Yoneda product

In algebra, the Yoneda product (named after Nobuo Yoneda) is the pairing between Ext groups of modules: $${\displaystyle \operatorname {Ext} ^{n}(M,N)\otimes \operatorname {Ext} ^{m}(L,M)\to \operatorname {Ext} ^{n+m}(L,N)}$$ induced by $${\displaystyle \operatorname {Hom} (N,M)\otimes \operatorname {Hom} (M,L)\to \operatorname {Hom} (N,L),\,f\otimes g\mapsto g\circ f.}$$

Specifically, for an element ${\displaystyle \xi \in \operatorname {Ext} ^{n}(M,N)}$, thought of as an extension $${\displaystyle \xi :0\rightarrow N\rightarrow E_{0}\rightarrow \cdots \rightarrow E_{n-1}\rightarrow M\rightarrow 0,}$$ and similarly $${\displaystyle \rho :0\rightarrow M\rightarrow F_{0}\rightarrow \cdots \rightarrow F_{m-1}\rightarrow L\rightarrow 0\in \operatorname {Ext} ^{m}(L,M),}$$ we form the Yoneda (cup) product $${\displaystyle \xi \smile \rho :0\rightarrow N\rightarrow E_{0}\rightarrow \cdots \rightarrow E_{n-1}\rightarrow F_{0}\rightarrow \cdots \rightarrow F_{m-1}\rightarrow L\rightarrow 0\in \operatorname {Ext} ^{m+n}(L,N).}$$

Note that the middle map ${\displaystyle E_{n-1}\rightarrow F_{0}}$ factors through the given maps to ${\displaystyle M}$.

We extend this definition to include ${\displaystyle m,n=0}$ using the usual functoriality of the ${\displaystyle \operatorname {Ext} ^{*}(\cdot ,\cdot )}$ groups.

Applications

Ext Algebras

Given a commutative ring ${\displaystyle R}$ and a module ${\displaystyle M}$, the Yoneda product defines a product structure on the groups ${\displaystyle {\text{Ext}}^{\bullet }(M,M)}$, where ${\displaystyle {\text{Ext}}^{0}(M,M)={\text{Hom}}_{R}(M,M)}$ is generally a non-commutative ring. This can be generalized to the case of sheaves of modules over a ringed space, or ringed topos.

Grothendieck duality

In Grothendieck's duality theory of coherent sheaves on a projective scheme ${\displaystyle i:X\hookrightarrow \mathbb {P} _{k}^{n}}$ of pure dimension ${\displaystyle r}$ over an algebraically closed field ${\displaystyle k}$, there is a pairing $${\displaystyle {\text{Ext}}_{{\mathcal {O}}_{X}}^{p}({\mathcal {O}}_{X},{\mathcal {F}})\times {\text{Ext}}_{{\mathcal {O}}_{X}}^{r-p}({\mathcal {F}},\omega _{X}^{\bullet })\to k}$$ where ${\displaystyle \omega _{X}}$ is the dualizing complex ${\displaystyle \omega _{X}={\mathcal {Ext}}_{{\mathcal {O}}_{\mathbb {P} }}^{n-r}(i_{*}{\mathcal {F}},\omega _{\mathbb {P} })}$ and ${\displaystyle \omega _{\mathbb {P} }={\mathcal {O}}_{\mathbb {P} }(-(n+1))}$ 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 $${\displaystyle X\xrightarrow {f} Y\to S}$$ and an ${\displaystyle S}$-extension ${\displaystyle j:Y\to Y'}$ of ${\displaystyle Y}$ by an ${\displaystyle {\mathcal {O}}_{Y}}$-module ${\displaystyle J}$, there is an obstruction class $${\displaystyle \omega (f,j)\in {\text{Ext}}^{2}(\mathbf {L} _{X/Y},f^{*}J)}$$ which can be described as the yoneda product $${\displaystyle \omega (f,j)=f^{*}(e(j))\cdot K(X/Y/S)}$$ where $${\displaystyle {\begin{aligned}K(X/Y/S)&\in {\text{Ext}}^{1}(\mathbf {L} _{X/Y},\mathbf {L} _{Y/S})\\f^{*}(e(j))&\in {\text{Ext}}^{1}(f^{*}\mathbf {L} _{Y/S},f^{*}J)\end{aligned}}}$$ and ${\displaystyle \mathbf {L} _{X/Y}}$ corresponds to the cotangent complex.

See also

• Ext functor
• Derived category
• Deformation theory
• Kodaira–Spencer map