero generators topologically generate the tame fundamental group of the special fibre and satisfy the displayed relation. The local standard tame coverings of Example 0EYF show that the image of each \(\sigma_j\) generates the corresponding tame inertia group. The same theorem identifies the maximal prime-to-\(p\) quotients, proving that there are no further relations there. The final formulation follows from the Galois correspondence.
Tricks in positive characteristic
In Piotr Achinger’s paper [Achinger] it is shown that an affine scheme in positive characteristic is always a \(K(\pi, 1)\). In this section we explain the more elementary parts of [Achinger]. Namely, we show that for a field \(k\) of positive characteristic an affine scheme étale over \(\mathbf{A}^n_k\) is actually finite étale over \(\mathbf{A}^n_k\) (by a different morphism). We also show that a closed immersion of connected affine schemes in positive characteristic induces an injective map on étale fundamental groups.
Let \(k\) be a field of characteristic \(p > 0\). Let \[k[x_1, \ldots, x_n] \longrightarrow A\] be a surjection of finite type \(k\)-algebras whose source is the polynomial algebra on \(x_1, \ldots, x_n\). Denote \(I \subset k[x_1, \ldots, x_n]\) the kernel so that we have \(A = k[x_1, \ldots, x_n]/I\). We do not assume \(A\) is nonzero (in other words, we allow the case where \(A\) is the zero ring and \(I = k[x_1, \ldots, x_n]\)). Finally, we assume given a finite étale ring map \(\pi : A \to B\).
Suppose given \(k, n, k[x_1, \ldots, x_n] \to A, I, \pi : A \to B\). Let \(C\) be a \(k\)-algebra. Consider commutative diagrams \[\xymatrix{ & B \\ C \ar[r] & C/\varphi(I)C \ar[u]^\tau \\ k[x_1, \ldots, x_n] \ar[u]^\varphi \ar[r] & A \ar[u] \ar@/_3em/[uu]_\pi }\] where \(\varphi\) is an étale \(k\)-algebra map and \(\tau\) is a surjective \(k\)-algebra map. Let \(C, \varphi, \tau\) be given. For any \(r \geq 0\) and \(y_1, \ldots, y_r \in C\) which generate \(C\) as an algebra over \(\Im(\varphi)\) let \(s = s(r, y_1, \ldots, y_r) \in \{0, \ldots, r\}\) be the maximal element such that \(y_i\) is integral over \(\Im(\varphi)\) for \(1 \leq i \leq s\). We define \(NF(C, \varphi, \tau)\) to be the minimum value of \(r - s = r - s(r, y_1, \ldots, y_r)\) for all choices of \(r\) and \(y_1, \ldots, y_r\) as above. Observe that \(NF(C, \varphi, \tau)\) is \(0\) if and only if \(\varphi\) is finite.
Lemma
In the situation above, if \(NF(C, \varphi, \tau) > 0\), then there exist an étale \(k\)-algebra map \(\varphi'\) and a surjective \(k\)-algebra map \(\tau'\) fitting into the commutative diagram \[\xymatrix{ & B \\ C \ar[r] & C/\varphi'(I)C \ar[u]_{\tau'} \\ k[x_1, \ldots, x_n] \ar[u]^{\varphi'} \ar[r] & A \ar[u] \ar@/_3em/[uu]_\pi }\] with \(NF(C, \varphi', \tau') < NF(C, \varphi, \tau)\).
Proof
Choose \(r \geq 0\) and \(y_1, \ldots, y_r \in C\) which generate \(C\) over \(\Im(\varphi)\) and let \(0 \leq s \leq r\) be such that \(y_1, \ldots, y_s\) are integral over \(\Im(\varphi)\) such that \(r - s = NF(C, \varphi, \tau) > 0\). Since \(B\) is finite over \(A\), the image of \(y_{s + 1}\) in \(B\) satisfies a monic polynomial over \(A\). Hence we can find \(d \geq 1\) and \(f_1, \ldots, f_d \in k[x_1, \ldots, x_n]\) such that \[z = y_{s + 1}^d + \varphi(f_1) y_{s + 1}^{d - 1} + \ldots + \varphi(f_d) \in J = \Ker(C \to C/\varphi(I)C \xrightarrow{\tau} B)\] Since \(\varphi : k[x_1, \ldots, x_n] \to C\) is étale, we can find a nonzero and nonconstant polynomial \(g \in k[T_1, \ldots, T_{n + 1}]\) such that \[g(\varphi(x_1), \ldots, \varphi(x_n), z) = 0 \quad\text{in}\quad C\] To see this you can use for example that \(C \otimes_{\varphi, k[x_1, \ldots, x_n]} k(x_1, \ldots, x_n)\) is a finite product of finite separable field extensions of \(k(x_1, \ldots, x_n)\) (see Algebra, Lemmas 00U3) and hence \(z\) satisfies a monic polynomial over \(k(x_1, \ldots, x_n)\). Clearing denominators we obtain \(g\).
