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 $f^{-1}(V_{i})$ are compact.^[1] 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,^[2] let A be a ring that does not satisfy the ascending chain conditions on radical ideals, and put $X=\operatorname {Spec} A$ . 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 $f:X\to Y$ 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 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.^[3]

References[edit]

^ This is the definition in Hartshorne.
^ Remark 1.5 in Vistoli
^ Schwede, Karl (2005), "Gluing schemes and a scheme without closed points", Recent progress in arithmetic and algebraic geometry, Contemp. Math., vol. 386, Amer. Math. Soc., Providence, RI, pp. 157–172, doi:10.1090/conm/386/07222, MR 2182775. See in particular Proposition 4.1.

Robin Hartshorne, Algebraic Geometry.
Angelo Vistoli, "Notes on Grothendieck topologies, fibered categories and descent theory." arXiv:math/0412512

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When is an irreducible scheme quasi-compact?

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[1] This is the definition in Hartshorne.

[2] Remark 1.5 in Vistoli

[3] Schwede, Karl (2005), "Gluing schemes and a scheme without closed points", Recent progress in arithmetic and algebraic geometry, Contemp. Math., vol. 386, Amer. Math. Soc., Providence, RI, pp. 157–172, doi:10.1090/conm/386/07222, MR 2182775. See in particular Proposition 4.1.

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