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.

See also

References

  1. Altman; Kleiman (1970). Grothendieck Duality. Lecture Notes in Mathematics. 146. p. 5. doi:10.1007/BFb0060932. ISBN 978-3-540-04935-7.
  2. Illusie, Luc. "Complexe cotangent; application a la theorie des deformations" (PDF). p. 163.
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