Yoneda product

Short description: Pairing in algebra between ext groups of modules

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

Specifically, for an element $\xi \in {\mathrm{Ext}}^n(M,N)$, thought of as an extension $$\xi :0\to N\to E_0\to \cdots \to E_{n-1}\to M\to 0,$$ and similarly $$\rho :0\to M\to F_0\to \cdots \to F_{m-1}\to L\to 0\in {\mathrm{Ext}}^m(L,M),$$ we form the Yoneda (cup) product $$\xi ⌣\rho :0\to N\to E_0\to \cdots \to E_{n-1}\to F_0\to \cdots \to F_{m-1}\to L\to 0\in {\mathrm{Ext}}^{m+n}(L,N).$$

Note that the middle map $E_{n-1}\to F_0$ factors through the given maps to $M$.

We extend this definition to include $m,n=0$ using the usual functoriality of the ${\mathrm{Ext}}^{*}(\cdot ,\cdot )$ 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 $i:X\hookrightarrow {\mathbb{\mathbb{P}}}_k^n$ of pure dimension $r$ over an algebraically closed field $k$, there is a pairing $${\text{Ext}}_{{{𝒪}}_X}^p({{𝒪}}_X,{ℱ})\times {\text{Ext}}_{{{𝒪}}_X}^{r-p}({ℱ},{\omega }_X^{\bullet })\to k$$ where ${\omega }_X$ is the dualizing complex ${\omega }_X={{ℰ}{𝓍}{𝓉}}_{{{𝒪}}_{\mathbb{\mathbb{P}}}}^{n-r}(i_{*}{ℱ},{\omega }_{\mathbb{\mathbb{P}}})$ and ${\omega }_{\mathbb{\mathbb{P}}}={{𝒪}}_{\mathbb{\mathbb{P}}}(-(n+1))$ given by the Yoneda pairing.

Deformation theory

The Yoneda product is useful for understanding the obstructions to a deformation of maps of ringed topoi. For example, given a composition of ringed topoi $$X\stackrel{f}{\to }Y\to S$$ and an $S$-extension $j:Y\to Y^{\prime }$ of $Y$ by an ${{𝒪}}_Y$-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 $$\begin{array}{rl}K(X/Y/S)& \in {\text{Ext}}^1({\mathbf{𝐋}}_{X/Y},{\mathbf{𝐋}}_{Y/S})\\ f^{*}(e(j))& \in {\text{Ext}}^1(f^{*}{\mathbf{𝐋}}_{Y/S},f^{*}J)\end{array}$$ and ${\mathbf{𝐋}}_{X/Y}$ corresponds to the cotangent complex.

See also

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

References

• May, J. Peter. "Notes on Tor and Ext". http://www.math.uchicago.edu/~may/MISC/TorExt.pdf. 

External links

• Universality of Ext functor using Yoneda extensions