Morphism of algebraic stacks

{{Short description|Type of functor}} {{Technical|date=November 2023}} In algebraic geometry, given algebraic stacks <math>p: X \to C, \, q: Y \to C</math> over a base category ''C'', a '''morphism <math>f: X \to Y</math> of algebraic stacks''' is a functor such that <math>q \circ f = p</math>.

More generally, one can also consider a morphism between prestacks (a stackification would be an example).

== Types == One particular important example is a presentation of a stack, which is widely used in the study of stacks.

An algebraic stack ''X'' is said to be '''smooth''' of dimension ''n'' - ''j'' if there is a smooth presentation <math>U \to X</math> of relative dimension ''j'' for some smooth scheme ''U'' of dimension ''n''. For example, if <math>\operatorname{Vect}_n</math> denotes the moduli stack of rank-''n'' vector bundles, then there is a presentation <math>\operatorname{Spec}(k) \to \operatorname{Vect}_n</math> given by the trivial bundle <math>\mathbb{A}^n_k</math> over <math>\operatorname{Spec}(k)</math>.

A '''quasi-affine morphism''' between algebraic stacks is a morphism that factorizes as a quasi-compact open immersion followed by an affine morphism.<ref>§ 8.6 of F. Meyer, [https://folk.uio.no/fredrme/algstacks.pdf Notes on algebraic stacks]</ref>

== Notes == {{reflist}}

== References == *Stacks Project, Ch, 83, [http://stacks.math.columbia.edu/download/stacks-morphisms.pdf Morphisms of algebraic stacks]

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Category:Stacks (mathematics)