Morphism of algebraic stacks explained

In algebraic geometry, given algebraic stacks

p: X \to C, q: Y \to C over a base category C, a morphism f: X \to Y of algebraic stacks is a functor such that 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 \to X of relative dimension j for some smooth scheme U of dimension n. For example, if \operatorname{Vect} _{n} denotes the moduli stack of rank-n vector bundles, then there is a presentation \operatorname{Spec}(k) \to \operatorname{Vect} _{n} given by the trivial bundle A ^{n} _{k} over \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]

References

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

Notes and References

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