The existence of \(g\) and Algebra, Lemma 051N produce integers \(e_1, e_2, \ldots, e_n \geq 1\) such that \(z\) is integral over the subring \(C'\) of \(C\) generated by \(t_1 = \varphi(x_1) + z^{pe_1}, \ldots, t_n = \varphi(x_n) + z^{pe_n}\). Of course, the elements \(\varphi(x_1), \ldots, \varphi(x_n)\) are also integral over \(C'\) as are the elements \(y_1, \ldots, y_s\). Finally, by our choice of \(z\) the element \(y_{s + 1}\) is integral over \(C'\) too.
Consider the ring map \[\varphi' : k[x_1, \ldots, x_n] \longrightarrow C, \quad x_i \longmapsto t_i\] with image \(C'\). Since \(\text{d}(\varphi(x_i)) = \text{d}(t_i) = \text{d}(\varphi'(x_i))\) in \(\Omega_{C/k}\) (and this is where we use the characteristic of \(k\) is \(p > 0\)) we conclude that \(\varphi'\) is étale because \(\varphi\) is étale, see Algebra, Lemma 0G1C. Observe that \(\varphi'(x_i) - \varphi(x_i) = t_i - \varphi(x_i) = z^{pe_i}\) is in the kernel \(J\) of the map \(C \to C/\varphi(I)C \to B\) by our choice of \(z\) as an element of \(J\). Hence for \(f \in I\) the element \[\varphi'(f) = f(t_1, \ldots, t_n) = f(\varphi(x_1) + z^{pe_1}, \ldots, \varphi(x_n) + z^{pe_n}) = \varphi(f) + \text{element of }(z)\] is in \(J\) as well. In other words, \(\varphi'(I)C \subset J\) and we obtain a surjection \[\tau' : C/\varphi'(I)C \longrightarrow C/J \cong B\] of algebras étale over \(A\). Finally, the algebra \(C\) is generated by the elements \(\varphi(x_1), \ldots, \varphi(x_n), y_1, \ldots, y_r\) over \(C' = \Im(\varphi')\) with \(\varphi(x_1), \ldots, \varphi(x_n), y_1, \ldots, y_{s + 1}\) integral over \(C' = \Im(\varphi')\). Hence \(NF(C, \varphi', \tau') < r - s = NF(C, \varphi, \tau)\). This finishes the proof.
Lemma
Let \(k\) be a field of characteristic \(p > 0\). Let \(X \to \mathbf{A}^n_k\) be an étale morphism with \(X\) affine. Then there exists a finite étale morphism \(X \to \mathbf{A}^n_k\).
Proof
Write \(X = \Spec(C)\). Set \(A = 0\) and denote \(I = k[x_1, \ldots, x_n]\). By assumption there exists some étale \(k\)-algebra map \(\varphi : k[x_1, \ldots, x_n] \to C\). Denote \(\tau : C/\varphi(I)C \to 0\) the unique surjection. We may choose \(\varphi\) and \(\tau\) such that \(N(C, \varphi, \tau)\) is minimal. By Lemma 0G1F we get \(N(C, \varphi, \tau) = 0\). Hence \(\varphi\) is finite étale.
Lemma
Let \(k\) be a field of characteristic \(p > 0\). Let \(Z \subset \mathbf{A}^n_k\) be a closed subscheme. Let \(Y \to Z\) be finite étale. There exists a finite étale morphism \(f : U \to \mathbf{A}^n_k\) such that there is an open and closed immersion \(Y \to f^{-1}(Z)\) over \(Z\).
Proof
Let us turn the problem into algebra. Write \(\mathbf{A}^n_k = \Spec(k[x_1, \ldots, x_n])\). Then \(Z = \Spec(A)\) where \(A = k[x_1, \ldots, x_n]/I\) for some ideal \(I \subset k[x_1, \ldots, x_n]\). Write \(Y = \Spec(B)\) so that \(Y \to Z\) corresponds to the finite étale \(k\)-algebra map \(A \to B\).
