182
7 Feynman Transform of a Modular Operad
is a morphism F( ˚
C ) → T of odd modular operads commuting with the differentials. Since F( ˚
C ) is free, such a morphism is determined by its restriction to the
module of generators, which is a morphism A : ˚
C → T of modular modules. By
the definition of the modular module ˚
C associated with the module comodule C , A
is precisely the family (7.31) satisfying (7.32).
It remains to express the condition that (7.34) commutes with the differentials,
i.e. that
d T α(S; g) = α(S; g)(∂ + d)
(7.35)
for each S ∈ Cor and g ∈ A, in terms of the family A. Since a composition of
a derivation with a morphism is again a derivation, it is enough by Theorem 7.1 to
verify (7.35) on the generators C . Let us apply α(S; g) to the first term in the righthand side of (7.28) defining ∂ on C (S; g). Using the fact that α extends A, we see
that
α(S; g)
g +s=g
• uv ι◦
uv
g =
g +s=g
• uv α(S {uv}; g
)ι◦
uv
g =
g +s=g
• uv A(S {uv}; g
)◦
uv
g
Applying α(S; g) to the second term in the right-hand side of (7.28) gives
α(S; g)
S 1 2 =S
g 1 +g 2 =g
a • b (ι ⊗ ι)
a
S 1 ; g 1
◦
b
S 2 ;g 2
=
S 1 2 =S
g 1 +g 2 =g
a • b
α(S 1 {a}; g 1 ) ⊗ α(S 2 {b}; g 2 )
(ι ⊗ ι)
a
S 1 ; g 1
◦
b
S 2 ;g 2
=
S 1 2 =S
g 1 +g 2 =g
a • b ◦
A(S 1 {a}; g 1 ) ⊗ A(S 2 {b}; g 2 )
◦
a
S 1 ; g 1
◦
b
S 2 ;g 2
.
Finally, α(S; g)d restricted to ˚
C clearly equals A(S; g)d C , so the right-hand side
of (7.33) equals α(S; g)(∂ + d) restricted to ˚
C . The fact that d T A(S; g) is
d T α(S; g) restricted to ˚
C (S; g) finishes the proof.
Remark 7.6 Theorem 8.2 below will use a skeletal version of Proposition 7.2. As
before, [n] := {1, . . . , n} for n ∈ N, and denote C (n; g) := C
[n]; g
and
T (n; g) := T
[n]; g
, g ∈ A. Each T (n; g) is a natural left Σ n -module, while
each C (n; g) is a natural right Σ n -module. We claim that the family (7.31) is
determined by the sequence
A sk =
A(n; g) : C (n; g) → T (n; g) | (S, g) ∈ N × A
(7.36)