ИСТИНА

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 non-trivial Noether and higher-stage Noether identities which are described as elements of homology groups of some chain complex. If a certain homology regularity condition holds, one can associate to a reducible degenerate Lagrangian the exact Koszul-Tate chain complex possessing the boundary operator whose nilpotentness is equivalent to all complete non-trivial Noether and higher-stage Noether identities. Second Noether theorems associate to 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 thus provides a BRST extension of an original Lagrangian system.