In algebra, the Yoneda product (named after Nobuo Yoneda) 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 (cup) 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.
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
[edit] 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.[1]
The Yoneda product is useful for understanding the obstructions to a deformation of maps of ringed topoi.[2] 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.