Algebraic stack - Knowlpedia

Definition

Let $S$ be a scheme. An algebraic stack (or Artin stack) over $S$ is a stack in groupoids on the big fppf site of $S$ such that:

its diagonal is representable by algebraic spaces; and
there is a scheme $U$ and a representable, smooth, surjective morphism $U\to\mathcal X$ .

The morphism $U\to\mathcal X$ is a smooth atlas. Some conventions impose additional separation or finiteness conditions, which must be stated separately.

Unlike a coarse moduli space, a stack retains the automorphism group of every object. This is essential for $\operatorname{Bun}_G$ , because a principal bundle can have nontrivial automorphisms.

Quotient example

If an algebraic group $G$ acts on a scheme $U$ , the quotient stack $[U/G]$ remembers stabilizers as isotropy groups. Even when an ordinary geometric quotient exists, it generally forgets this information.

Derived warning

Modern geometric Langlands often replaces ordinary algebraic stacks by derived algebraic stacks, whose structure sheaves retain homotopical information. That is an additional structure, not part of the definition above.

References

Michael Artin, “Versal deformations and algebraic stacks,” Inventiones Mathematicae 27 (1974), 165–189. DOI.
Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, Springer,
DOI.