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.
Applications[edit]
Ext Algebras[edit]
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]
Deformation theory[edit]
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.
See also[edit]
References[edit]
External links[edit]