Manuscript XV-D@1 · XV-D.tex · theorem · 967136eb

Theorem: The two kernels

The null defect in the one source-complete module › Normal form, extraction, and support · lines 767–792 · proof included or linked

XV-D@1#thm:kernels, XV-D@1#eq:Hpunct, XV-D@1#eq:Hfull

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Let $T$ be an extension of $T^{\circ}$ . Then

H H_{T} = ker Rec^{\circ} = {a \in P_{r}^{jet} : β_{γ} (a) ∣_{X^{\circ}} T^{\circ} = 0 in D^{'} (X^{\circ}) for all ∣ γ ∣ \leq r}, = ker Rec_{T} = {a \in P_{r}^{jet} : β_{γ} (a) T = 0 in D^{'} (X) for all ∣ γ ∣ \leq r},

and $H_{T} \subseteq H$ . Moreover $r \circ Rec_{T} = Rec^{\circ}$ , so every $Rec_{T}$ is a preliminary lift for the punctured reconstruction.

Proof. (eq:Hpunct) and (eq:Hfull) follow from Lemma lem:normalform and Lemma lem:extraction(i) on $X^{\circ}$ and on $X$ respectively. For the compatibility, apply Lemma lem:extraction(iii) to (eq:normalform): $Rec_{T} (a) ∣_{U^{\circ}} = \sum E^{\circ}_{γ} (β_{γ} (a) ∣_{X^{\circ}} T ∣_{X^{\circ}}) = Rec^{\circ} (a)$ . If $Rec_{T} (a) = 0$ then its restriction $Rec^{\circ} (a) = 0$ , giving $H_{T} \subseteq H$ .

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XV-D@1#eq:Hpunct · line 770
$H H_{T} = ker Rec^{\circ} = {a \in P_{r}^{jet} : β_{γ} (a) ∣_{X^{\circ}} T^{\circ} = 0 in D^{'} (X^{\circ}) for all ∣ γ ∣ \leq r}, = ker Rec_{T} = {a \in P_{r}^{jet} : β_{γ} (a) T = 0 in D^{'} (X) for all ∣ γ ∣ \leq r},$
XV-D@1#eq:Hfull · line 770
$H H_{T} = ker Rec^{\circ} = {a \in P_{r}^{jet} : β_{γ} (a) ∣_{X^{\circ}} T^{\circ} = 0 in D^{'} (X^{\circ}) for all ∣ γ ∣ \leq r}, = ker Rec_{T} = {a \in P_{r}^{jet} : β_{γ} (a) T = 0 in D^{'} (X) for all ∣ γ ∣ \leq r},$

Theorem: The two kernels

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