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.

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 $$i:X \hookrightarrow \mathbb{P}^n_k$$ of pure dimension $$r$$ over an algebraically closed field $$k$$, there is a pairing $$\text{Ext}^p_{\mathcal{O}_X}(\mathcal{O}_X, \mathcal{F}) \times \text{Ext}^{r-p}_{\mathcal{O}_X}(\mathcal{F},\omega_X^\bullet) \to k$$ 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.

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 $$X \xrightarrow{f} Y \to S$$ 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 $$\begin{align} K(X/Y/S) &\in \text{Ext}^1(\mathbf{L}_{X/Y}, \mathbf{L}_{Y/S}) \\ f^*(e(j)) &\in \text{Ext}^1(f^*\mathbf{L}_{Y/S}, f^*J) \end{align}$$ and $$\mathbf{L}_{X/Y}$$ corresponds to the cotangent complex.