arbitrary functions on the extended phase space as a derivation. One easily verifies that $\delta^2 = 0$.

Given ${{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (s)} \\ \Omega \end{array}}}$ with $s \leq p$, one solves (\[delta-om=d\]) for ${{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p+1)} \\ \Omega \end{array}}}$. This can always be done because $\delta
{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}} = 0$, and because $\delta$ is acyclic in positive degree. One then goes on to ${{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p+2)} \\ \Omega \end{array}}}$ etc... until one reaches the complete expression for $\Omega$. The last function ${{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}}$ that can be non zero is ${{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (n-1)} \\ \Omega \end{array}}}$ where $n$ is the number of constraints. Indeed, the product $\eta^{a_1} \cdots \eta^{a_n} \eta^{a_{n+1}}$ of $n+1$ anticommuting ghost variables in ${\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (n)} \\ \Omega \end{array}}$ is zero. The function ${\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p+1)} \\ \Omega \end{array}}$ is determined by (\[delta-om=d\]) up to a $\delta$-exact term. This amounts to making a canonical transformation in the extended phase space.

First Class Polynomial Poisson algebras
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The standard BRST construction recalled in the previous section assumes that the $G_a$’s are realized as functions on some phase space, and allows the $C^c{}_{ab}$ in $$[G_a, 