nilpotency condition becomes, in terms of ${{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}}$, $$\label{delta-om=d}
\delta {{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p+1)} \\ \Omega \end{array}}} + {\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}} = 0$$ where ${\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}}$ involves only the lower order ${{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (s)} \\ \Omega \end{array}}}$ with $s \leq p$ and is defined by $$\label{d-p}
{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}} = 1 / 2 \left[
  \sum^p_{k=0} [{{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}} , {{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p-k)} \\ \Omega \end{array}}}]_{\mbox{orig}} +
  \sum^{p-1}_{k=0} [ {{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k+1)} \\ \Omega \end{array}}} ,
  {{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p-k)} \\ \Omega \end{array}}}]_{{{\cal P}}, \eta}
\right].$$

Here, the bracket $[\; , \;]_{\mbox{orig}}$ refers to the Poisson bracket in the original phase space, which only acts on the $q^i$ and $p_i$, and not on the ghosts, whereas $[\; , \;]_{{{\cal P}}, \eta}$ refers to the Poisson bracket acting only on the ghost and ghost momenta arguments and not on the original phase space variables. The “Koszul” differential $\delta$ in (\[delta-om=d\]) is defined by $$\label{koszul}
\delta q^i = \delta p_i = 0, \quad \delta \eta^a = 0, \quad \delta
{{\cal P}}_a = - G_a$$ and is extended to 