  We discuss intrinsic aspects of Krupka's approach to finite-order variational
sequences. We give intrinsic isomorphisms of the quotient subsheaves of the
short finite-order variational sequence with sheaves of forms on jet spaces of
suitable order, obtaining a new finite-order (short exact) variational sequence
which is made by sheaves of polynomial differential operators. Moreover, we
present an intrinsic formulation for the Helmholtz condition of local
variationality using a technique introduced by Kolar that we have adapted to
our context. Finally, we provide the minimal order solution to the inverse
problem of the calculus of variations and a solution of the problem of the
variationally trivial Lagrangian.
