# 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 $\operatorname {Ext} ^{*}(\_,\_)$ groups.