By Algebra, Lemma 04D1 there exists an étale ring map \[\varphi : k[x_1, \ldots, x_n] \to C\] and a surjective \(A\)-algebra map \(\tau : C/\varphi(I)C \to B\). (We can even choose \(C, \varphi, \tau\) such that \(\tau\) is an isomorphism, but we won’t use this). We may choose \(\varphi\) and \(\tau\) such that \(N(C, \varphi, \tau)\) is minimal. By Lemma 0G1F we get \(N(C, \varphi, \tau) = 0\). Hence \(\varphi\) is finite étale.
Let \(f : U = \Spec(C) \to \mathbf{A}^n_k\) be the finite étale morphism corresponding to \(\varphi\). The morphism \(Y \to f^{-1}(Z) = \Spec(C/\varphi(I)C)\) induced by \(\tau\) is a closed immersion as \(\tau\) is surjective and open as it is an étale morphism by Morphisms, Lemma 02GW. This finishes the proof.
Here is the main result.
Proposition
Let \(p\) be a prime number. Let \(i : Z \to X\) be a closed immersion of connected affine schemes over \(\mathbf{F}_p\). For any geometric point \(\overline{z}\) of \(Z\) the map \[\pi_1(Z, \overline{z}) \to \pi_1(X, \overline{z})\] is injective.
Proof
Let \(Y \to Z\) be a finite étale morphism. It suffices to construct a finite étale morphism \(f : U \to X\) such that \(Y\) is isomorphic to an open and closed subscheme of \(f^{-1}(Z)\), see Lemma 0BN7. Write \(Y = \Spec(A)\) and \(X = \Spec(R)\) so the closed immersion \(Y \to X\) is given by a surjection \(R \to A\). We may write \(A = \colim A_i\) as the filtered colimit of its \(\mathbf{F}_p\)-subalgebras of finite type. By Lemma 0BTV we can find an \(i\) and a finite étale morphism \(Y_i \to Z_i = \Spec(A_i)\) such that \(Y = Z \times_{Z_i} Y_i\).
Choose a surjection \(\mathbf{F}_p[x_1, \ldots, x_n] \to A_i\). This determines a closed immersion \[Z_i = \Spec(A_i) \longrightarrow X_i = \mathbf{A}^n_{\mathbf{F}_p} = \Spec(\mathbf{F}_p[x_1, \ldots, x_n])\] By the universal property of polynomial algebras and since \(R \to A\) is surjective, we can find a commutative diagram \[\xymatrix{ \mathbf{F}_p[x_1, \ldots, x_n] \ar[r] \ar[d] & A_i \ar[d] \\ R \ar[r] & A }\] of \(\mathbf{F}_p\)-algebras. Thus we have a commutative diagram \[\xymatrix{ Y_i \ar[r] & Z_i \ar[r] & X_i \\ Y \ar[u] \ar[r] & Z \ar[u] \ar[r] & X \ar[u] }\] whose right square is cartesian. Clearly, if we can find \(f_i : U_i \to X_i\) finite étale such that \(Y_i\) is isomorphic to an open and closed subscheme of \(f_i^{-1}(Z_i)\), then the base change \(f : U \to X\) of \(f_i\) by \(X \to X_i\) is a solution to our problem. Thus we conclude by applying Lemma 0G1H to \(Y_i \to Z_i \to X_i = \mathbf{A}^n_{\mathbf{F}_p}\).
When we discuss the pro-étale fundamental group the general case will be of interest.↩︎
Namely, given a morphism \(f\) of \(\mathcal{C}\) if \(F(f)\) is an isomorphism, then \(f\) is an isomorphism.↩︎
This means that \(F\) commutes with finite limits and colimits, see Categories, Section 0033.↩︎
This is possible because \(\kappa(\mathfrak m^{sh})\) is a separable algebraic closure of \(\kappa\) and \(\kappa(\mathfrak m^{sep})\) is an algebraic closure of \(\kappa\) by Lemma 0BSP.↩︎
Namely, \(T\) is proper over \(k\) (being closed in \(X\)) and affine (being closed in the affine scheme \(X \setminus Y\), see Morphisms, Lemma 0EKE) and hence finite over \(k\) (Morphisms, Lemma 01WN). Thus \(T\) is a finite set of closed points.↩︎