  In the Batalin - Fradkin - Vilkovisky approach to quantization of gauge
theories a principal role is given to the BRST charge which can be constructed
as a series in Grassmannian (ghost) variables with coefficients given by
generalized structure functions of constraints algebra. Alternatively, the BRST
charge can be derived making use of the Noether theorem and global BRST
invariance of the effective action. In the case of Yang - Mills fields the both
methods lead to the same expression for the BRST charge, but it is not valid in
the case of General Relativity. It is illustrated by examples of an isotropic
cosmological model as well as by spherically-symmetric gravitational model
which imitates the full theory of gravity much better. The consideration is
based on Hamiltonian formulation of General Relativity in extended phase space.
At the quantum level the structure of the BRST charge is of great importance
since BRST invariant quantum states are believed to be physical states. Thus,
the definition of the BRST charge at the classical level is inseparably related
to our attempts to find a true way to quantize Gravity.
