The Stacks project

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 ( 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 (

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 $\mathop{\mathrm{Spec}}(R) \to \mathfrak X$. $\square$

Comments (5)

Comment #1556 by Matthew Emerton on

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

Comment #1941 by Brian Conrad on

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

Comment #3790 by slogan_bot on

Suggested slogan: "Formal schemes are fpqc sheaves"

There are also:

  • 12 comment(s) on Section 86.2: Formal schemes à la EGA

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 0AI2. Beware of the difference between the letter 'O' and the digit '0'.