It is shown that the new formula for the field theory Poisson brackets arise
naturally in the extension of the formal variational calculus incorporating
divergences. The linear spaces of local functionals, evolutionary vector
fields, functional forms, multi-vectors and differential operators become
graded with respect to divergences. The bilinear operations, such as the action
of vector fields on functionals, the commutator of vector fields, the interior
product of forms and vectors and the Schouten-Nijenhuis bracket are compatible
with the grading. A definition of the adjoint graded operator is proposed and
antisymmetric operators are constructed with the help of boundary terms. The
fulfilment of the Jacobi identity for the new Poisson brackets is shown to be
equivalent to vanishing of the Schouten-Nijenhuis bracket of the Poisson
bivector with itself. It is demonstrated, as an example, that the second
structure of the Korteweg-de Vries equation is not Hamiltonian with respect to
the new brackets until special boundary conditions are prescribed.