Difference between revisions of "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==== | ====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 | + | 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
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