Theorem: Free causal BRST differential

The derivation $s_0$ preserves the free equation ideal and all relations (eq:CCR). It induces a nilpotent graded derivation of ${\mathfrak{A}}_0$.

Proof. First consider the equations. Their variations are

The antighost equation varies into $\square b$. Taking the divergence of the first equation and using (eq:dividentities) shows that $\square b=0$ belongs to the equation ideal. The scalar and ghost equations are otherwise inert.

For the causal relations, the only cancellation between nonzero blocks occurs in the initially zero $r$–$\bar{c}$ relation. In distributional component notation,

The exchanged relation follows by graded antisymmetry. Variations of the $r$–$r$, $r$–$b$, and $r$–$c$ relations involve zero ghost cross blocks or the zero $cc$ block. Variations of the $\bar{c}$–$\bar{c}$ and $b$–$\bar{c}$ relations use zero $b$-ghost blocks and the zero $bb$ block. All scalar cross relations are inert. These checks exhaust the generator pairs.

Thus $s_0$ is well defined on the quotient. Lemma lem:s0nil proves its nilpotence there.