In algebraic geometry, a morphism between schemes is said to be quasi-compact if Y can be covered by open affine subschemes such that the pre-images are compact.This is the definition in Hartshorne. If f is quasi-compact, then the pre-image of a compact open subscheme (e.g., open affine subscheme) under f is compact.
It is not enough that Y admits a covering by compact open subschemes whose pre-images are compact. To give an example,Remark 1.5 in Vistoli let A be a ring that does not satisfy the ascending chain conditions on radical ideals, and put . Then X contains an open subset U that is not compact. Let Y be the scheme obtained by gluing two X'''s along U. X, Y are both compact. If is the inclusion of one of the copies of X, then the pre-image of the other X, open affine in Y, is U—not compact. Hence, f'' is not quasi-compact.
A morphism from a quasi-compact scheme to an affine scheme is quasi-compact.
Let be a quasi-compact morphism between schemes. Then is closed if and only if it is stable under specialization.
The composition of quasi-compact morphisms is quasi-compact. The base change of a quasi-compact morphism is quasi-compact.
An affine scheme is quasi-compact. In fact, a scheme is quasi-compact if and only if it is a finite union of open affine subschemes. Serre's criterion gives a necessary and sufficient condition for a quasi-compact scheme to be affine.
A quasi-compact scheme has at least one closed point.. See in particular Proposition 4.1.
See also fpqc morphism
References
Angelo Vistoli, "Notes on Grothendieck topologies, fibered categories and descent theory."
External links When is an irreducible scheme quasi-compact?