Higher-Stage Noether Identities and Second Noether Theorems

The direct and inverse second Noether theorems are formulated in a general case of reducible degenerate Grassmann-graded Lagrangian theory of even and odd variables on graded bundles. Such Lagrangian theory is characterized by a hierarchy of nontrivial higher-stage Noether identities which is described in the homology terms. If a certain homology regularity condition holds, one can associate with a reducible degenerate Lagrangian the exact Koszul–Tate chain complex possessing the boundary operator whose nilpotentness is equivalent to all complete nontrivial Noether and higher-stage Noether identities. The second Noether theorems associate with the above-mentioned Koszul–Tate complex a certain cochain sequence whose ascent operator consists of the gauge and higher-order gauge symmetries of a Lagrangian system. If gauge symmetries are algebraically closed, this operator is extended to the nilpotent BRST operator which brings the above-mentioned cochain sequence into the BRST complex and provides a BRST extension of an original Lagrangian.