  The Hochschild cohomology of a differential graded algebra, or a differential
graded category, admits a natural map to the graded center of its homology
category: the characteristic homomorphism. We interpret it as an edge
homomorphism in a spectral sequence. This gives a conceptual explanation of the
failure of the characteristic homomorphism to be injective or surjective, in
general. To illustrate this, we discuss modules over the dual numbers, coherent
sheaves over algebraic curves, as well as examples related to free loop spaces
and string topology.
