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A morphism of schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a0110501.png" /> such that the pre-image of any open affine subscheme in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a0110502.png" /> is an affine [[Scheme|scheme]]. The scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a0110503.png" /> is called an affine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a0110505.png" />-scheme.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a0110506.png" /> be a scheme, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a0110507.png" /> be a quasi-coherent sheaf of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a0110508.png" />-algebras and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a0110509.png" /> be open affine subschemes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105010.png" /> which form a covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105011.png" />. Then the glueing of the affine schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105012.png" /> determines an affine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105013.png" />-scheme, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105014.png" />. Conversely, any affine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105015.png" />-scheme definable by an affine morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105016.png" /> is isomorphic (as a scheme over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105017.png" />) to the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105018.png" />. The set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105019.png" />-morphisms of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105020.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105021.png" /> into the affine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105022.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105023.png" /> is in bijective correspondence with the homomorphisms of the sheaves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105024.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105025.png" />.
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A morphism of schemes  $  f: X \rightarrow S $
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such that the pre-image of any open affine subscheme in  $  S $
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is an affine [[Scheme|scheme]]. The scheme  $  X $
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is called an affine  $  S $-
 +
scheme.
 +
 
 +
Let  $  S $
 +
be a scheme, let $  A $
 +
be a quasi-coherent sheaf of $  {\mathcal O} _ {S} $-
 +
algebras and let $  U _ {i} $
 +
be open affine subschemes in $  S $
 +
which form a covering of $  S $.  
 +
Then the glueing of the affine schemes $  { \mathop{\rm Spec} }  \Gamma (U _ {i} , A) $
 +
determines an affine $  S $-
 +
scheme, denoted by $  { \mathop{\rm Spec} }  A $.  
 +
Conversely, any affine $  S $-
 +
scheme definable by an affine morphism $  f: X \rightarrow S $
 +
is isomorphic (as a scheme over $  S $)  
 +
to the scheme $  { \mathop{\rm Spec} }  f _ {*} ( {\mathcal O} _ {X} ) $.  
 +
The set of $  S $-
 +
morphisms of an $  S $-
 +
scheme $  f: Z \rightarrow S $
 +
into the affine $  S $-
 +
scheme $  { \mathop{\rm Spec} }  A $
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is in bijective correspondence with the homomorphisms of the sheaves of $  {\mathcal O} _ {S} $-
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algebras $  A \rightarrow f _ {*} ( {\mathcal O} _ {Z} ) $.
  
 
Closed imbeddings of schemes or arbitrary morphisms of affine schemes are affine morphisms; other examples of affine morphisms are entire morphisms and finite morphisms. Thus the morphism of normalization of a scheme is an affine morphism. Under composition and base change the property of a morphism to be an affine morphism is preserved.
 
Closed imbeddings of schemes or arbitrary morphisms of affine schemes are affine morphisms; other examples of affine morphisms are entire morphisms and finite morphisms. Thus the morphism of normalization of a scheme is an affine morphism. Under composition and base change the property of a morphism to be an affine morphism is preserved.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Grothendieck, "The cohomology theory of abstract algebraic varieties" , ''Proc. Internat. Math. Congress Edinburgh, 1958'' , Cambridge Univ. Press (1960) pp. 103–118 {{MR|0130879}} {{ZBL|0119.36902}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J. Dieudonné, A. Grothendieck, "Elements de géometrie algébrique" ''Publ. Math. IHES'' , '''4''' (1960) {{MR|0217083}} {{MR|0163908}} {{ZBL|0203.23301}} {{ZBL|0136.15901}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Grothendieck, "The cohomology theory of abstract algebraic varieties" , ''Proc. Internat. Math. Congress Edinburgh, 1958'' , Cambridge Univ. Press (1960) pp. 103–118 {{MR|0130879}} {{ZBL|0119.36902}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J. Dieudonné, A. Grothendieck, "Elements de géometrie algébrique" ''Publ. Math. IHES'' , '''4''' (1960) {{MR|0217083}} {{MR|0163908}} {{ZBL|0203.23301}} {{ZBL|0136.15901}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105026.png" /> is a finite morphism if there exist a covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105028.png" /> by affine open subschemes such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105029.png" /> is affine for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105030.png" /> and such that the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105031.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105032.png" /> is finitely generated as a module over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105033.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105034.png" />. The morphism is entire if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105035.png" /> is entire over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105036.png" />, i.e. if every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105037.png" /> integral over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105038.png" />, which means that it is a root of a monic polynomial with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105039.png" />, or, equivalently, if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105040.png" /> the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105041.png" /> is a finitely-generated module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011050/a01105042.png" />.
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$  f : X \rightarrow S $
 +
is a finite morphism if there exist a covering $  ( S _  \alpha  ) $
 +
of $  S $
 +
by affine open subschemes such that $  f  ^ {-1} ( S _  \alpha  ) $
 +
is affine for all $  \alpha $
 +
and such that the ring $  B _  \alpha  $
 +
of $  f  ^ {-1} ( S _  \alpha  ) $
 +
is finitely generated as a module over the ring $  A _  \alpha  $
 +
of $  S _  \alpha  $.  
 +
The morphism is entire if $  B _  \alpha  $
 +
is entire over $  A _  \alpha  $,  
 +
i.e. if every $  x \in B _  \alpha  $
 +
integral over $  A _  \alpha  $,  
 +
which means that it is a root of a monic polynomial with coefficients in $  A _  \alpha  $,  
 +
or, equivalently, if for each $  x \in B _  \alpha  $
 +
the module $  A _  \alpha  [ x ] $
 +
is a finitely-generated module over $  A _  \alpha  $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>

