Morphism of algebraic stacks
In algebraic geometry, given algebraic stacks [math]\displaystyle{ p: X \to C, \, q: Y \to C }[/math] over a base category C, a morphism [math]\displaystyle{ f: X \to Y }[/math] of algebraic stacks is a functor such that [math]\displaystyle{ 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]\displaystyle{ U \to X }[/math] of relative dimension j for some smooth scheme U of dimension n. For example, if [math]\displaystyle{ \operatorname{Vect}_n }[/math] denotes the moduli stack of rank-n vector bundles, then there is a presentation [math]\displaystyle{ \operatorname{Spec}(k) \to \operatorname{Vect}_n }[/math] given by the trivial bundle [math]\displaystyle{ \mathbb{A}^n_k }[/math] over [math]\displaystyle{ \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.[1]
Notes
- ↑ § 8.6 of F. Meyer, Notes on algebraic stacks
References
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