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 , 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
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.[1]
Deformation theory
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.
References
- Altman; Kleiman (1970). Grothendieck Duality. Lecture Notes in Mathematics. 146. p. 5. doi:10.1007/BFb0060932. ISBN 978-3-540-04935-7.
- Illusie, Luc. "Complexe cotangent; application a la theorie des deformations" (PDF). p. 163.
- May, J. Peter. "Notes on Tor and Ext" (PDF).