In algebraic geometry, given algebraic stacks p : X → C , q : Y → C {\displaystyle p:X\to C,\,q:Y\to C} over a base category C, a morphism f : X → Y {\displaystyle f:X\to Y} of algebraic stacks is a functor such that q ∘ f = p {\displaystyle q\circ f=p} .
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 U → X {\displaystyle U\to X} of relative dimension j for some smooth scheme U of dimension n. For example, if Vect n {\displaystyle \operatorname {Vect} _{n}} denotes the moduli stack of rank-n vector bundles, then there is a presentation Spec ( k ) → Vect n {\displaystyle \operatorname {Spec} (k)\to \operatorname {Vect} _{n}} given by the trivial bundle A k n {\displaystyle \mathbb {A} _{k}^{n}} over Spec ( k ) {\displaystyle \operatorname {Spec} (k)} .
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
References
§ 8.6 of F. Meyer, Notes on algebraic stacks https://folk.uio.no/fredrme/algstacks.pdf ↩