The Stacks project

112.8 Definition of algebraic stacks

An algebraic stack is a stack in groupoids over the category of schemes with the fppf topology that has a diagonal representable by algebraic spaces and is the target of a surjective smooth morphism from a scheme. See Algebraic Stacks, Section 93.12. A “Deligne-Mumford stack” is an algebraic stack for which there exists a scheme and a surjective étale morphism from that scheme to it as in the paper [DM] of Deligne and Mumford, see Algebraic Stacks, Definition 93.12.2. We will reserve the term “Artin stack” for a stack such as in the papers by Artin, see [ArtinI], [ArtinII], and [ArtinVersal]. A possible definition is that an Artin stack is an algebraic stack $\mathcal{X}$ over a locally Noetherian scheme $S$ such that $\mathcal{X} \to S$ is locally of finite type1.

[1] Namely, these are exactly the algebraic stacks over $S$ satisfying Artin's axioms [-1], [0], [1], [2], [3], [4], [5] of Artin's Axioms, Section 97.14.

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Your email address will not be published.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 02BK. Beware of the difference between the letter 'O' and the digit '0'.