Artin's criterion explained

In mathematics, Artin's criteria^{[1] [2] [3]} are a collection of related necessary and sufficient conditions on deformation functors which prove the representability of these functors as either Algebraic spaces^[4] 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.^[5]

Notation and technical notes

Throughout this article, let

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

A stack

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

\lim _{→} F(X _{i)} \to F(\lim _{→} X _{i)}

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

A limit preserving stack

F over Sch/S is called an algebraic stack if

1. For any pair of elements

x \in F(X), y \in F(Y) the fiber product X x _{F} Y is represented as an algebraic space

1. There is a scheme

X \to S locally of finite type, and an element x \in F(X) which is smooth and surjective such that for any y \in F(Y) the induced map X x _{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

Notes and References

1. Artin. M.. September 1974. Versal deformations and algebraic stacks. Inventiones Mathematicae. 27. 3. 165–189. 10.1007/bf01390174. 122887093. 0020-9910.
2. Artin. M.. January 1970. Algebraization of Formal Moduli: II. Existence of Modifications. The Annals of Mathematics. 91. 1. 88–135. 10.2307/1970602. 1970602. 0003-486X.
3. Artin. M.. January 1969. Algebraic approximation of structures over complete local rings. Publications Mathématiques de l'IHÉS. 36. 1. 23–58. 10.1007/bf02684596. 4617543. 0073-8301.
4. 1306.4599. Jack. Hall. David. Rydh. Artin's criteria for algebraicity revisited. Algebra & Number Theory. 2019. 13. 4. 749–796. 10.2140/ant.2019.13.749. 119597571.
5. Knudsen. Finn F.. 1983-12-01. The projectivity of the moduli space of stable curves, II: The stacks $M_$. Mathematica Scandinavica. en. 52. 161–199. 10.7146/math.scand.a-12001. 1903-1807. free.