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.
be a scheme of finite-type over a field
will be a category fibered in groupoids,
will be the groupoid lying over
is called limit preserving if it is compatible with filtered direct limits in
, meaning given a filtered system
there is an equivalence of categories
is called an algebraic element if it is the henselization of an
A limit preserving stack
is called an algebraic stack if
This algebraic geometry–related article is a stub.