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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.
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A scheme isomorphic to an open compact subscheme of an [[Affine scheme|affine scheme]]. A compact scheme  $  X $
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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} ) $
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is an open imbedding; and 2) any [[Quasi-coherent sheaf|quasi-coherent sheaf]] of  $  {\mathcal O} _ {X} $-
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modules is generated by global sections. A morphism of schemes  $  f :  X \rightarrow Y $
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is called quasi-affine if for any open affine subscheme  $  U $
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in  $  Y $
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the inverse image  $  f ^ { - 1 } ( U) $
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is a quasi-affine scheme.
  
 
====Comments====
 
====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" />.
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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====
 
====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>
 
<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>

Latest revision as of 08:09, 6 June 2020


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
How to Cite This Entry:
Quasi-affine scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-affine_scheme&oldid=23944
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article