Noether identities of a differential operator: the Koszul-Tate complex

Given a generic Lagrangian system, its Euler–Lagrange operator obeys Noether identities which need not be independent, but satisfy first-stage Noether identities, and so on. This construction is generalized to arbitrary differential operators on a smooth fiber bundle. Namely, if a certain necessary and sufficient condition holds, one can associate to a differential operator the exact chain complex with the boundary operator whose nilpotency restarts all the Noether identities characterizing the degeneracy of an original differential operator.