In algebra, the Yoneda product (named after Nobuo Yoneda ) is the pairing between Ext groups of modules :
Ext
n
(
M
,
N
)
⊗
Ext
m
(
L
,
M
)
→
Ext
n
+
m
(
L
,
N
)
{\displaystyle \operatorname {Ext} ^{n}(M,N)\otimes \operatorname {Ext} ^{m}(L,M)\to \operatorname {Ext} ^{n+m}(L,N)}
induced by
Hom
(
N
,
M
)
⊗
Hom
(
M
,
L
)
→
Hom
(
N
,
L
)
,
f
⊗
g
↦
g
∘
f
.
{\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
ξ
∈
Ext
n
(
M
,
N
)
{\displaystyle \xi \in \operatorname {Ext} ^{n}(M,N)}
ξ
:
0
→
N
→
E
0
→
⋯
→
E
n
−
1
→
M
→
0
,
{\displaystyle \xi :0\rightarrow N\rightarrow E_{0}\rightarrow \cdots \rightarrow E_{n-1}\rightarrow M\rightarrow 0,}
and similarly
ρ
:
0
→
M
→
F
0
→
⋯
→
F
m
−
1
→
L
→
0
∈
Ext
m
(
L
,
M
)
,
{\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
ξ
⌣
ρ
:
0
→
N
→
E
0
→
⋯
→
E
n
−
1
→
F
0
→
⋯
→
F
m
−
1
→
L
→
0
∈
Ext
m
+
n
(
L
,
N
)
.
{\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
E
n
−
1
→
F
0
{\displaystyle E_{n-1}\rightarrow F_{0}}
M
{\displaystyle M}
We extend this definition to include
m
,
n
=
0
{\displaystyle m,n=0}
functoriality of the
Ext
∗
(
⋅
,
⋅
)
{\displaystyle \operatorname {Ext} ^{*}(\cdot ,\cdot )}
Applications Ext Algebras Given a commutative ring
R
{\displaystyle R}
M
{\displaystyle M}
Ext
∙
(
M
,
M
)
{\displaystyle {\text{Ext}}^{\bullet }(M,M)}
Ext
0
(
M
,
M
)
=
Hom
R
(
M
,
M
)
{\displaystyle {\text{Ext}}^{0}(M,M)={\text{Hom}}_{R}(M,M)}
ringed space , or ringed topos.
Grothendieck duality In Grothendieck's duality theory of coherent sheaves on a projective scheme
i
:
X
↪
P
k
n
{\displaystyle i:X\hookrightarrow \mathbb {P} _{k}^{n}}
r
{\displaystyle r}
k
{\displaystyle k}
Ext
O
X
p
(
O
X
,
F
)
×
Ext
O
X
r
−
p
(
F
,
ω
X
∙
)
→
k
{\displaystyle {\text{Ext}}_{{\mathcal {O}}_{X}}^{p}({\mathcal {O}}_{X},{\mathcal {F}})\times {\text{Ext}}_{{\mathcal {O}}_{X}}^{r-p}({\mathcal {F}},\omega _{X}^{\bullet })\to k}
where
ω
X
{\displaystyle \omega _{X}}
is the dualizing complex
ω
X
=
E
x
t
O
P
n
−
r
(
i
∗
F
,
ω
P
)
{\displaystyle \omega _{X}={\mathcal {Ext}}_{{\mathcal {O}}_{\mathbb {P} }}^{n-r}(i_{*}{\mathcal {F}},\omega _{\mathbb {P} })}
and
ω
P
=
O
P
(
−
(
n
+
1
)
)
{\displaystyle \omega _{\mathbb {P} }={\mathcal {O}}_{\mathbb {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
→
f
Y
→
S
{\displaystyle X\xrightarrow {f} Y\to S}
and an
S
{\displaystyle S}
-extension
j
:
Y
→
Y
′
{\displaystyle j:Y\to Y'}
of
Y
{\displaystyle Y}
by an
O
Y
{\displaystyle {\mathcal {O}}_{Y}}
-module
J
{\displaystyle J}
, there is an obstruction class
ω
(
f
,
j
)
∈
Ext
2
(
L
X
/
Y
,
f
∗
J
)
{\displaystyle \omega (f,j)\in {\text{Ext}}^{2}(\mathbf {L} _{X/Y},f^{*}J)}
which can be described as the yoneda product
ω
(
f
,
j
)
=
f
∗
(
e
(
j
)
)
⋅
K
(
X
/
Y
/
S
)
{\displaystyle \omega (f,j)=f^{*}(e(j))\cdot K(X/Y/S)}
where
K
(
X
/
Y
/
S
)
∈
Ext
1
(
L
X
/
Y
,
L
Y
/
S
)
f
∗
(
e
(
j
)
)
∈
Ext
1
(
f
∗
L
Y
/
S
,
f
∗
J
)
{\displaystyle {\begin{aligned}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{aligned}}}
and
L
X
/
Y
{\displaystyle \mathbf {L} _{X/Y}}
corresponds to the
cotangent complex .
See also References
^ Altman; Kleiman (1970). Grothendieck Duality. Lecture Notes in Mathematics. Vol. 146. p. 5. doi :10.1007/BFb0060932. ISBN 978-3-540-04935-7 ^ Illusie, Luc. "Complexe cotangent; application a la theorie des deformations" (PDF) . p. 163. External links Universality of Ext functor using Yoneda extensions 
