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}(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.

References

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