  A generic degenerate Lagrangian system of even and odd fields is examined in
algebraic 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. We show that, if a
certain necessary and sufficient condition holds, one can associate to a
degenerate Lagrangian system the exact Koszul-Tate complex with the boundary
operator whose nilpotency condition restarts all its Noether and higher-stage
Noether identities. This complex provides a sufficient analysis of the
degeneracy of a Lagrangian system for the purpose of its BV quantization.
