View source for Quasi-affine scheme

A scheme isomorphic to an open compact subscheme of an [[Affine scheme|affine scheme]]. A compact scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076360/q0763601.png" /> is quasi-affine if and only if one of the following equivalent conditions holds: 1) the canonical morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076360/q0763602.png" /> is an open imbedding; and 2) any [[Quasi-coherent sheaf|quasi-coherent sheaf]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076360/q0763603.png" />-modules is generated by global sections. A morphism of schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076360/q0763604.png" /> is called quasi-affine if for any open affine subscheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076360/q0763605.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076360/q0763606.png" /> the inverse image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076360/q0763607.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076360/q0763608.png" />.

====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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Hartshorne,   "Algebraic geometry" , Springer  (1977)  pp. 3, 21</TD></TR></table>