Yoneda product explained

In algebra, the Yoneda product (named after Nobuo Yoneda) is the pairing between Ext groups of modules: $\operatorname^n(M, N) \otimes \operatorname^m(L, M) \to \operatorname^(L, N)$ induced by $\operatorname(N, M) \otimes \operatorname(M, L) \to \operatorname(N, L),\, f \otimes g \mapsto g \circ f.$

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_ \rightarrow M \rightarrow 0,

and similarly

\rho : 0 \rightarrow M \rightarrow F_0\rightarrow \cdots \rightarrow F_ \rightarrow L \rightarrow 0 \in \operatorname^m(L, M),

we form the Yoneda (cup) product

\xi \smile \rho : 0 \rightarrow N \rightarrow E_0 \rightarrow \cdots \rightarrow E_ \rightarrow F_0 \rightarrow \cdots \rightarrow F_ \rightarrow L \rightarrow 0 \in \operatorname^(L, N).

Note that the middle map

E_n-1 → F₀

factors through the given maps to

We extend this definition to include

m,n=0

using the usual functoriality of the

\operatorname{Ext}^{*( ⋅ , ⋅ )}

groups.

Applications

Ext Algebras

Given a commutative ring

and a module

, the Yoneda product defines a product structure on the groups

Ext^\bullet(M,M)

, where

Ext^0(M,M)=Hom_R(M,M)

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

i:X\hookrightarrow

	n
P
	k

of pure dimension

over an algebraically closed field

, there is a pairing

\text^p_(\mathcal_X, \mathcal) \times\text^_(\mathcal,\omega_X^\bullet) \to k

where

\omega_X

is the dualizing complex

\omega_X=l{Ext}_l{O_P

}^(i_*\mathcal,\omega_) and

\omega_P=l{O}_P(-(n+1))

given by the Yoneda pairing.^[1]

Deformation theory

The Yoneda product is useful for understanding the obstructions to a deformation of maps of ringed topoi.^[2] For example, given a composition of ringed topoi $X \xrightarrow Y \to S$ and an

-extension

j:Y\toY'

by an

l{O}_Y

-module

, there is an obstruction class

\omega(f,j) \in \text^2(\mathbf_, f^*J)

which can be described as the yoneda product

\omega(f,j) = f^*(e(j))\cdot K(X/Y/S)

where

\beginK(X/Y/S) &\in \text^1(\mathbf_, \mathbf_) \\f^*(e(j)) &\in \text^1(f^*\mathbf_, f^*J) \end

and

L_X/Y

corresponds to the cotangent complex.

References

Book: Altman. Grothendieck Duality . Kleiman . Lecture Notes in Mathematics. 1970. 146 . 978-3-540-04935-7 . 5 . 10.1007/BFb0060932.
Web site: Complexe cotangent; application a la theorie des deformations . Illusie . Luc . 163.

Web site: J. Peter. May . J. Peter May . Notes on Tor and Ext.

External links

Universality of Ext functor using Yoneda extensions