  The divergence map, an important ingredient in the algebraic description of
the Turaev cobracket on a connected oriented compact surface with boundary, is
reformulated in the context of non-commutative geometry using a flat connection
on the space of 1-forms on a formally smooth associative algebra. We then
extend this construction to the case of associative algebras with any finite
cohomological dimension, which allows us to give a similar algebraic
description of the Turaev cobracket on a closed surface. We also look into a
relation between the Satoh trace and the divergence map on a free Lie algebra
via geometry over Lie operad.
