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.

See also

• Artin approximation theorem
• Schlessinger's theorem

References

1. Artin, M. (September 1974). "Versal deformations and algebraic stacks". Inventiones Mathematicae. 27 (3): 165–189. doi:10.1007/bf01390174. ISSN 0020-9910.
2. Artin, M. (2015-12-31), "Algebraization of formal moduli: I", Global Analysis: Papers in Honor of K. Kodaira (PMS-29), Princeton: Princeton University Press, pp. 21–72, ISBN 978-1-4008-7123-0, retrieved 2020-06-15
3. Artin, M. (January 1970). "Algebraization of Formal Moduli: II. Existence of Modifications". The Annals of Mathematics. 91 (1): 88. doi:10.2307/1970602. ISSN 0003-486X.
4. Artin, M. (January 1969). "Algebraic approximation of structures over complete local rings". Publications mathématiques de l'IHÉS. 36 (1): 23–58. doi:10.1007/bf02684596. ISSN 0073-8301.
5. Hall, Jack; Rydh, David (2013). "Artin's criteria for algebraicity revisited". arXiv:1306.4599 [math.AG].
6. Deligne, P.; Rapoport, M., "Les schémas de modules de courbes elliptiques", Lecture Notes in Mathematics, Springer Berlin Heidelberg, pp. 143–316, ISBN 978-3-540-06558-6, retrieved 2020-06-15
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.

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