sheaf extension
In sheaf theory (an area of mathematics), a sheaf extension is a way of describing a sheaf in terms of a subsheaf and a quotient sheaf, analogous to a how a group extension describes a group in terms of a subgroup, and a quotient group.
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[edit] 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
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
where the second arrow is the inclusion and the fourth arrow is the projection onto the second summand. This extension is sometimes called trivial.
[edit] 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 group 
, where the identity element in corresponds to the trivial extension.
In the case where H is the structure sheaf 
, we have 
, so the group of extensions of by F is also isomorphic to the first sheaf cohomology group with coefficients in F.
[edit] Generalization
The definition of an extension and the correspondence between extensions and Ext groups can be generalized to abelian categories, of which sheaves of modules are special instances.
[edit] See also
[edit] References
- Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, OCLC 13348052, MR0463157, in the algebraic-geometric setting, i.e. referring to the Zariski topology
- Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, ISBN 978-0-521-55987-4, OCLC 36131259, MR1269324
