The Stacks project

Lemma 35.10.7. Let $\{ T_ i \to T\} $ be an fpqc covering, see Topologies, Definition 34.9.1. Then $\{ T_ i \to T\} $ is a universal effective epimorphism in the category of schemes, see Sites, Definition 7.12.1. In other words, every representable functor on the category of schemes satisfies the sheaf condition for the fpqc topology, see Topologies, Definition 34.9.12.

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 $h_ S = \mathop{Mor}\nolimits _{\mathit{Sch}}( - , S)$ 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 34.9.13 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 35.10.6 the morphism $X \to Y$ is universally submersive. Hence we may apply Lemma 35.10.5 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 35.10.1 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 29.24.12 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 $\mathop{\mathrm{Spec}}(R) \to S$. $\square$


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 023Q. Beware of the difference between the letter 'O' and the digit '0'.