Morphism of algebraic stacks

In algebraic geometry, given algebraic stacks over a base category C, a morphism of algebraic stacks is a functor such that .

More generally, one can also consider a morphism between prestacks; for this, see prestack#Morphisms (a stackification would be an example.)


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 of relative dimension j for some smooth scheme U of dimension n. For example, if denotes the moduli stack of rank-n vector bundles, then there is a presentation given by the trivial bundle over .

A quasi-affine morphism between algebraic stacks is a morphism that factorizes as a quasi-compact open immersion followed by an affine morphism.[1]


  1. § 8.6 of F. Meyer, Notes on algebraic stacks


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