The main result of this section is that a universal homeomorphism of connected schemes induces an isomorphism on fundamental groups. See Proposition 58.8.4.
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 58.8.1. Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated schemes such that the base change functor $\textit{FÉt}_ Y \to \textit{FÉt}_ X$ 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 28.2.4. By Topology, Lemma 5.12.10 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 5.22.5) we conclude that $\pi _0(X) \to \pi _0(Y)$ is a homeomorphism by Topology, Lemma 5.17.8. This proves (1). Part (2) is immediate. $\square$
The following lemma tells us that the fundamental group of a henselian pair is the fundamental group of the closed subset.
Lemma 58.8.2. Let $(A, I)$ be a henselian pair. Set $X = \mathop{\mathrm{Spec}}(A)$ and $Z = \mathop{\mathrm{Spec}}(A/I)$. The functor is an equivalence of categories.
\[ \textit{FÉt}_ X \longrightarrow \textit{FÉt}_ Z,\quad U \longmapsto U \times _ X Z \]
Proof. This is a translation of More on Algebra, Lemma 15.13.2. $\square$
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 58.8.3. Let $X \subset X'$ be a thickening of schemes. The functor is an equivalence of categories.
\[ \textit{FÉt}_{X'} \longrightarrow \textit{FÉt}_ X,\quad U' \longmapsto U' \times _{X'} X \]
Proof. For a discussion of thickenings see More on Morphisms, Section 37.2. 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 37.3.4. Now, if $X \subset X'$ is a finite order thickening then this remark combined with Étale Morphisms, Theorem 41.15.2 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 27.2. 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 $X' = \mathop{\mathrm{Spec}}(A')$ and $X = \mathop{\mathrm{Spec}}(A'/I')$ where $I'$ is a locally nilpotent ideal. Then $(A', I')$ is a henselian pair (More on Algebra, Lemma 15.11.2) and the result follows from Lemma 58.8.2 (which is much easier in this case). $\square$
The “correct” way to prove the following proposition would be to deduce it from the invariance of the étale site, see Étale Cohomology, Theorem 59.45.2.
Proposition 58.8.4. Let $f : X \to Y$ be a universal homeomorphism of schemes. Then 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.
\[ \textit{FÉt}_ Y \longrightarrow \textit{FÉt}_ X,\quad V \longmapsto V \times _ Y X \]
Proof. Recall that a universal homeomorphism is the same thing as an integral, universally injective, surjective morphism, see Morphisms, Lemma 29.45.5. In particular, the diagonal $\Delta : X \to X \times _ Y X$ is a thickening by Morphisms, Lemma 29.10.2. Thus by Lemma 58.8.3 we see that given a finite étale morphism $U \to X$ there is a unique isomorphism
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 41.20.6 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 $\textit{FÉt}_ Y \to \textit{FÉt}_ X$ is essentially surjective.
Arguing in the same manner as above we see that given $V_1 \to Y$ and $V_2 \to Y$ in $\textit{FÉt}_ Y$ 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 41.20.3. $\square$
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