Quasi-compact Morphism

In algebraic geometry, a morphism $f: X \to Y$ between schemes is said to be quasi-compact if ''Y'' can be covered by open affine subschemes $V_i$ such that the pre-images $f^(V_i)$ are quasi-compact (as topological space). If ''f'' is quasi-compact, then the pre-image of a quasi-compact open subscheme (e.g., open affine subscheme) under ''f'' is quasi-compact.

It is not enough that ''Y'' admits a covering by quasi-compact open subschemes whose pre-images are quasi-compact. To give an example, let ''A'' be a ring that does not satisfy the ascending chain conditions on radical ideals, and put $X = \operatorname A$. ''X'' contains an open subset ''U'' that is not quasi-compact. Let ''Y'' be the scheme obtained by gluing two ''Xs along ''U''. ''X'', ''Y'' are both quasi-compact. If $f: X \to Y$ 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 quasi-compact. Hence, ''f'' is not quasi-compact.

A morphism from a quasi-compact scheme to an affine scheme is quasi-compact.

Let $f: X \to Y$ be a quasi-compact morphism between schemes. Then $f(X)$ 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

*Hartshorne, ''Algebraic Geometry''.
*Angelo Vistoli, "Notes on Grothendieck topologies, fibered categories and descent theory."

External links

When is an irreducible scheme quasi-compact?

Morphisms of schemes

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