Manuscript XVIII-A@2 · XVIII-A-v2.tex · proposition · 0f4336ec

Proposition: Formal-order separation

Descent, class transfer, and formal orders · lines 908–928 · proof included or linked

XVIII-A@2#prop:hbar

Let $A$ be a vector space or an $ℏ$ -independent quotient. A nonzero $a_{0} \in A$ cannot be cancelled in $A [[ℏ]]$ by positive formal powers:

a_{0} + ℓ \geq 1 \sum ℏ^{ℓ} a_{ℓ} \neq = 0.

If $Q_{ℓ}$ are well-defined linear maps on the declared coefficient domain and

Q = Q_{0} + ℏ Q_{1} + \dots, O = O_{0} + ℏ O_{1} + \dots,

then $QO = 0$ implies $Q_{0} O_{0} = 0$ .

Proof. Formal series are equal coefficientwise. The coefficient at $ℏ^{0}$ proves both assertions.

Displayed formulas in this passage

unlabeled · line 913
$a_{0} + ℓ \geq 1 \sum ℏ^{ℓ} a_{ℓ} \neq = 0.$
unlabeled · line 918
$Q = Q_{0} + ℏ Q_{1} + \dots, O = O_{0} + ℏ O_{1} + \dots,$