Theorem 54.45.1. 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.

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 40.15.2. We first prove the result for morphisms and then we state the result for categories.

Theorem 54.45.1. 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

\[ \mathop{Mor}\nolimits _ S(Y, X) \longrightarrow \mathop{Mor}\nolimits _{S'}(Y', X') \]

is bijective.

**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

54.45.1.1

\begin{equation} \label{etale-cohomology-equation-descent-etale-forget} \begin{matrix} \text{descent data }(X', \varphi ')\text{ relative to }S'/S
\\ \text{ with }X'\text{ étale over }S'
\end{matrix} \longrightarrow \text{schemes }X'\text{ étale over }S' \end{equation}

We claim this functor is an equivalence. On the other hand, the functor

54.45.1.2

\begin{equation} \label{etale-cohomology-equation-descent-etale} \text{schemes }X\text{ étale over }S \longrightarrow \begin{matrix} \text{descent data }(X', \varphi ')\text{ relative to }S'/S
\\ \text{ with }X'\text{ étale over }S'
\end{matrix} \end{equation}

is fully faithful by Étale Morphisms, Lemma 40.20.3. 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 28.43.5. In particular, the diagonal $\Delta : S' \to S' \times _ S S'$ is a thickening by Morphisms, Lemma 28.10.2. Thus by Étale Morphisms, Theorem 40.15.1 we see that given $X' \to S'$ étale there is a unique isomorphism

\[ \varphi ' : X' \times _ S S' \to S' \times _ S X' \]

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 (54.45.1.1). This proves the claim and finishes the proof of the theorem. $\square$

Theorem 54.45.2. 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 28.43.5). The functor

\[ V \longmapsto V_ X = X \times _ Y V \]

defines an equivalence of categories

\[ \{ \text{schemes }V\text{ étale over }Y \} \leftrightarrow \{ \text{schemes }U\text{ étale over }X \} \]

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 54.45.1 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

\[ X \times _ Y V_{i, i'} \cong U_ i \cap U_{i'} \cong X \times _ Y V_{i', i} \]

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 25.14. Applying Schemes, Lemma 25.14.2 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 54.45.1 we showed that $U$ comes with a unique descent datum $(U, \varphi )$ relative to $X/Y$. By Étale Morphisms, Proposition 40.20.6 (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 54.45.1 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

\[ X \times _ Y V_{i, i'} \cong U_ i \cap U_{i'} \cong X \times _ Y V_{i', i} \]

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 25.14. Applying Schemes, Lemma 25.14.2 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

54.45.2.1

\begin{equation} \label{etale-cohomology-equation-affine-etale} \{ \text{affine schemes }V\text{ étale over }Y \} \leftrightarrow \{ \text{affine schemes }U\text{ étale over }X \} \end{equation}

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 28.34.6 and 28.54.10. Hence we can do induction on the integer $n$ bounding the degree of the fibres of $U \to X$. See Morphisms, Lemma 28.54.9 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 40.14.1), and the result is clear. Assume $n > 1$.

By Lemma 54.44.4 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

\[ \xymatrix{ U \ar[d] & U \times _ Y W \ar[l] \ar[d] & W_ X \amalg R \ar@{=}[l]\\ X \ar[d] & W_ X \ar[l] \ar[d] \\ Y & W \ar[l] } \]

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 (54.45.2.1) 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 34.34.1 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 34.20.29. Moreover, there is an isomorphism $V_ X \cong U$ which comes from descending the isomorphism

\[ V_ X \times _ X W_ X = X \times _ Y V \times _ Y W = (X \times _ Y W) \times _ W (W \times _ Y V) \cong W_ X \times _ W V'' \cong U \times _ Y W \]

which we have by construction. Some details omitted. $\square$

Remark 54.45.3. In the situation of Theorem 54.45.2 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 31.11.2.

Proposition 54.45.4 (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 54.45.2. 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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