Yoneda product

In algebra, the Yoneda product 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}(M, N) \otimes \operatorname{Hom}(L, M) \to \operatorname{Hom}(L, N),\, f \otimes g \mapsto f \circ g.

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_1 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.


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