The Stacks project

57.10 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 57.10.1. Let $f : X \to Y$ be a morphism of schemes. If $f(X)$ is dense in $Y$ then the base change functor $\textit{FÉt}_ Y \to \textit{FÉt}_ X$ is faithful.

Proof. Since the category of finite étale coverings has an internal hom (Lemma 57.5.4) 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 26.21.5) 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 41.6.1) hence they are equal. $\square$

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 15.98.5. There is a partial converse in Properties, Lemma 28.15.3.

Lemma 57.10.2. Let $(A, \mathfrak m)$ be a local ring. Set $X = \mathop{\mathrm{Spec}}(A)$ and let $U = X \setminus \{ \mathfrak m\} $. If the punctured spectrum of the strict henselization of $A$ is connected, then

\[ \textit{FÉt}_ X \longrightarrow \textit{FÉt}_ U,\quad Y \longmapsto Y \times _ X U \]

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 57.5.4) 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 $Y$. Let $A^{sh}$ be the strict henselization of $A$ and denote $X^{sh} = \mathop{\mathrm{Spec}}(A^{sh})$, $U^{sh} = U \times _ X X^{sh}$, $Y^{sh} = Y \times _ X X^{sh}$. 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 $X' = \mathop{\mathrm{Spec}}(A')$ 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 10.38.19). Thus $t'_1 = t'_2$ by Lemma 57.10.1. Hence $t^{sh}$ descends to a morphism $t : X \to Y$ for example by Descent, Lemma 35.10.7. $\square$

In view of Lemma 57.10.2 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 51.3.1.

Lemma 57.10.3. Let $X$ be a scheme. Let $U \subset X$ be a dense open. Assume

  1. the underlying topological space of $X$ is Noetherian, and

  2. for every $x \in X \setminus U$ the punctured spectrum of the strict henselization of $\mathcal{O}_{X, x}$ is connected.

Then $\textit{FÉt}_ X \to \textit{Fét}_ U$ 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 57.10.1.

Let $x \in X \setminus U$ be a generic point of an irreducible component of $X \setminus U$. Set $V = U \times _ X \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$. By our choice of $x$ this is the punctured spectrum of $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$. By Lemma 57.10.2 we can extend the morphism $\varphi _ V : (Y_1)_ V \to (Y_2)_ V$ uniquely to a morphism $(Y_1)_{\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})} \to (Y_2)_{\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})}$. By Limits, Lemma 32.18.3 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. $\square$

Lemma 57.10.4. Let $X$ be a scheme. Let $U \subset X$ be a dense open. Assume

  1. $U \to X$ is quasi-compact,

  2. every point of $X \setminus U$ is closed, and

  3. for every $x \in X \setminus U$ the punctured spectrum of the strict henselization of $\mathcal{O}_{X, x}$ is connected.

Then $\textit{FÉt}_ X \to \textit{Fét}_ U$ 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 57.10.1.

Let $x \in X \setminus U$. Set $V = U \times _ X \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$. Since every point of $X \setminus U$ is closed $V$ is the punctured spectrum of $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$. By Lemma 57.10.2 we can extend the morphism $\varphi _ V : (Y_1)_ V \to (Y_2)_ V$ uniquely to a morphism $(Y_1)_{\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})} \to (Y_2)_{\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})}$. By Limits, Lemma 32.18.3 (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 57.10.1). Thus we can extend $\varphi $ to $\bigcup U'_ x = X$ as desired. $\square$

Lemma 57.10.5. Let $X$ be a scheme. Let $U \subset X$ be a dense open. Assume

  1. every quasi-compact open of $X$ has finitely many irreducible components,

  2. for every $x \in X \setminus U$ the punctured spectrum of the strict henselization of $\mathcal{O}_{X, x}$ is connected.

Then $\textit{FÉt}_ X \to \textit{Fét}_ U$ 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 57.10.1. 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 $V = U \times _ X \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$. By our choice of $x$ this is the punctured spectrum of $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$. By Lemma 57.10.2 we can extend the morphism $\varphi _ V : (Y_1)_ V \to (Y_2)_ V$ (uniquely) to a morphism $(Y_1)_{\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})} \to (Y_2)_{\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})}$. 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 $V' = W' \times _ X \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$. By Limits, Lemma 32.18.3 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 57.10.1 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 $X$. $\square$

Lemma 57.10.6. Let $(A, \mathfrak m)$ be a local ring. Set $X = \mathop{\mathrm{Spec}}(A)$ 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 57.4.3 part (1).

Proof. The map $\pi _1(U) \to \pi _1(X)$ is surjective by Lemmas 57.10.2 and 57.4.1.

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

To finish the proof, it suffices according to Lemma 57.4.3 to show the following: Given a finite étale morphism $V \to U$ 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 57.10.4 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 35.34.1). 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 35.32.11) 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 35.20.29 and 35.20.23. $\square$

Let $X$ be an irreducible scheme. Let $\eta \in X$ be the generic point. The canonical morphism $\eta \to X$ induces a canonical map

57.10.6.1
\begin{equation} \label{pione-equation-inclusion-generic-point} \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 57.6.3.

Lemma 57.10.7. 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 (57.10.6.1) $\pi _1(\eta , \overline{\eta }) \to \pi _1(X, \overline{\eta })$ is surjective as well.

Proof. By Lemma 57.8.3 we may replace $X$ by its reduction. Thus we may assume that $X$ is an integral scheme. By Lemma 57.4.1 the assertion of the lemma translates into the statement that the functors $\textit{FÉt}_ X \to \textit{FÉt}_ U$ and $\textit{FÉt}_ X \to \textit{FÉt}_\eta $ are fully faithful.

The result for $\textit{FÉt}_ X \to \textit{FÉt}_ U$ follows from Lemma 57.10.5 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 15.98.5.

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{FÉt}_\eta $ is the colimit of the categories $\textit{FÉt}_ U$ over such $U$, see Limits, Lemmas 32.10.1, 32.8.3, and 32.8.10. A formal argument then shows that fully faithfulness for $\textit{FÉt}_ X \to \textit{FÉt}_\eta $ follows from the fully faithfulness of the functors $\textit{FÉt}_ X \to \textit{FÉt}_ U$. $\square$

Lemma 57.10.8. Let $X$ be a scheme. Let $x_1, \ldots , x_ n \in X$ be a finite number of closed points such that

  1. $U = X \setminus \{ x_1, \ldots , x_ n\} $ is connected and is a retrocompact open of $X$, and

  2. 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 57.10.4 and 57.4.1. 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

\[ \xymatrix{ U \ar[d] & \mathop{\mathrm{Spec}}(A) \setminus \{ \mathfrak m\} \ar[l] \ar[d] & U^{sh} \ar[d] \ar[l] \\ X & \mathop{\mathrm{Spec}}(A) \ar[l] & \mathop{\mathrm{Spec}}(A^{sh}) \ar[l] } \]

By Lemma 57.4.3 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 57.10.6 we know the corresponding statement for finite étale schemes over the punctured spectrum of $A$. However, by Limits, Lemma 32.18.1 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. $\square$


Comments (2)

Comment #2972 by Michael Neururer on

Before (52.2.9.7.1), shouldn't eta be the generic point of X, rather than "the geometric point".


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