In the following theorem we show that the small étale site is a topological invariant in the following sense: If $f : X \to Y$ is a morphism of schemes which is a universal homeomorphism, then $X_{\acute{e}tale}\cong Y_{\acute{e}tale}$ as sites. This improves the result of Étale Morphisms, Theorem 41.15.2. We first prove the result for morphisms and then we state the result for categories.
Theorem 59.45.1. Let $X$ and $Y$ be two schemes over a base scheme $S$. Let $S' \to S$ be a universal homeomorphism. Denote $X'$ (resp. $Y'$) the base change to $S'$. If $X$ is étale over $S$, then the map is bijective.
\[ \mathop{\mathrm{Mor}}\nolimits _ S(Y, X) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{S'}(Y', X') \]
Proof. After base changing via $Y \to S$, we may assume that $Y = S$. Thus we may and do assume both $X$ and $Y$ are étale over $S$. In other words, the theorem states that the base change functor is a fully faithful functor from the category of schemes étale over $S$ to the category of schemes étale over $S'$.
Consider the forgetful functor
We claim this functor is an equivalence. On the other hand, the functor
is fully faithful by Étale Morphisms, Lemma 41.20.3. Thus the claim implies the theorem.
Proof of the claim. 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 $\Delta : S' \to S' \times _ S S'$ is a thickening by Morphisms, Lemma 29.10.2. Thus by Étale Morphisms, Theorem 41.15.1 we see that given $X' \to S'$ étale there is a unique isomorphism
of schemes étale over $S' \times _ S S'$ which pulls back under $\Delta $ to $\text{id} : X' \to X'$ over $S'$. Since $S' \to S' \times _ S S' \times _ S S'$ is a thickening as well (it is bijective and a closed immersion) we conclude that $(X', \varphi ')$ is a descent datum relative to $S'/S$. The canonical nature of the construction of $\varphi '$ shows that it is compatible with morphisms between schemes étale over $S'$. In other words, we obtain a quasi-inverse $X' \mapsto (X', \varphi ')$ of the functor (59.45.1.1). This proves the claim and finishes the proof of the theorem. $\square$
Theorem 59.45.2. Let $f : X \to Y$ be a morphism of schemes. Assume $f$ is integral, universally injective and surjective (i.e., $f$ is a universal homeomorphism, see Morphisms, Lemma 29.45.5). The functor defines an equivalence of categories
\[ V \longmapsto V_ X = X \times _ Y V \]
\[ \{ \text{schemes }V\text{ étale over }Y \} \leftrightarrow \{ \text{schemes }U\text{ étale over }X \} \]
We give two proofs. The first uses effectivity of descent for quasi-compact, separated, étale morphisms relative to surjective integral morphisms. The second uses the material on properties (A), (B), and (C) discussed earlier in the chapter.
First proof. By Theorem 59.45.1 we see that the functor is fully faithful. It remains to show that the functor is essentially surjective. Let $U \to X$ be an étale morphism of schemes.
Suppose that the result holds if $U$ and $Y$ are affine. In that case, we choose an affine open covering $U = \bigcup U_ i$ such that each $U_ i$ maps into an affine open of $Y$. By assumption (affine case) we can find étale morphisms $V_ i \to Y$ such that $X \times _ Y V_ i \cong U_ i$ as schemes over $X$. Let $V_{i, i'} \subset V_ i$ be the open subscheme whose underlying topological space corresponds to $U_ i \cap U_{i'}$. Because we have isomorphisms
as schemes over $X$ we see by fully faithfulness that we obtain isomorphisms $\theta _{i, i'} : V_{i, i'} \to V_{i', i}$ of schemes over $Y$. We omit the verification that these isomorphisms satisfy the cocycle condition of Schemes, Section 26.14. Applying Schemes, Lemma 26.14.2 we obtain a scheme $V \to Y$ by glueing the schemes $V_ i$ along the identifications $\theta _{i, i'}$. It is clear that $V \to Y$ is étale and $X \times _ Y V \cong U$ by construction.
Thus it suffices to show the lemma in case $U$ and $Y$ are affine. Recall that in the proof of Theorem 59.45.1 we showed that $U$ comes with a unique descent datum $(U, \varphi )$ relative to $X/Y$. By Étale Morphisms, Proposition 41.20.6 (which applies because $U \to X$ is quasi-compact and separated as well as étale by our reduction to the affine case) there exists an étale morphism $V \to Y$ such that $X \times _ Y V \cong U$ and the proof is complete. $\square$
Second proof. By Theorem 59.45.1 we see that the functor is fully faithful. It remains to show that the functor is essentially surjective. Let $U \to X$ be an étale morphism of schemes.
