Quasi-compact morphism

In algebraic geometry, a morphism $f:\; X\; \backslash to\; Y$ 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.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 $X\; =\; \backslash operatorname\{Spec\}\; A$. 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 $f:\; X\; \backslash to\; Y$ 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 $f:\; X\; \backslash to\; Y$ 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.

{{expert needed|Mathematics |talk=Quasi-compact schemes vs. quasi-compact morphisms|date=August 2023}}

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.{{citation

| last = Schwede | first = Karl

| contribution = Gluing schemes and a scheme without closed points

| doi = 10.1090/conm/386/07222

| mr = 2182775

| pages = 157–172

| publisher = Amer. Math. Soc., Providence, RI

| series = Contemp. Math.

| title = Recent progress in arithmetic and algebraic geometry

| volume = 386

| year = 2005| isbn = 978-0-8218-3401-5

}}. See in particular Proposition 4.1.