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 |
Quasi-affine scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-affine_scheme&oldid=23944