  We address the problem of extending an original field Lagrangian to ghosts
and antifields in order to satisfy the master equation in the framework of the
BV quantization of Lagrangian field systems. This extension essentially depends
on the degeneracy of an original Lagrangian whose Euler-Lagrange operator
generally obeys the Noether identities which need not be independent, but
satisfy the first-stage Noether identities, and so on. A generic Lagrangian
system of even and odd fields on an arbitrary smooth manifold is examined in
the algebraic terms of the Grassmann-graded variational bicomplex. We state the
necessary and sufficient condition for the existence of the exact antifield
Koszul-tate complex whose boundary operator provides all the Noether and
higher-stage Noether identities of an original Lagrangian system. The Noether
inverse second theorem that we prove associates to this Koszul-Tate complex the
sequence of ghosts whose ascent operator provides the gauge and higher-stage
gauge supersymmetries of an original Lagrangian. We show that an original
Lagrangian is extended to a solution of the master equation if this ascent
operator admits a nilpotent extension and only if it is extended to an operator
nilpotent on the shell.
