# 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 proving their 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.

7. ^ Knudsen, Finn F. (1983-12-01). "The projectivity of the moduli space of stable curves, II: The stacks $M_{g,n}$". Mathematica Scandinavica. 52: 161–199. doi:10.7146/math.scand.a-12001. ISSN 1903-1807.