View source for Quasi-affine scheme

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A scheme isomorphic to an open compact subscheme of an [[Affine scheme|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|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|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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Grothendieck, "Étude globale élémentaire de quelques classes de morphismes" ''Publ. Math. IHES'' , '''8''' (1961) pp. Sect. 5.1 {{MR|0217084}} {{MR|0163909}} {{ZBL|0118.36206}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 3, 21 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>