Quasi-compact morphism

In algebraic geometry, a morphism formula_1 between schemes is said to be quasi-compact if "Y" can be covered by open affine subschemes formula_2 such that the pre-images formula_3 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 formula_4. "X" contains an open subset "U" that is not quasi-compact. Let "Y" be the scheme obtained by gluing two "X"'s along "U". "X", "Y" are both quasi-compact. If formula_1 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 formula_1 be a quasi-compact morphism between schemes. Then formula_7 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.