Artin's criterion

In mathematics, Artin's criteria^[1][2][3][4] are a collection of related necessary and sufficient conditions on deformation functors which prove the representability of these functors as either Algebraic spaces^[5] or as Algebraic stacks. In particular, these conditions are used in the construction of the moduli stack of elliptic curves^[6] and the construction of the moduli stack of pointed curves.^[7]

Notation and technical notes

Throughout this article, let ${\displaystyle S}$ be a scheme of finite-type over a field ${\displaystyle k}$ or an excellent DVR. ${\displaystyle p:F\to (Sch/S)}$ will be a category fibered in groupoids, ${\displaystyle F(X)}$ will be the groupoid lying over ${\displaystyle X\to S}$.

A stack ${\displaystyle F}$ is called limit preserving if it is compatible with filtered direct limits in ${\displaystyle Sch/S}$, meaning given a filtered system ${\displaystyle \{X_{i}\}_{i\in I}}$ there is an equivalence of categories

${\displaystyle \lim _{\rightarrow }F(X_{i})\to F(\lim _{\rightarrow }X_{i})}$

An element of ${\displaystyle x\in F(X)}$ is called an algebraic element if it is the henselization of an ${\displaystyle {\mathcal {O}}_{S}}$-algebra of finite type.

A limit preserving stack ${\displaystyle F}$ over ${\displaystyle Sch/S}$ is called an algebraic stack if

1. For any pair of elements ${\displaystyle x\in F(X),y\in F(Y)}$ the fiber product ${\displaystyle X\times _{F}Y}$ is represented as an algebraic space
2. There is a scheme ${\displaystyle X\to S}$ locally of finite type, and an element ${\displaystyle x\in F(X)}$ which is smooth and surjective such that for any ${\displaystyle y\in F(Y)}$ the induced map ${\displaystyle X\times _{F}Y\to Y}$ 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