In algebra, the Yoneda product (named after Nobuo Yoneda) is the pairing between Ext groups of modules:
![{\displaystyle \operatorname {Ext} ^{n}(M,N)\otimes \operatorname {Ext} ^{m}(L,M)\to \operatorname {Ext} ^{n+m}(L,N)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4971072ed3ac24b6f5512b200ea18fd122a31b2a)
induced by
![{\displaystyle \operatorname {Hom} (N,M)\otimes \operatorname {Hom} (M,L)\to \operatorname {Hom} (N,L),\,f\otimes g\mapsto g\circ f.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c984f3dab79eae0c75725f2c84ea87e79d4e9f45)
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
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de47f3235b199f785f64371d32b0a157d809b91b)
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
![{\displaystyle X\xrightarrow {f} Y\to S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/897b1a2d90f4e0a283ea130e45bc146e40dc0cbf)
and an
-extension
of
by an
-module
, there is an obstruction class
![{\displaystyle \omega (f,j)\in {\text{Ext}}^{2}(\mathbf {L} _{X/Y},f^{*}J)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9a75ea5adbbbd78237cf6ba78482a78c78738c2)
which can be described as the yoneda product
![{\displaystyle \omega (f,j)=f^{*}(e(j))\cdot K(X/Y/S)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e7678d70d3010a29d502ffbb4be064cb5e4f264)
where
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c09b61e3e96ac8bea94c8f3fc2178bcff408fee)
and
corresponds to the cotangent complex.
See also
References
External links