Lemma: Finiteness and locality

For $F,G\in {\mathfrak{A}}_W$, $\left(F,G\right)\in {\mathfrak{A}}_W$. The extended action $S_{\mathrm{ext}}$ is a formal bulk functional, not an element of ${\mathfrak{A}}_W$, but for every $F\in {\mathfrak{A}}_W$ the bracket $\left(S_{\mathrm{ext}},F\right)$ is a well-defined element of ${\mathfrak{A}}_W$, and it depends on $F$ only through its restriction to ${\mathfrak{F}}_W$.

Proof. The density of $\left(F,G\right)$ is a finite sum of products of Euler derivatives of $\mathcal{F}$ and $\mathcal{G}$; each factor inherits the compact coefficient support, so the product is a local density with compact support. For $\left(S_{\mathrm{ext}},F\right)$, every term pairs a derivative of $S_{\mathrm{ext}}$ with a derivative of $F$, and the latter is compactly supported in ${M_T}^{\circ }$; the Euler derivatives of $S_{\mathrm{ext}}$ are local expressions in the fields and antifields (XII-C's $e_{\mathrm{D}},E_{\varphi }$ and the antifield-linear terms of (eq:Sext)), so the pairing is finite and local. Dependence on the restriction to ${\mathfrak{F}}_W$: the $c$-derivative of $F$ enters only paired with ${\delta }^{\mathrm{L}}{S}_{\mathrm{ext}}/\delta c^{\ast }=sc\in {\mathfrak{g}}_W\otimes \Lambda$, a direction tangent to ${\mathfrak{F}}_W$; all other slots of ${\mathfrak{F}}_W$ are unrestricted.