My new article: G.Sardanashvily, Higher-stage Noether identities and second Noether theorems, Advances in Mathematical Physics, v 2015 (2015) 127481(#)
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 non-trivial higher-stage Noether identities which is described in the homology terms. If a certain homology regularity conditions 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. The 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 provides a BRST extension of an original Lagrangian.