Yoneda product

In algebra, the Yoneda product (named after Nobuo Yoneda) is the pairing between Ext groups of modules:
induced by
Specifically, for an element formula_3, thought of as an extension 
and similarly 
we form the Yoneda (cup) product 
Note that the middle map formula_7 factors through the given maps to formula_8.
We extend this definition to include formula_9 using the usual functoriality of the formula_10 groups.
Applications.
Ext Algebras.
Given a commutative ring formula_11 and a module formula_8, the Yoneda product defines a product structure on the groups formula_13, where formula_14 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 formula_15 of pure dimension formula_16 over an algebraically closed field formula_17, there is a pairingformula_18where formula_19 is the dualizing complex formula_20 and formula_21 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 topoiformula_22and an formula_23-extension formula_24 of formula_25 by an formula_26-module formula_27, there is an obstruction classformula_28which can be described as the yoneda productformula_29whereformula_30and formula_31 corresponds to the cotangent complex.