Suppose that the result holds if $U$ and $Y$ are affine. In that case, we choose an affine open covering $U = \bigcup U_ i$ such that each $U_ i$ maps into an affine open of $Y$. By assumption (affine case) we can find étale morphisms $V_ i \to Y$ such that $X \times _ Y V_ i \cong U_ i$ as schemes over $X$. Let $V_{i, i'} \subset V_ i$ be the open subscheme whose underlying topological space corresponds to $U_ i \cap U_{i'}$. Because we have isomorphisms
as schemes over $X$ we see by fully faithfulness that we obtain isomorphisms $\theta _{i, i'} : V_{i, i'} \to V_{i', i}$ of schemes over $Y$. We omit the verification that these isomorphisms satisfy the cocycle condition of Schemes, Section 26.14. Applying Schemes, Lemma 26.14.2 we obtain a scheme $V \to Y$ by glueing the schemes $V_ i$ along the identifications $\theta _{i, i'}$. It is clear that $V \to Y$ is étale and $X \times _ Y V \cong U$ by construction.
Thus it suffices to prove that the functor
is essentially surjective when $X$ and $Y$ are affine.
Let $U \to X$ be an affine scheme étale over $X$. We have to find $V \to Y$ étale (and affine) such that $X \times _ Y V$ is isomorphic to $U$ over $X$. Note that an étale morphism of affines has universally bounded fibres, see Morphisms, Lemmas 29.36.6 and 29.57.9. Hence we can do induction on the integer $n$ bounding the degree of the fibres of $U \to X$. See Morphisms, Lemma 29.57.8 for a description of this integer in the case of an étale morphism. If $n = 1$, then $U \to X$ is an open immersion (see Étale Morphisms, Theorem 41.14.1), and the result is clear. Assume $n > 1$.
By Lemma 59.44.4 there exists an étale morphism of schemes $W \to Y$ and a surjective morphism $W_ X \to U$ over $X$. As $U$ is quasi-compact we may replace $W$ by a disjoint union of finitely many affine opens of $W$, hence we may assume that $W$ is affine as well. Here is a diagram
The disjoint union decomposition arises because by construction the étale morphism of affine schemes $U \times _ Y W \to W_ X$ has a section. OK, and now we see that the morphism $R \to X \times _ Y W$ is an étale morphism of affine schemes whose fibres have degree universally bounded by $n - 1$. Hence by induction assumption there exists a scheme $V' \to W$ étale such that $R \cong W_ X \times _ W V'$. Taking $V'' = W \amalg V'$ we find a scheme $V''$ étale over $W$ whose base change to $W_ X$ is isomorphic to $U \times _ Y W$ over $X \times _ Y W$.
At this point we can use descent to find $V$ over $Y$ whose base change to $X$ is isomorphic to $U$ over $X$. Namely, by the fully faithfulness of the functor (59.45.2.1) corresponding to the universal homeomorphism $X \times _ Y (W \times _ Y W) \to (W \times _ Y W)$ there exists a unique isomorphism $\varphi : V'' \times _ Y W \to W \times _ Y V''$ whose base change to $X \times _ Y (W \times _ Y W)$ is the canonical descent datum for $U \times _ Y W$ over $X \times _ Y W$. In particular $\varphi $ satisfies the cocycle condition. Hence by Descent, Lemma 35.37.1 we see that $\varphi $ is effective (recall that all schemes above are affine). Thus we obtain $V \to Y$ and an isomorphism $V'' \cong W \times _ Y V$ such that the canonical descent datum on $W \times _ Y V/W/Y$ agrees with $\varphi $. Note that $V \to Y$ is étale, by Descent, Lemma 35.23.29. Moreover, there is an isomorphism $V_ X \cong U$ which comes from descending the isomorphism
which we have by construction. Some details omitted. $\square$
Remark 59.45.3. In the situation of Theorem 59.45.2 it is also true that $V \mapsto V_ X$ induces an equivalence between those étale morphisms $V \to Y$ with $V$ affine and those étale morphisms $U \to X$ with $U$ affine. This follows for example from Limits, Proposition 32.11.2.
Proposition 59.45.4 (Topological invariance of étale cohomology). Let $X_0 \to X$ be a universal homeomorphism of schemes (for example the closed immersion defined by a nilpotent sheaf of ideals). Then
the étale sites $X_{\acute{e}tale}$ and $(X_0)_{\acute{e}tale}$ are isomorphic,
the étale topoi $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ and $\mathop{\mathit{Sh}}\nolimits ((X_0)_{\acute{e}tale})$ are equivalent, and
$H^ q_{\acute{e}tale}(X, \mathcal{F}) = H^ q_{\acute{e}tale}(X_0, \mathcal{F}|_{X_0})$ for all $q$ and for any abelian sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$.
Proof. The equivalence of categories $X_{\acute{e}tale}\to (X_0)_{\acute{e}tale}$ is given by Theorem 59.45.2. We omit the proof that under this equivalence the étale coverings correspond. Hence (1) holds. Parts (2) and (3) follow formally from (1). $\square$
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