  For $\mathcal{O}$ a reduced operad, a generalized divergence from the
derivations of a free $\mathcal{O}$-algebra to a suitable trace space is
constructed. In the case of the Lie operad, this corresponds to Satoh's trace
map and, for the associative operad, to the double divergence of Alekseev,
Kawazumi, Kuno and Naef. The generalized divergence is shown to be a
$1$-cocycle for the usual Lie algebra structure on derivations. These results
place the previous constructions into a unified framework; moreover, they are
natural with respect to the operad.
  An important new ingredient is the use of naturality with respect to the
category of finite-rank free modules and split monomorphisms over a commutative
ring $R$. This allows the notion of torsion for such functors to be exploited.
  Supposing that the ring $R$ is a PID and that the operad $\mathcal{O}$ is
binary, the main result relates the kernel of the generalized divergence to the
sub Lie algebra of the Lie algebra of derivations that is generated by the
elements of degree one with respect to the grading induced by arity.

Algebraic Topology