7.1 Modules and Derivations
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The T -module sM is called the suspension of M. The desuspension s −1 M is
defined analogously. We define s k M for an arbitrary integer k ∈ Z by iteration.
One has the expected:
Lemma 7.3 The modular module sM with the above operations is a T -module.
Proof. A straightforward verification. Let us, for instance, check axiom (vii) of
Definition 7.1. We have, by the definition of the structure operations of sM and
by (7.2) for the suspension M,
a
L
• b (• uv ⊗ 1) = −↓ a
L
• b (1 ⊗ ↑)(• uv ⊗ 1) = ↓ a
L
• b (• uv ⊗ 1)(1 ⊗ ↑)
= − ↓ uv a
L
• b (1 ⊗ ↑).
Similarly,
− uv a
L
• b = −↓ uv ↑↓ a
L
• b (1 ⊗ ↑) = −↓ uv a
L
• b (1 ⊗ ↑),
so a
L
• b (• uv ⊗ 1) = − uv a
L
• b as required.
Remark 7.2 The minus sign in (7.7) resp. (7.8) defining a
L
• b resp. ab is crucial
for establishing axioms (iii) and (vii) of Definition 7.1 for sM. These axioms are
not homogeneous with respect to the number of operations, so they are not invariant
under the change
a
L
• b → − a
L
• b and/or uv → − uv .
The other axioms are not sign-sensitive. We encourage the reader to analyze the
proof of Lemma 7.3 and verify that without the minus sign in (7.8), the equality
a
L
• b (• uv ⊗ 1) = − uv a
L
• b would not hold.
Lemmas 7.2 and 7.3 imply that T ⊕ s k M is an odd modular operad for each
integer k. The following lemma characterizes derivations via homomorphisms.
Lemma 7.4 Let θ : T → M a degree k morphism of modular modules. Associate
to it a degree 0 morphism θ : T → s −k M of modular modules by
θ := ↓ k θ
and another degree 0 morphism Θ : T → T ⊕ s −k M of modular modules given
by
Θ := (1 T , θ).