  A generic degenerate Lagrangian system of even and odd variables on an
arbitrary smooth manifold is examined in terms of the Grassmann-graded
variational bicomplex. Its Euler-Lagrange operator obeys Noether identities
which need not be independent, but satisfy first-stage Noether identities, and
so on. However, non-trivial higher-stage Noether identities are ill defined,
unless a certain homology condition holds. We show that, under this condition,
there exists the exact Koszul-Tate chain complex whose boundary operator
produces all non-trivial Noether and higher-stage Noether identities of an
original Lagrangian system. Noether's inverse second theorem that we prove
associates to this complex a cochain sequence whose ascent operator provides
all gauge and higher-stage gauge supersymmetries of an original Lagrangian.
