The Stacks project

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

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

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
\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
\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 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

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

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

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