Yoneda product

In algebra, the Yoneda product (named after Nobuo Yoneda) is the pairing between Ext groups of modules: [math]\displaystyle{ \operatorname{Ext}^n(M, N) \otimes \operatorname{Ext}^m(L, M) \to \operatorname{Ext}^{n+m}(L, N) }[/math] induced by [math]\displaystyle{ \operatorname{Hom}(N, M) \otimes \operatorname{Hom}(M, L) \to \operatorname{Hom}(N, L),\, f \otimes g \mapsto g \circ f. }[/math]

Specifically, for an element [math]\displaystyle{ \xi \in \operatorname{Ext}^n(M, N) }[/math], thought of as an extension [math]\displaystyle{ \xi : 0 \rightarrow N \rightarrow E_0 \rightarrow \cdots \rightarrow E_{n-1} \rightarrow M \rightarrow 0 , }[/math] and similarly [math]\displaystyle{ \rho : 0 \rightarrow M \rightarrow F_0\rightarrow \cdots \rightarrow F_{m-1} \rightarrow L \rightarrow 0 \in \operatorname{Ext}^m(L, M), }[/math] we form the Yoneda (cup) product [math]\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). }[/math]

Note that the middle map [math]\displaystyle{ E_{n-1} \rightarrow F_0 }[/math] factors through the given maps to [math]\displaystyle{ M }[/math].

We extend this definition to include [math]\displaystyle{ m, n = 0 }[/math] using the usual functoriality of the [math]\displaystyle{ \operatorname{Ext}^*(\cdot,\cdot) }[/math] groups.

Applications

Ext Algebras

Given a commutative ring [math]\displaystyle{ R }[/math] and a module [math]\displaystyle{ M }[/math], the Yoneda product defines a product structure on the groups [math]\displaystyle{ \text{Ext}^\bullet(M,M) }[/math], where [math]\displaystyle{ \text{Ext}^0(M,M) = \text{Hom}_R(M,M) }[/math] 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 [math]\displaystyle{ i:X \hookrightarrow \mathbb{P}^n_k }[/math] of pure dimension [math]\displaystyle{ r }[/math] over an algebraically closed field [math]\displaystyle{ k }[/math], there is a pairing [math]\displaystyle{ \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 }[/math] where [math]\displaystyle{ \omega_X }[/math] is the dualizing complex [math]\displaystyle{ \omega_X = \mathcal{Ext}_{\mathcal{O}_\mathbb{P}}^{n-r}(i_*\mathcal{F},\omega_{\mathbb{P}}) }[/math] and [math]\displaystyle{ \omega_{\mathbb{P}} = \mathcal{O}_\mathbb{P}(-(n+1)) }[/math] 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 [math]\displaystyle{ X \xrightarrow{f} Y \to S }[/math] and an [math]\displaystyle{ S }[/math]-extension [math]\displaystyle{ j:Y \to Y' }[/math] of [math]\displaystyle{ Y }[/math] by an [math]\displaystyle{ \mathcal{O}_Y }[/math]-module [math]\displaystyle{ J }[/math], there is an obstruction class [math]\displaystyle{ \omega(f,j) \in \text{Ext}^2(\mathbf{L}_{X/Y}, f^*J) }[/math] which can be described as the yoneda product [math]\displaystyle{ \omega(f,j) = f^*(e(j))\cdot K(X/Y/S) }[/math] where [math]\displaystyle{ \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} }[/math] and [math]\displaystyle{ \mathbf{L}_{X/Y} }[/math] corresponds to the cotangent complex.

See also

• Ext functor
• Derived category
• Deformation theory
• Kodaira–Spencer map

References

• May, J. Peter. "Notes on Tor and Ext". http://www.math.uchicago.edu/~may/MISC/TorExt.pdf. 

External links

• Universality of Ext functor using Yoneda extensions