  The first and 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 and the corresponding
higher-stage gauge symmetries which are described in the homology terms. In
these terms, the second Noether theorems associate to the Koszul - Tate chain
complex of higher-stage Noether identities the gauge cochain sequence whose
ascent operator provides higher-order gauge symmetries of Lagrangian theory. If
gauge symmetries are algebraically closed, this operator is extended to the
nilpotent BRST operator which brings the gauge cochain sequence into the BRST
complex. In this framework, the first Noether theorem is formulated as a
straightforward corollary of the first variational formula. It associates to
any variational Lagrangian symmetry the conserved current whose total
differential vanishes on-shell. We prove in a general setting that a conserved
current of a gauge symmetry is reduced to a total differential on-shell. The
physically relevant examples of gauge theory on principal bundles, gauge
gravitational theory on natural bundles, topological Chern - Simons field
theory and topological BF theory are present. The last one exemplifies a
reducible Lagrangian system.
