Quasi-affine scheme

A scheme isomorphic to an open compact subscheme of an affine scheme. A compact scheme is quasi-affine if and only if one of the following equivalent conditions holds: 1) the canonical morphism is an open imbedding; and 2) any quasi-coherent sheaf of -modules is generated by global sections. A morphism of schemes is called quasi-affine if for any open affine subscheme in the inverse image is a quasi-affine scheme.

Comments

A quasi-affine variety is an open subvariety of an affine algebraic variety. (As an open subspace of a Noetherian space it is automatically compact.) An example of a quasi-affine variety that is not affine is .

References

[a1]	A. Grothendieck, "Étude globale élémentaire de quelques classes de morphismes" Publ. Math. IHES , 8 (1961) pp. Sect. 5.1 MR0217084 MR0163909 Zbl 0118.36206
[a2]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 3, 21 MR0463157 Zbl 0367.14001