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In algebraic geometry, given algebraic stacks p:X→C,q:Y→Cdisplaystyle p:Xto C,,q:Yto C over a base category C, a morphism f:X→Ydisplaystyle f:Xto Y of algebraic stacks is a functor such that q∘f=pdisplaystyle qcirc f=p.

More generally, one can also consider a morphism between prestacks; for this, see prestack#Morphisms (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→Xdisplaystyle Uto X of relative dimension j for some smooth scheme U of dimension n. For example, if Vectndisplaystyle operatorname Vect _n denotes the moduli stack of rank-n vector bundles, then there is a presentation Spec⁡(k)→Vectndisplaystyle operatorname Spec (k)to operatorname Vect _n given by the trivial bundle Akndisplaystyle 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

• 
  Stacks Project, Ch, 83, Morphisms of algebraic stacks
  

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