This is an
Abstract of my invited lecture: Bedlewo , Poland ).

**Noether theorems in a general setting. Reducible graded Lagrangians,**in the Conference:**(10-16 May 2015,***Geometry of Jets and Fields*
Noether
theorems are formulated in a general case of reducible degenerate
Grassmann-graded Lagrangian theory of even and odd variables on graded bundles.
A problem is that any Euler-Lagrange operator satisfies Noether identities,
which therefore must be separated into the trivial and non-trivial ones. These
Noether identities can obey first-stage Noether identities, which in turn are
subject to the second-stage ones, and so on. Thus, there is a hierarchy of
non-trivial Noether and higher-stage Noether identities. This hierarchy is
described in 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. Since this complex is necessarily Grassmann-graded,
Lagrangian theory on graded bundles is considered from the beginning, and is
formulated in terms of the Grassmann-graded variational bicomplex. Its
cohomology defines a first variational formula whose straightforward corollary
is the first Noether theorem. 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 ascent operator is prolonged to the ilpotent BRST operator which
brings the gauge cochain sequence into a BRST complex, and thus provides a BRST
extension of an original Lagrangian system. [G.Sardanashvily,

**]***arXiv: 1411.2910*