Formal schemes are fpqc sheaves

Lemma 86.2.2. Let $\mathfrak X$ be a formal scheme. The functor of points $h_\mathfrak X$ (see Lemma 86.2.1) satisfies the sheaf condition for fpqc coverings.

Proof. Topologies, Lemma 34.9.13 reduces us to the case of a Zariski covering and a covering $\{ \mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)\}$ with $R \to S$ faithfully flat. We observed in the proof of Lemma 86.2.1 that $h_\mathfrak X$ satisfies the sheaf condition for Zariski coverings.

Suppose that $R \to S$ is a faithfully flat ring map. Denote $\pi : \mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ the corresponding morphism of schemes. It is surjective and flat. Let $f : \mathop{\mathrm{Spec}}(S) \to \mathfrak X$ be a morphism such that $f \circ \text{pr}_1 = f \circ \text{pr}_2$ as maps $\mathop{\mathrm{Spec}}(S \otimes _ R S) \to \mathfrak X$. By Descent, 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 $g : \mathop{\mathrm{Spec}}(R) \to \mathfrak X$. By Morphisms, Lemma 29.25.12 and the fact that $f$ is continuous we see that $g$ is continuous.

Pick $y \in \mathop{\mathrm{Spec}}(R)$. Choose $\mathfrak U \subset \mathfrak X$ an affine formal open subscheme containing $g(y)$. Say $\mathfrak U = \text{Spf}(A)$ for some admissible topological ring $A$. By the above we may choose an $r \in R$ such that $y \in D(r) \subset g^{-1}(\mathfrak U)$. The restriction of $f$ to $\pi ^{-1}(D(r))$ into $\mathfrak U$ corresponds to a continuous ring map $A \to S_ r$ by (86.2.0.1). The two induced ring maps $A \to S_ r \otimes _{R_ r} S_ r = (S \otimes _ R S)_ r$ are equal by assumption on $f$. Note that $R_ r \to S_ r$ is faithfully flat. By Descent, Lemma 35.3.6 the equalizer of the two arrows $S_ r \to S_ r \otimes _{R_ r} S_ r$ is $R_ r$. We conclude that $A \to S_ r$ factors uniquely through a map $A \to R_ r$ which is also continuous as it has the same (open) kernel as the map $A \to S_ r$. This map in turn gives a morphism $D(r) \to \mathfrak U$ by (86.2.0.1).

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

Comment #1556 by Matthew Emerton on

In line 3 from the bottom of the second-last para, should the second $S_r$ actually be $R_r$?

Comment #1941 by Brian Conrad on

On line 3 of 2nd to last paragraph, $x$ should be $y$. (There is no $x$ anywhere else in this proof.)

Comment #3790 by slogan_bot on

Suggested slogan: "Formal schemes are fpqc sheaves"

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