Manuscript X-C-ST@1 · X-C-ST.tex · theorem · b679954e

Theorem: Background identity through second order

Three-term variation and the completed identity › The completed background identity · lines 1638–1669 · proof included or linked

X-C-ST@1#thm:ward, X-C-ST@1#eq:ward, X-C-ST@1#eq:ideal-representative

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For every $F \in G_{p}$ , with the free field held fixed,

A [F^{*} q] = A [q] in A_{0} \otimes D_{2} .

Equivalently,

A_{1} [h^{'}] = A_{1} [h], A_{2} [h^{'}, k^{'}] = A_{2} [h, k] .

Off shell, the compact tests obey

e [q^{'}] - e [q] = P_{m} ((F^{*} - 1) s [q]),

not necessarily equality as tests.

Proof. Lemma lem:theta gives $Θ_{F} = 0$ on the inherited coordinate plateau. Equation (eq:plateau) gives $g = 1$ on a neighbourhood of both generator supports, so Lemma lem:sigma gives $Σ_{F} = 0$ . Proposition prop:variation reduces to (eq:ideal-representative). Every coefficient on its right belongs to $P_{m} D$ , and therefore has zero free-field class.

Displayed formulas in this passage

X-C-ST@1#eq:ward · line 1641
$A [F^{*} q] = A [q] in A_{0} \otimes D_{2} .$
X-C-ST@1#eq:ideal-representative · line 1652
$e [q^{'}] - e [q] = P_{m} ((F^{*} - 1) s [q]),$
unlabeled · line 1647
$A_{1} [h^{'}] = A_{1} [h], A_{2} [h^{'}, k^{'}] = A_{2} [h, k] .$

Nearby in this section

section: The completed background identity