Yoneda product

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

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

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

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

Applications

Ext Algebras

Given a commutative ring $R$ and a module $M$ , the Yoneda product defines a product structure on the groups ${\text{Ext}}^{\bullet }(M,M)$ , where ${\text{Ext}}^{0}(M,M)={\text{Hom}}_{R}(M,M)$ 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 $i:X\hookrightarrow \mathbb {P} _{k}^{n}$ of pure dimension $r$ over an algebraically closed field $k$ , there is a pairing ${\text{Ext}}_{{\mathcal {O}}_{X}}^{p}({\mathcal {O}}_{X},{\mathcal {F}})\times {\text{Ext}}_{{\mathcal {O}}_{X}}^{r-p}({\mathcal {F}},\omega _{X}^{\bullet })\to k$ where $\omega _{X}$ is the dualizing complex $\omega _{X}={\mathcal {Ext}}_{{\mathcal {O}}_{\mathbb {P} }}^{n-r}(i_{*}{\mathcal {F}},\omega _{\mathbb {P} })$ and $\omega _{\mathbb {P} }={\mathcal {O}}_{\mathbb {P} }(-(n+1))$ 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 $X\xrightarrow {f} Y\to S$ and an $S$ -extension $j:Y\to Y'$ of $Y$ by an ${\mathcal {O}}_{Y}$ -module $J$ , there is an obstruction class $\omega (f,j)\in {\text{Ext}}^{2}(\mathbf {L} _{X/Y},f^{*}J)$ which can be described as the yoneda product $\omega (f,j)=f^{*}(e(j))\cdot K(X/Y/S)$ where ${\begin{aligned}K(X/Y/S)&\in {\text{Ext}}^{1}(\mathbf {L} _{X/Y},\mathbf {L} _{Y/S})\\f^{*}(e(j))&\in {\text{Ext}}^{1}(f^{*}\mathbf {L} _{Y/S},f^{*}J)\end{aligned}}$ and $\mathbf {L} _{X/Y}$ corresponds to the cotangent complex.