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.

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

$0 \rightarrow F \rightarrow G \rightarrow H \rightarrow 0.$

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

$0 \rightarrow F \rightarrow F \bigoplus H \rightarrow H \rightarrow 0$

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 group $\mathrm{Ext}^1(H,F)$, where the identity element in $\mathrm{Ext}^1(H,F)$ corresponds to the trivial extension.

In the case where H is the structure sheaf $\mathcal{O}_X$, we have $H^1(X, F) \cong \mathrm{Ext}^1(\mathcal{O}_X,F)$, so the group of extensions of $\mathcal{O}_X$ 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 categories, of which sheaves of modules are special instances.

See also

• Ext functor

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