Revision as of 16:09, 1 April 2020


A morphism of schemes $ f: X \rightarrow S $ such that the pre-image of any open affine subscheme in $ S $ is an affine scheme. The scheme $ X $ is called an affine $ S $- scheme.

Let $ S $ be a scheme, let $ A $ be a quasi-coherent sheaf of $ {\mathcal O} _ {S} $- algebras and let $ U _ {i} $ be open affine subschemes in $ S $ which form a covering of $ S $. Then the glueing of the affine schemes $ { \mathop{\rm Spec} } \Gamma (U _ {i} , A) $ determines an affine $ S $- scheme, denoted by $ { \mathop{\rm Spec} } A $. Conversely, any affine $ S $- scheme definable by an affine morphism $ f: X \rightarrow S $ is isomorphic (as a scheme over $ S $) to the scheme $ { \mathop{\rm Spec} } f _ {*} ( {\mathcal O} _ {X} ) $. The set of $ S $- morphisms of an $ S $- scheme $ f: Z \rightarrow S $ into the affine $ S $- scheme $ { \mathop{\rm Spec} } A $ is in bijective correspondence with the homomorphisms of the sheaves of $ {\mathcal O} _ {S} $- algebras $ A \rightarrow f _ {*} ( {\mathcal O} _ {Z} ) $.

Closed imbeddings of schemes or arbitrary morphisms of affine schemes are affine morphisms; other examples of affine morphisms are entire morphisms and finite morphisms. Thus the morphism of normalization of a scheme is an affine morphism. Under composition and base change the property of a morphism to be an affine morphism is preserved.

References

[1] A. Grothendieck, "The cohomology theory of abstract algebraic varieties" , Proc. Internat. Math. Congress Edinburgh, 1958 , Cambridge Univ. Press (1960) pp. 103–118 MR0130879 Zbl 0119.36902
[2] J. Dieudonné, A. Grothendieck, "Elements de géometrie algébrique" Publ. Math. IHES , 4 (1960) MR0217083 MR0163908 Zbl 0203.23301 Zbl 0136.15901

Comments

$ f : X \rightarrow S $ is a finite morphism if there exist a covering $ ( S _ \alpha ) $ of $ S $ by affine open subschemes such that $ f ^ {-1} ( S _ \alpha ) $ is affine for all $ \alpha $ and such that the ring $ B _ \alpha $ of $ f ^ {-1} ( S _ \alpha ) $ is finitely generated as a module over the ring $ A _ \alpha $ of $ S _ \alpha $. The morphism is entire if $ B _ \alpha $ is entire over $ A _ \alpha $, i.e. if every $ x \in B _ \alpha $ integral over $ A _ \alpha $, which means that it is a root of a monic polynomial with coefficients in $ A _ \alpha $, or, equivalently, if for each $ x \in B _ \alpha $ the module $ A _ \alpha [ x ] $ is a finitely-generated module over $ A _ \alpha $.

References

[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
How to Cite This Entry:
Affine morphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affine_morphism&oldid=23740
This article was adapted from an original article by V.I. DanilovI.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article