Proposition: Relative obstruction in the unrestricted puncture-supported class

Let ${\mathcal{A}}_Z=\mathrm{Hom}({{\mathcal{P}}_r}^{\mathrm{jet}},V_{Z\times \{0\}})$. For every extension $T$, the relative obstruction class ${\mathrm{Ob}}_{{\mathcal{A}}_Z}({F_0}^T)=[{d_0}^T]\in \mathrm{Hom}(H,Y)/\{a{|}_H:a\in {\mathcal{A}}_Z\}$ vanishes. The cancelling correction constructed in the proof is algebraic; it is not asserted to be continuous, to be a change of extension, or to lie in any inherited class.

Proof. By Theorem thm:defect(ii), ${d_0}^T(H)\subseteq V_{Z\times \{0\}}$. Choose an algebraic complement ${{\mathcal{P}}_r}^{\mathrm{jet}}=H\oplus L$ with projection ${\pi }_H$ and set $a=-{d_0}^T{\pi }_H\in {\mathcal{A}}_Z$. Then $({F_0}^T+a){|}_H={d_0}^T-{d_0}^T=0$. This is the algebraic criterion of XV-B XVB instantiated with the actual spaces of this module.