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.