The Stacks project

Proof. For every quasi-compact and quasi-separated object $U_0$ of $(X_0)_{spaces, {\acute{e}tale}}$ consider the condition $P$ that

is an isomorphism where $U = X \times _{X_0} U_0$ and $U_ i = X_ i \times _{X_0} U_0$. We will prove $P$ holds for each $U_0$.

Suppose that $(U_0 \subset W_0, V_0 \to W_0)$ is an elementary distinguished square in $(X_0)_{spaces, {\acute{e}tale}}$ and $P$ holds for $U_0, V_0, U_0 \times _{W_0} V_0$. Then $P$ holds for $W_0$ by Mayer-Vietoris for hom in the derived category, see Derived Categories of Spaces, Lemma 75.10.4.

We first consider $U_0 = W_0 \times _{Y_0} X_0$ with $W_0$ a quasi-compact and quasi-separated object of $(Y_0)_{spaces, {\acute{e}tale}}$. By the induction principle of Derived Categories of Spaces, Lemma 75.9.3 applied to these $W_0$ and the previous paragraph, we find that it is enough to prove $P$ for $U_0 = W_0 \times _{Y_0} X_0$ with $W_0$ affine. In other words, we have reduced to the case where $Y_0$ is affine. Next, we apply the induction principle again, this time to all quasi-compact and quasi-separated opens of $X_0$, to reduce to the case where $X_0$ is affine as well.

If $X_0$ and $Y_0$ are affine, then we are back in the case of schemes which is proved in Derived Categories of Schemes, Lemma 36.35.8. The reader may use Derived Categories of Spaces, Lemmas 75.13.6, 75.4.2, 75.13.2, and 75.13.4 to accomplish the translation of the statement into a statement involving only schemes and derived categories of modules on schemes. $\square$