Quasi-affine scheme

A scheme isomorphic to an open compact subscheme of an affine scheme. A compact scheme $ X $ is quasi-affine if and only if one of the following equivalent conditions holds: 1) the canonical morphism $ X \mapsto \mathop{\rm Spec} \Gamma ( X , {\mathcal O} _ {X} ) $ is an open imbedding; and 2) any quasi-coherent sheaf of $ {\mathcal O} _ {X} $- modules is generated by global sections. A morphism of schemes $ f : X \rightarrow Y $ is called quasi-affine if for any open affine subscheme $ U $ in $ Y $ the inverse image $ f ^ { - 1 } ( U) $ 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 $ \mathbf C ^ {2} \setminus \{ ( 0, 0) \} $.

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