In algebra, the Yoneda product (named after Nobuo Yoneda) is the pairing between Ext groups of modules:
Specifically, for an element
, thought of as an extension
we form the Yoneda (cup) product
ξ ⌣ ρ : 0 →
Note that the middle map
factors through the given maps to
We extend this definition to include
using the usual functoriality of the
Given a commutative ring
, the Yoneda product defines a product structure on the groups
{\displaystyle {\text{Ext}}^{0}(M,M)={\text{Hom}}_{R}(M,M)}
is generally a non-commutative ring.
This can be generalized to the case of sheaves of modules over a ringed space, or ringed topos.
In Grothendieck's duality theory of coherent sheaves on a projective scheme
of pure dimension
over an algebraically closed field
ω
{\displaystyle {\text{Ext}}_{{\mathcal {O}}_{X}}^{p}({\mathcal {O}}_{X},{\mathcal {F}})\times {\text{Ext}}_{{\mathcal {O}}_{X}}^{r-p}({\mathcal {F}},\omega _{X}^{\bullet })\to k}
ω
is the dualizing complex
ω
{\displaystyle \omega _{X}={\mathcal {Ext}}_{{\mathcal {O}}_{\mathbb {P} }}^{n-r}(i_{*}{\mathcal {F}},\omega _{\mathbb {P} })}
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
, there is an obstruction class
which can be described as the yoneda product
{\displaystyle {\begin{aligned}K(X/Y/S)&\in {\text{Ext}}^{1}(\mathbf {L} _{X/Y},\mathbf {L} _{Y/S})\\f^{*}(e(j))&\in {\text{Ext}}^{1}(f^{*}\mathbf {L} _{Y/S},f^{*}J)\end{aligned}}}
corresponds to the cotangent complex.