Artin's criterion

In mathematics, Artin's criteria are a collection of related necessary and sufficient conditions on deformation functors which prove the representability of these functors as either algebraic spaces or as algebraic stacks. In particular, these conditions are used in the construction of the moduli stack of elliptic curves and the construction of the moduli stack of pointed curves.

Notation and technical notes

Throughout this article, let be a scheme of finite-type over a field or an excellent DVR. will be a category fibered in groupoids, will be the groupoid lying over .

A stack is called limit preserving if it is compatible with filtered direct limits in , meaning given a filtered system there is an equivalence of categories<blockquote></blockquote>An element of is called an algebraic element if it is the henselization of an -algebra of finite type.

A limit preserving stack over is called an algebraic stack if

1. For any pair of elements the fiber product is represented as an algebraic space
2. There is a scheme locally of finite type, and an element which is smooth and surjective such that for any the induced map is smooth and surjective.

See also

• Artin approximation theorem
• Schlessinger's theorem

References

• Deformation theory and algebraic stacks - overview of Artin's papers and related research