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 [math]\displaystyle{ S }[/math] be a scheme of finite-type over a field [math]\displaystyle{ k }[/math] or an excellent DVR. [math]\displaystyle{ p:F \to (Sch/S) }[/math] will be a category fibered in groupoids, [math]\displaystyle{ F(X) }[/math] will be the groupoid lying over [math]\displaystyle{ X \to S }[/math].
A stack [math]\displaystyle{ F }[/math] is called limit preserving if it is compatible with filtered direct limits in [math]\displaystyle{ Sch/S }[/math], meaning given a filtered system [math]\displaystyle{ \{X_i\}_{i\in I} }[/math] there is an equivalence of categories
[math]\displaystyle{ \lim_{\rightarrow}F(X_i) \to F(\lim_{\rightarrow}X_i) }[/math]
An element of [math]\displaystyle{ x \in F(X) }[/math] is called an algebraic element if it is the henselization of an [math]\displaystyle{ \mathcal{O}_S }[/math]-algebra of finite type.
A limit preserving stack [math]\displaystyle{ F }[/math] over [math]\displaystyle{ Sch/S }[/math] is called an algebraic stack if
- For any pair of elements [math]\displaystyle{ x \in F(X), y \in F(Y) }[/math] the fiber product [math]\displaystyle{ X\times_F Y }[/math] is represented as an algebraic space
- There is a scheme [math]\displaystyle{ X \to S }[/math] locally of finite type, and an element [math]\displaystyle{ x \in F(X) }[/math] which is smooth and surjective such that for any [math]\displaystyle{ y \in F(Y) }[/math] the induced map [math]\displaystyle{ X\times_F Y \to Y }[/math] is smooth and surjective.
See also
References
- ↑ Artin, M. (September 1974). "Versal deformations and algebraic stacks". Inventiones Mathematicae 27 (3): 165–189. doi:10.1007/bf01390174. ISSN 0020-9910.
- ↑ 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, doi:10.1515/9781400871230-003, ISBN 978-1-4008-7123-0
- ↑ Artin, M. (January 1970). "Algebraization of Formal Moduli: II. Existence of Modifications". The Annals of Mathematics 91 (1): 88–135. doi:10.2307/1970602. ISSN 0003-486X.
- ↑ 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. http://www.numdam.org/item/PMIHES_1969__36__23_0/.
- ↑ Hall, Jack; Rydh, David (2019). "Artin's criteria for algebraicity revisited". Algebra & Number Theory 13 (4): 749–796. doi:10.2140/ant.2019.13.749.
- ↑ Deligne, P.; Rapoport, M. (1973), Les schémas de modules de courbes elliptiques, Lecture Notes in Mathematics, 349, Springer Berlin Heidelberg, pp. 143–316, doi:10.1007/bfb0066716, ISBN 978-3-540-06558-6
- ↑ Knudsen, Finn F. (1983-12-01). "The projectivity of the moduli space of stable curves, II: The stacks $M_{g,n}$" (in en). Mathematica Scandinavica 52: 161–199. doi:10.7146/math.scand.a-12001. ISSN 1903-1807. https://www.mscand.dk/article/view/12001.
- Deformation theory and algebraic stacks - overview of Artin's papers and related research
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