Manuscript XV-B@2 · XV-B.tex · section · 4d3f12e8

The punctured module and its natural lift

Natural null defects and point-puncture corrections › The punctured module and its natural lift · lines 1235–1282

XV-B@2#eq:puncturednull, XV-B@2#eq:naturallift, XV-B@2#eq:naturaldefect

Retain $U = X \times V_{q}$ , let $Z \subset X$ be relatively closed, and set

X^{\circ} = X ∖ Z, U^{\circ} = X^{\circ} \times V_{q} .

Fix $T^{\circ} \in D^{'} (X^{\circ})$ and use the presentation space

P = ∣ α ∣ \leq r ⨁ C^{\infty} (U) .

Thus the coefficients are smooth across $Z$ ; we are not using the larger space of arbitrary smooth coefficients on $U^{\circ}$ .

Define

Rec^{\circ} (a) = ∣ α ∣ \leq r \sum a_{α} ∣_{U^{\circ}} E_{α} T^{\circ}, H^{\circ} = ker Rec^{\circ} .

The complete criterion of Theorem thm:jetnull reads

a \in H^{\circ} ⟺ f_{γ} (a) T^{\circ} = 0 on X^{\circ} for every ∣ γ ∣ \leq r .

If $T_{0} \in D^{'} (X)$ extends $T^{\circ}$ , the natural preliminary lift is

F_{T_{0}} (a) = ∣ α ∣ \leq r \sum a_{α} E_{α} T_{0} .

For a null presentation,

F_{T_{0}} (a) = ∣ γ ∣ \leq r \sum E_{γ} (f_{γ} (a) T_{0}), supp (f_{γ} (a) T_{0}) \subset Z .

Consequently the natural null defect is supported on $Z \times {0}$ .

Displayed formulas in this passage

XV-B@2#eq:puncturednull · line 1257
$a \in H^{\circ} ⟺ f_{γ} (a) T^{\circ} = 0 on X^{\circ} for every ∣ γ ∣ \leq r .$
XV-B@2#eq:naturallift · line 1267
$F_{T_{0}} (a) = ∣ α ∣ \leq r \sum a_{α} E_{α} T_{0} .$
XV-B@2#eq:naturaldefect · line 1272
$F_{T_{0}} (a) = ∣ γ ∣ \leq r \sum E_{γ} (f_{γ} (a) T_{0}), supp (f_{γ} (a) T_{0}) \subset Z .$
unlabeled · line 1239
$X^{\circ} = X ∖ Z, U^{\circ} = X^{\circ} \times V_{q} .$
unlabeled · line 1243
$P = ∣ α ∣ \leq r ⨁ C^{\infty} (U) .$
unlabeled · line 1250
$Rec^{\circ} (a) = ∣ α ∣ \leq r \sum a_{α} ∣_{U^{\circ}} E_{α} T^{\circ}, H^{\circ} = ker Rec^{\circ} .$