  For classical relativistic field theory in Minkowski space-time, the addition
of a superpotential term to a conserved current density is trivial in the sense
that it does not modify the local conservation law nor change the conserved
charge, though it may allow us to obtain a current density with some improved
properties. The addition of a total derivative term to a Lagrangian density is
also trivial in the sense that it does not modify the equations of motion of
the theory. These facts suggest that both operations are related and possibly
equivalent to each other for any global symmetry of an action functional. We
address this question following the study of two quite different (and well
known) instances: the Callan-Coleman-Jackiw improvement of the canonical
energy-momentum tensor for scalar and vector fields (providing an on-shell
traceless energy-momentum tensor) and the construction of a current density
satisfying a zero curvature condition for two-dimensional sigma models on
deformed spaces (notably the squashed three-sphere and warped AdS spaces).
These instances correspond to fairly different implementations of the general
results. An appendix addresses the precise relationship between the approaches
to local conservation laws based on active and passive symmetry
transformations, respectively.
