G_b] = C^c{}_{ab} G_c$$ to be functions of $q^i$ and $p_i$. However, when the $C^c{}_{ab}$’s depend on the $q$’s and $p$’s only through the $G_a$’s themselves, as is the case when the $G_a$’s form a first class polynomial Poisson algebra, one can define the BRST generator directly in the algebra $\Bbb{C}\,({{\cal P}}_a) \otimes \Bbb{C}\,(G_a) \otimes \Bbb{C}\,(\eta^a)$ of polynomials in the $G$’s, the $\eta$’s and the ${{\cal P}}$’s without any reference to the explicit realization of the $G$’s as phase space functions[^3]. That is, the BRST generator can be associated with the Poisson algebra itself.

The reason for which this can be done is that both the Koszul differential $\delta$ defined by (\[koszul\]) [*and*]{} the ${\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}}$ in (\[d-p\]) involve only $G_a$ and not $q^i$ or $p_i$ individually. Thus, ${\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p+1)} \\ \Omega \end{array}}$ can be taken to depend only on $G_a$. The BRST generator is defined accordingly in the algebra $\Bbb{C}\,({{\cal P}}_a)
\otimes \Bbb{C}\,(G_a) \otimes \Bbb{C}\,(\eta_a)$.

One can give an explicit solution of (\[delta-om=d\]) in terms of the homotopy $\sigma$ defined on the generators by $$\sigma G_a = - {{\cal P}}_a, \quad \sigma {{\cal P}}_a = \sigma G_a = 0$$ and extended to 