The Stacks project

Remark 26.15.5. Suppose the functor $F$ is defined on all locally ringed spaces, and if conditions of Lemma 26.15.4 are replaced by the following:

  1. $F$ satisfies the sheaf property on the category of locally ringed spaces,

  2. there exists a set $I$ and a collection of subfunctors $F_ i \subset F$ such that

    1. each $F_ i$ is representable by a scheme,

    2. each $F_ i \subset F$ is representable by open immersions on the category of locally ringed spaces, and

    3. the collection $(F_ i)_{i \in I}$ covers $F$ as a functor on the category of locally ringed spaces.

We leave it to the reader to spell this out further. Then the end result is that the functor $F$ is representable in the category of locally ringed spaces and that the representing object is a scheme.

Comments (0)

There are also:

  • 5 comment(s) on Section 26.15: A representability criterion

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Your email address will not be published.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