  We consider systems of local variational problems defining non vanishing
cohomolgy classes. In particular, we prove that the conserved current
associated with a generalized symmetry, assumed to be also a symmetry of the
variation of the corresponding local inverse problem, is variationally
equivalent to the variation of the strong Noether current for the corresponding
local system of Lagrangians. This current is conserved and a sufficient
condition will be identified in order such a current be global.
