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Difference between revisions of "Quasi-coherent sheaf"

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A [[Sheaf|sheaf]] of modules locally defined by generators and relations. More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q0764101.png" /> be a topological space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q0764102.png" /> be a sheaf of rings on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q0764103.png" />; a sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q0764104.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q0764105.png" />-modules is called quasi-coherent if for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q0764106.png" /> there is an open neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q0764107.png" /> and an exact sequence of sheaves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q0764108.png" />-modules
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q0764109.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q07641010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q07641011.png" /> are certain sets, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q07641012.png" /> denotes the restriction of a sheaf to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q07641013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q07641014.png" /> is the direct sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q07641015.png" /> copies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q07641016.png" />. A quasi-coherent sheaf is similarly defined on a [[Topologized category|topologized category]] with a sheaf of rings.
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A [[Sheaf|sheaf]] of modules locally defined by generators and relations. More precisely, let  $  X $
 +
be a topological space and let  $  {\mathcal A} $
 +
be a sheaf of rings on  $  X $;
 +
a sheaf $  {\mathcal F} $
 +
of $  {\mathcal A} $-
 +
modules is called quasi-coherent if for any point  $  x \in X $
 +
there is an open neighbourhood  $  U $
 +
and an exact sequence of sheaves of $  ( {\mathcal A} \mid  _ {U} ) $-
 +
modules
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q07641017.png" /> is an affine scheme, then the association <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q07641018.png" /> gives rise to an equivalence of the category of quasi-coherent sheaves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q07641019.png" />-modules and the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076410/q07641020.png" />-modules. As a result of this, quasi-coherent sheaves find broad application in the theory of schemes (see also [[Coherent sheaf|Coherent sheaf]]; [[Scheme|Scheme]]).
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$$
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{\mathcal A} | _ {U}  ^ {(} I)  \rightarrow  {\mathcal A} | _ {U}  ^ {(} J) \rightarrow \
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{\mathcal F} \mid  _ {U}  \rightarrow  0 ,
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$$
  
 +
where  $  I $
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and  $  J $
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are certain sets,  $  \mid  _ {U} $
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denotes the restriction of a sheaf to  $  U $
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and  $  {\mathcal A}  ^ {(} I) $
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is the direct sum of  $  I $
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copies of  $  {\mathcal A} $.
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A quasi-coherent sheaf is similarly defined on a [[Topologized category|topologized category]] with a sheaf of rings.
  
 +
If  $  ( X , {\mathcal A} ) $
 +
is an affine scheme, then the association  $  {\mathcal F} \mapsto \Gamma ( X , {\mathcal F} ) $
 +
gives rise to an equivalence of the category of quasi-coherent sheaves of  $  {\mathcal A} $-
 +
modules and the category of  $  \Gamma ( X , {\mathcal A} ) $-
 +
modules. As a result of this, quasi-coherent sheaves find broad application in the theory of schemes (see also [[Coherent sheaf|Coherent sheaf]]; [[Scheme|Scheme]]).
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 111–115; 126 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 111–115; 126 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>

Latest revision as of 08:09, 6 June 2020


A sheaf of modules locally defined by generators and relations. More precisely, let $ X $ be a topological space and let $ {\mathcal A} $ be a sheaf of rings on $ X $; a sheaf $ {\mathcal F} $ of $ {\mathcal A} $- modules is called quasi-coherent if for any point $ x \in X $ there is an open neighbourhood $ U $ and an exact sequence of sheaves of $ ( {\mathcal A} \mid _ {U} ) $- modules

$$ {\mathcal A} | _ {U} ^ {(} I) \rightarrow {\mathcal A} | _ {U} ^ {(} J) \rightarrow \ {\mathcal F} \mid _ {U} \rightarrow 0 , $$

where $ I $ and $ J $ are certain sets, $ \mid _ {U} $ denotes the restriction of a sheaf to $ U $ and $ {\mathcal A} ^ {(} I) $ is the direct sum of $ I $ copies of $ {\mathcal A} $. A quasi-coherent sheaf is similarly defined on a topologized category with a sheaf of rings.

If $ ( X , {\mathcal A} ) $ is an affine scheme, then the association $ {\mathcal F} \mapsto \Gamma ( X , {\mathcal F} ) $ gives rise to an equivalence of the category of quasi-coherent sheaves of $ {\mathcal A} $- modules and the category of $ \Gamma ( X , {\mathcal A} ) $- modules. As a result of this, quasi-coherent sheaves find broad application in the theory of schemes (see also Coherent sheaf; Scheme).

Comments

References

[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 111–115; 126 MR0463157 Zbl 0367.14001
How to Cite This Entry:
Quasi-coherent sheaf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-coherent_sheaf&oldid=48377
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article