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In [[sheaf theory]] (an area of mathematics), a '''sheaf extension''' is a way of describing a [[sheaf (mathematics)|sheaf]] in terms of a [[subsheaf]] and a [[quotient sheaf]], analogous to a how a [[group extension]] describes a [[group (mathematics)|group]] in terms of a [[subgroup]], and a [[quotient group]].
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==Definition==
Let ''X'' be a scheme, and let ''F'', ''H'' be sheaves (of modules) on ''X''. An '''extension''' of ''H'' by ''F'' is a [[short exact sequence]] of sheaves
 
: <math>0 \rightarrow F \rightarrow G \rightarrow H \rightarrow 0.</math>
 
Note that an extension is not determined by the sheaf ''G'' alone: The morphisms are also important.
 
A simple example of an extension of ''H'' by ''F'' is the sequence
 
: <math>0 \rightarrow F \rightarrow F \bigoplus H \rightarrow H \rightarrow 0 </math>
 
where the second arrow is the inclusion and the fourth arrow is the projection onto the second summand. This extension is sometimes called '''trivial'''.
 
==Properties==
As with group extensions, if we fix ''F'' and ''H'', then all (equivalence classes of) possible extensions of ''H'' by ''F'' form an [[abelian group]]. This group is isomorphic to the [[Ext functor|Ext group]] <math>\mathrm{Ext}^1(H,F)</math>, where the identity element in <math>\mathrm{Ext}^1(H,F)</math> corresponds to the trivial extension.  
 
In the case where ''H'' is the [[structure sheaf]] <math> \mathcal{O}_X </math>, we have <math>H^1(X, F) \cong \mathrm{Ext}^1(\mathcal{O}_X,F) </math>, so the group of extensions of <math>\mathcal{O}_X</math> by ''F'' is also isomorphic to the first [[sheaf cohomology]] group with coefficients in ''F''.  
 
==Generalization==
 
The definition of an extension and the correspondence between extensions and Ext groups can be generalized to [[abelian category|abelian categories]], of which sheaves of modules are special instances.
 
== See also ==
*[[Ext functor]]
 
==References==
* {{Citation | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=[[Algebraic Geometry (book)|Algebraic Geometry]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90244-9 | oclc=13348052 | id={{MathSciNet | id = 0463157}} | year=1977}}, in the algebraic-geometric setting, i.e. referring to the [[Zariski topology]]
* {{Weibel IHA}}
 
 
[[Category:Sheaf theory]]
[[Category:Algebraic geometry]]
[[Category:Homological algebra]]

Revision as of 13:29, 5 March 2014

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