The Stacks project

Proof. We are going to use Schemes, Lemma 26.15.4.

First we check that $F$ satisfies the sheaf property for the Zariski topology. Namely, suppose that $T$ is a scheme, that $T = \bigcup _{i \in I} U_ i$ is an open covering, and that $(f_ i, \varphi _ i) \in F(U_ i)$ such that $(f_ i, \varphi _ i)|_{U_ i \cap U_ j} = (f_ j, \varphi _ j)|_{U_ i \cap U_ j}$. This implies that the morphisms $f_ i : U_ i \to S$ glue to a morphism of schemes $f : T \to S$ such that $f|_{U_ i} = f_ i$, see Schemes, Section 26.14. Thus $f_ i^*\mathcal{A} = f^*\mathcal{A}|_{U_ i}$ and by assumption the morphisms $\varphi _ i$ agree on $U_ i \cap U_ j$. Hence by Sheaves, Section 6.33 these glue to a morphism of $\mathcal{O}_ T$-algebras $f^*\mathcal{A} \to \mathcal{O}_ T$. This proves that $F$ satisfies the sheaf condition with respect to the Zariski topology.

Let $S = \bigcup _{i \in I} U_ i$ be an affine open covering. Let $F_ i \subset F$ be the subfunctor consisting of those pairs $(f : T \to S, \varphi )$ such that $f(T) \subset U_ i$.

We have to show each $F_ i$ is representable. This is the case because $F_ i$ is identified with the functor associated to $U_ i$ equipped with the quasi-coherent $\mathcal{O}_{U_ i}$-algebra $\mathcal{A}|_{U_ i}$, by Lemma 27.4.1. Thus the result follows from Lemma 27.4.2.

Next we show that $F_ i \subset F$ is representable by open immersions. Let $(f : T \to S, \varphi ) \in F(T)$. Consider $V_ i = f^{-1}(U_ i)$. It follows from the definition of $F_ i$ that given $a : T' \to T$ we gave $a^*(f, \varphi ) \in F_ i(T')$ if and only if $a(T') \subset V_ i$. This is what we were required to show.

Finally, we have to show that the collection $(F_ i)_{i \in I}$ covers $F$. Let $(f : T \to S, \varphi ) \in F(T)$. Consider $V_ i = f^{-1}(U_ i)$. Since $S = \bigcup _{i \in I} U_ i$ is an open covering of $S$ we see that $T = \bigcup _{i \in I} V_ i$ is an open covering of $T$. Moreover $(f, \varphi )|_{V_ i} \in F_ i(V_ i)$. This finishes the proof of the lemma. $\square$


Comments (0)

There are also:

  • 14 comment(s) on Section 27.4: Relative spectrum as a functor

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Your email address will not be published.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 01LU. Beware of the difference between the letter 'O' and the digit '0'.