Smooth Stack

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 $\operatorname_n$ denotes the moduli stack of rank-''n'' vector bundles, then there is a presentation $\operatorname(k) \to \operatorname_n$ given by the trivial bundle $\mathbb^n_k$ over $\operatorname(k)$.

A quasi-affine morphism between algebraic stacks is a morphism that factorizes as a quasi-compact open immersion followed by an affine morphism.§ 8.6 of F. Meyer
Notes on algebraic stacks
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Notes

References

*Stacks Project, Ch, 83
Morphisms of algebraic stacks

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Algebraic geometry