An operad describes a category of algebras and a (co)homology theory for
these algebras may be formulated using the homological algebra of operads. A
morphism of operads f:OβP describes a functor
allowing a P-algebra to be viewed as an O-algebra. We
show that the O-algebra (co)homology of a P-algebra may
be represented by a certain operadic comodule. Thus filtrations of this
comodule result in spectral sequences computing the (co)homology.
As a demonstration we study operads with a filtered distributive law; for the
associative operad we obtain a new proof of the Hodge decomposition of the
Hochschild cohomology of a commutative algebra. This generalises to many other
operads and as an illustration we compute the post-Lie cohomology of a Lie
algebra.Comment: 32 page