Theorem: Descending extensions

Let $T$ extend $T^{\circ }$ with $\mathrm{Ann}(T)=\mathrm{Ann}(T^{\circ })$. An extension $T^{\prime }$ satisfies $\mathrm{Ann}(T^{\prime })=\mathrm{Ann}(T^{\circ })$ iff $T^{\prime }-T\in {\mathcal{S}}_Z(T^{\circ })$. Hence the descending extensions form the affine space $T+{\mathcal{S}}_Z(T^{\circ })$, the descending extension lifts are ${F_0}^T+\{{\mathsf{Rec}}_s:s\in {\mathcal{S}}_Z(T^{\circ })\}$, and for ${\mathcal{A}}_{\mathrm{ext}}=\{{\mathsf{Rec}}_s:\mathrm{supp}s\subseteq Z\}$ the space ${\mathcal{A}}_{\mathrm{ext},0}=\{a\in {\mathcal{A}}_{\mathrm{ext}}:a{|}_H=0\}$ of XV-B's Theorem thm:lifting(iii) is $\{{\mathsf{Rec}}_s:s\in {\mathcal{S}}_Z(T^{\circ })\}$.

Proof. $T^{\prime }$ extends $T^{\circ }$ iff $s=T^{\prime }-T$ is supported in $Z$. By (R2), $\mathrm{Ann}(T^{\prime })=\mathrm{Ann}(T^{\circ })$ iff ${\mathsf{Rec}}_{T^{\prime }}{|}_H=0$; since ${\mathsf{Rec}}_{T^{\prime }}={\mathsf{Rec}}_T+{\mathsf{Rec}}_s$ and ${\mathsf{Rec}}_T{|}_H=0$, this is ${\mathsf{Rec}}_s{|}_H=0$, i.e. $s\in {\mathcal{S}}_Z(T^{\circ })$ by Lemma lem:recs.