between schemes is said to be quasi-compact if Y can be covered by open affine subschemes
[1] If f is quasi-compact, then the pre-image of a compact open subscheme (e.g., open affine subscheme) under f is compact.
To give an example,[2] let A be a ring that does not satisfy the ascending chain conditions on radical ideals, and put
Let Y be the scheme obtained by gluing two X's along U. X, Y are both compact.
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.
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] This abstract algebra-related article is a stub.