Yoneda product

In algebra, the Yoneda product is the pairing between Ext groups of modules:
induced by
Specifically, for an element, thought of as an extension
and similarly
we form the Yoneda product
Note that the middle map factors through the given maps to.
We extend this definition to include using the usual functoriality of the groups.

Applications

Ext Algebras

Given a commutative ring and a module, the Yoneda product defines a product structure on the groups, where is generally a non-commutative ring. This can be generalized to the case of sheaves of modules over a ringed space, or ringed topos.

Grothendieck duality

In Grothendieck's duality theory of coherent sheaves on a projective scheme of pure dimension over an algebraically closed field, there is a pairing where is the dualizing complex and given by the Yoneda pairing.

Deformation theory

The Yoneda product is useful for understanding the obstructions to a deformation of maps of ringed topoi. For example, given a composition of ringed topoi and an -extension of by an -module, there is an obstruction class which can be described as the yoneda product
where
and corresponds to the cotangent complex.