The Stacks project

Proof. Let $S$ be a scheme. We have to show the following: Given morphisms $\varphi _ i : T_ i \to S$ such that $\varphi _ i|_{T_ i \times _ T T_ j} = \varphi _ j|_{T_ i \times _ T T_ j}$ there exists a unique morphism $T \to S$ which restricts to $\varphi _ i$ on each $T_ i$. In other words, we have to show that the functor $h_ S = \mathop{\mathrm{Mor}}\nolimits _{\mathit{Sch}}( - , S)$ satisfies the sheaf property for the fpqc topology.

If $\{ T_ i \to T\} $ is a Zariski covering, then this follows from Schemes, Lemma 26.14.1. Thus Topologies, Lemma 34.9.14 reduces us to the case of a covering $\{ X \to Y\} $ given by a single surjective flat morphism of affines.

First proof. By Lemma 35.8.1 we have the sheaf condition for quasi-coherent modules for $\{ X \to Y\} $. By Lemma 35.13.6 the morphism $X \to Y$ is universally submersive. Hence we may apply Lemma 35.13.5 to see that $\{ X \to Y\} $ is a universal effective epimorphism.

Second proof. Let $R \to A$ be the faithfully flat ring map corresponding to our surjective flat morphism $\pi : X \to Y$. Let $f : X \to S$ be a morphism such that $f \circ \text{pr}_1 = f \circ \text{pr}_2$ as morphisms $X \times _ Y X = \mathop{\mathrm{Spec}}(A \otimes _ R A) \to S$. By Lemma 35.13.1 we see that as a map on the underlying sets $f$ is of the form $f = g \circ \pi $ for some (set theoretic) map $g : \mathop{\mathrm{Spec}}(R) \to S$. By Morphisms, Lemma 29.25.12 and the fact that $f$ is continuous we see that $g$ is continuous.

Pick $y \in Y = \mathop{\mathrm{Spec}}(R)$. Choose $U \subset S$ affine open containing $g(y)$. Say $U = \mathop{\mathrm{Spec}}(B)$. By the above we may choose an $r \in R$ such that $y \in D(r) \subset g^{-1}(U)$. The restriction of $f$ to $\pi ^{-1}(D(r))$ into $U$ corresponds to a ring map $B \to A_ r$. The two induced ring maps $B \to A_ r \otimes _{R_ r} A_ r = (A \otimes _ R A)_ r$ are equal by assumption on $f$. Note that $R_ r \to A_ r$ is faithfully flat. By Lemma 35.3.6 the equalizer of the two arrows $A_ r \to A_ r \otimes _{R_ r} A_ r$ is $R_ r$. We conclude that $B \to A_ r$ factors uniquely through a map $B \to R_ r$. This map in turn gives a morphism of schemes $D(r) \to U \to S$, see Schemes, Lemma 26.6.4.

What have we proved so far? We have shown that for any prime $\mathfrak p \subset R$, there exists a standard affine open $D(r) \subset \mathop{\mathrm{Spec}}(R)$ such that the morphism $f|_{\pi ^{-1}(D(r))} : \pi ^{-1}(D(r)) \to S$ factors uniquely through some morphism of schemes $D(r) \to S$. We omit the verification that these morphisms glue to the desired morphism $\mathop{\mathrm{Spec}}(R) \to S$. $\square$