Every Euler - Lagrange operator obeys Noether identities which, however, can be trivial. If they are not trivial, a Lagrange system is called degenerate. Nontrivial Noether identities always satisfy first-order Noether identity, which also may be trivial, and etc. If first-order Noether identities are not-trivial, a degenerate Lagrangian system is called reducible. A problem was to separate trivial and non-trivial Noether identities of zero and higher orders.
This separation was effected in terms of cohomology. As a result, we described a generic reducible degenerate Lagrangian system, whose Euler - Lagrange operator satisfies non-trivial Noether identities which are not independent, but are subject to non-trivial first-order Noether identities, satisfying, in turn, non-trivila second-order Noether identities, etc. Under a certain cohomology condition, the hierarchy of these Noether identities is described by the exact cochain complex of odd and even antifields, called the Kozul - Tate complex [119,127].
This description was extended to Noether identities for an arbitrary differential operator .
Having received the hierarchy of Noether identities for reducible degenerate Lagrangian systems, I believed natural to generalize to it the second Noether theorem, linking the Noether identities with the gauge transformations of zero and higher order. This has been done. Generalized second Noether theorem corresponds to the Koszul – Tate complex some cochain sequence. Its ascent operator, called the gauge operator, consists of a gauge symmetry of a Lagrangian and gauge symmetries of first and higher orders, which are parameterized by odd and even ghost fields [129,133]. This cochain sequence and the Kozul - Tate complex of Noether identitie fully characterize the degeneration of a Lagrangian system, which is necessary for its quantization.
G.Giachetta, L.Mangiarotti, G.Sardanashvily, Advanced Classical Field Theory (WS, 2009)