Artin's Criterion
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In mathematics, Artin's criteria are a collection of related
necessary and sufficient condition In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
s on deformation functors which prove the representability of these functors as either
algebraic space In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively, schemes are given by gluing together affine schemes using the Zariski topology, ...
s or as
algebraic stack In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's re ...
s. In particular, these conditions are used in the construction of the
moduli stack of elliptic curves In mathematics, the moduli stack of elliptic curves, denoted as \mathcal_ or \mathcal_, is an algebraic stack over \text(\mathbb) classifying elliptic curves. Note that it is a special case of the moduli stack of algebraic curves \mathcal_. In part ...
and the construction of the moduli stack of pointed curves.


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 Stack may refer to: Places * Stack Island, an island game reserve in Bass Strait, south-eastern Australia, in Tasmania’s Hunter Island Group * Blue Stack Mountains, in Co. Donegal, Ireland People * Stack (surname) (including a list of people ...
, 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 \_ 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 \mathcal_S-algebra of finite type. A limit preserving stack F over Sch/S is called an algebraic stack if # For any pair of elements x \in F(X), y \in F(Y) the fiber product X\times_F Y is represented as an algebraic space # 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\times_F Y \to Y is smooth and surjective.


See also

*
Artin approximation theorem In mathematics, the Artin approximation theorem is a fundamental result of in deformation theory which implies that formal power series with coefficients in a field ''k'' are well-approximated by the algebraic functions on ''k''. More precisely, ...
*
Schlessinger's theorem In algebra, Schlessinger's theorem is a theorem in deformation theory introduced by that gives conditions for a functor of artinian local rings to be pro-representable, refining an earlier theorem of Grothendieck. Definitions Λ is a complete No ...


References


Deformation theory and algebraic stacks
- overview of Artin's papers and related research Algebraic geometry {{algebraic-geometry-stub