Manuscript X-L-ST@1 · X-L-ST.tex · theorem · 20a32a22

Theorem: Conditional restricted quotient

Conditional corrections and quantum-presentation tests › An exact classical quotient criterion · lines 1158–1188 · proof included or linked

X-L-ST@1#thm:conditionalquotient, X-L-ST@1#eq:idealtest, X-L-ST@1#eq:niltest

Back to manuscript

Let $D$ be a declared graded subalgebra of $F_{ml} [[κ]]$ , and let $J \subset D$ be a graded ideal. Let $C$ be an odd ghost-number-one derivation of $D$ , beginning at positive order in $κ$ , and define $d = s + C$ .

Then $d$ induces a square-zero derivation of $D / J$ if and only if

(s + C) J \subseteq J, (s C + C s + C^{2}) D \subseteq J .

Both conditions refer to the entire declared ideal and domain, not to a selected witness.

Proof. A derivation descends through a quotient exactly when it preserves the defining ideal, proving (eq:idealtest). Since $s^{2} = 0$ ,

d^{2} = s C + C s + C^{2} .

Its induced square vanishes exactly under (eq:niltest).

Displayed formulas in this passage

X-L-ST@1#eq:idealtest · line 1169
$(s + C) J \subseteq J, (s C + C s + C^{2}) D \subseteq J .$
X-L-ST@1#eq:niltest · line 1169
$(s + C) J \subseteq J, (s C + C s + C^{2}) D \subseteq J .$
unlabeled · line 1183
$d^{2} = s C + C s + C^{2} .$

Nearby in this section

section: An exact classical quotient criterion