Symmetric Monoidal Categories for Operads - Index of

210 13 The (Co)homology of Algebras over Operads
DerP(QA,E)=Hom UP(QA)Mod(Ω 1 P (QA),E)=Hom UP(A)Mod(T 1 P (QA),E)
and E ⊗ UP(QA) Ω 1 P (QA) =E ⊗ UP(A) T 1 P (QA),
where T 1 P (QA) =UP(A) ⊗ UP(QA) Ω 1 P (QA).
The idea is to apply the standard hypercohomology spectral sequence
E st
2 =Ext s UP(A)(Ht(T 1 P (QA)),E),
respectively the standard hyperhomology spectral sequence
E 2 st =TorUP(A) s (E,Ht(T 1 P (QA))),
associated to this dg-hom, respectively tensor product, over the enveloping
algebra UP(A).
The object T 1 P (QA) forms a cofibrant left UP(A)-module by lemma 13.1.10.
As a consequence, the hypercohomology spectral sequence abuts to the cohomology
H ∗ (Hom UP(A)Mod(T 1 P (QA),E)), the hyperhomology spectral sequence
abuts to the homology H∗(E⊗ UP(A)T 1 P (QA)). By definition of T 1 P (QA),
we also have Ht(T 1 P (QA)) = H P t (A, UP(A)). Thus we obtain finally:
13.2.2 Proposition (compare with [1, Propositions 5.3.2-5.3.3]). We have
universal coefficient spectral sequences
E 2 s,t =Tor UP(A)
s (E,H P t (A, UP(A))) ⇒ H P s+t(A, E)
and E s,t
2 =Exts UP(A) (HP t (A, UP(A)),E) ⇒ H s−t
P (A, E). ⊓⊔
As a byproduct:
13.2.3 Proposition (compare with [1, Remark 5.3.4] and with [64]). If the
homology H P ∗ (A, UP(A)) vanishes in degree ∗ > 0, then we have
H P ∗ (A, E) =TorUP(A) ∗ (E,Ω 1 P (A)) and H∗ P (A, E) =Ext∗UP(A) (Ω1 P (A),E).
Proof. The condition implies that the universal coefficients spectral sequences
degenerate and the identities of the proposition follow. ⊓⊔
Conversely, if the homology HP ∗ (A, E) is given by a Tor-functor over the
enveloping algebra UP(A), then we have necessarily HP ∗(A, UP(A)) = 0, for
∗ > 0, since Tor UP(A)
∗ (UP(A),F)vanishesindegree∗ > 0 for every left UP(A)module
F .
13.2.4 Classical Examples. The identities of proposition 13.2.3 are known
to hold for the Hochschild (co)homology (case of the associative operad P = A)
and for the Chevalley-Eilenberg (co)homology (case of the Lie operad P = L).
The reader is referred to his favorite textbook. In §17.3, we prove that the
assertion about Hochschild (co)homology is implied by the deeper property,
checked in §10.3, that the module Ω1 A forms a free right A-module, and similarly
as regards the Chevalley (co)homology.