## 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 . 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.

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