The Stacks project

54.45 Topological invariance of the small étale site

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. $\square$

reference

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

  1. the étale sites $X_{\acute{e}tale}$ and $(X_0)_{\acute{e}tale}$ are isomorphic,

  2. the étale topoi $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ and $\mathop{\mathit{Sh}}\nolimits ((X_0)_{\acute{e}tale})$ are equivalent, and

  3. $H^ q_{\acute{e}tale}(X, \mathcal{F}) = H^ q_{\acute{e}tale}(X_0, \mathcal{F}|_{X_0})$ 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). $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04DY. Beware of the difference between the letter 'O' and the digit '0'.