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In algebra, the Yoneda product 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
(
M
,
N
)
⊗
Hom
(
L
,
M
)
→
Hom
(
L
,
N
)
,
f
⊗
g
↦
f
∘
g
.
{\displaystyle \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
ξ
∈
Ext
n
(
M
,
N
)
{\displaystyle \xi \in \operatorname {Ext} ^{n}(M,N)}
, thought of as an extension
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
1
{\displaystyle E_{n-1}\rightarrow F_{1}}
factors through the given maps to
M
{\displaystyle M}
.
We extend this definition to include
m
,
n
=
0
{\displaystyle m,n=0}
using the usual functoriality of the
Ext
∗
(
_
,
_
)
{\displaystyle \operatorname {Ext} ^{*}(\_,\_)}
groups.
References
External links