Corollary: Extension dependence and cancellation by extension

Let $T,T^{\prime }$ be extensions of $T^{\circ }$ and $s=T^{\prime }-T$, so $\mathrm{supp}s\subseteq Z$. Then

and ${d_0}^{T^{\prime }}={d_0}^T+{\mathsf{Rec}}_s{|}_H$. The defect of ${F_0}^T$ is cancelled by a change of extension if and only if there is an extension $T^{\prime }$ with $H_{T^{\prime }}=H$. In Proposition prop:example, $s=-{\partial }_{x_1}{\delta }_0$ cancels the defect of ${F_0}^{T_2}$.

Proof. Linearity of $T\mapsto {\mathsf{Rec}}_T$ gives ${F_0}^{T^{\prime }}-{F_0}^T={\mathsf{Rec}}_s$. By (eq:normalform), ${\mathsf{Rec}}_s(a)=\sum {\mathsf{E}}_{\gamma }({\beta }_{\gamma }(a)s)$ with each coefficient supported in $Z$, so Lemma lem:extraction(ii) gives the support statement. Cancellation means ${d_0}^{T^{\prime }}=0$, which by Theorem thm:defect(iii) is $H_{T^{\prime }}=H$. In the example, $T_2+s=T_1$ and $H_{T_1}=H$.