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Artin's criterion

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In mathematics, Artin's criteria are a collection of related necessary and sufficient conditions on deformation functors which prove the representability of these functors as either Algebraic spaces 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.

Notation and technical notes

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

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

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

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

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

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

See also

References

  1. Artin, M. (September 1974). "Versal deformations and algebraic stacks". Inventiones Mathematicae. 27 (3): 165–189. doi:10.1007/bf01390174. ISSN 0020-9910. S2CID 122887093.
  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, doi:10.1515/9781400871230-003, ISBN 978-1-4008-7123-0
  3. 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. JSTOR 1970602.
  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. S2CID 4617543.
  5. Hall, Jack; Rydh, David (2019). "Artin's criteria for algebraicity revisited". Algebra & Number Theory. 13 (4): 749–796. arXiv:1306.4599. doi:10.2140/ant.2019.13.749. S2CID 119597571.
  6. Deligne, P.; Rapoport, M. (1973), Les schémas de modules de courbes elliptiques, Lecture Notes in Mathematics, vol. 349, Springer Berlin Heidelberg, pp. 143–316, doi:10.1007/bfb0066716, ISBN 978-3-540-06558-6
  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.


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