Quasi-compact morphism
In algebraic geometry, a morphism between
It is not enough that Y admits a covering by compact open subschemes whose pre-images are compact. To give an example,
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.
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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.[3]
See also
References
- Robin Hartshorne, Algebraic Geometry.
- arXiv:math/0412512