Sheaf extension

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In Sheaf theory (a branch of the mathematics area of algebraic geometry), 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.

Contents 1 Definition 2 Properties 3 Generalization 4 See also 5 References

[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 Ext¹(H,F), where the identity element in Ext¹(H,F) 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

• Ext functor

[edit] References

• Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, MR0463157, ISBN 978-0-387-90244-9, OCLC 13348052 , 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, MR1269324, ISBN 978-0-521-55987-4, OCLC 36131259