Manuscript XV-C@1 · XV-C.tex · section · 5afd124d

The inherited elementary null identity

Complete null tests in a finite normal-jet module › The inherited elementary null identity · lines 651–693

XV-C@1#eq:qidentity, XV-C@1#eq:inheritednull, XV-C@1#eq:actualnulltest

Let $e_{j}$ denote a coordinate multi-index. Directly from (eq:insert),

q_{j} E_{α} T = {- α_{j} E_{α - e_{j}} T, 0, α_{j} > 0, α_{j} = 0.

For $α = e_{j}$ this gives

q_{j} E_{e_{j}} T + E_{0} T = 0.

The supplied XIV-J chart uses the corresponding one-variable reconstruction convention

Rec (M_{R}) = δ (q) R (r), Rec (i θ M_{R}) = δ^{'} (q) R (r) .

Consequently its reported null presentation is

n = (q i θ + 1) M_{R}, Rec (n) = 0.

Theorem thm:jetnull includes this identity in its local model Jnull.

For a candidate $F$ defined on the two supplied summands, descent requires the concrete equality

F (q i θ M_{R}) + F (M_{R}) = 0 on the full domain .

Canonical agreement establishes only its restriction to the proved canonical domain.

Displayed formulas in this passage

XV-C@1#eq:qidentity · line 655
$q_{j} E_{α} T = {- α_{j} E_{α - e_{j}} T, 0, α_{j} > 0, α_{j} = 0.$
XV-C@1#eq:inheritednull · line 676
$n = (q i θ + 1) M_{R}, Rec (n) = 0.$
XV-C@1#eq:actualnulltest · line 686
$F (q i θ M_{R}) + F (M_{R}) = 0 on the full domain .$
unlabeled · line 665
$q_{j} E_{e_{j}} T + E_{0} T = 0.$
unlabeled · line 671
$Rec (M_{R}) = δ (q) R (r), Rec (i θ M_{R}) = δ^{'} (q) R (r) .$