Cohomology Annihilators of Truncated Polynomial Rings

We compute every cohomology-annihilator ideal and its least nonnegative stabilization index for \(S[x]/(x^n)\), where \(n\ge 2\) and \(S\) is a polynomial or formal power-series ring over any field. In base dimension zero, the eventual ideal is \((x^{\lfloor n/2\rfloor})\) and the index is 1. In base dimension one, the ideal is \((x^{n-1})\) and the index is 2 in every characteristic; this formula holds over every nonfield principal ideal domain. In base dimension \(d\ge 2\), the ideal is \((x^{n-1})\), with index \(d+1\), when \(n\) is invertible in the field; it is zero, with index 0, when the characteristic divides \(n\). Maps through projectives are described by an explicit transfer. A classical symmetric-power nilpotent operator gives a trace and colon-ideal obstruction in positive characteristic, and finite-free induction treats every divisible truncation degree. The field comparison also yields strict failures of equality under polynomial extension and localization.

Version 1.1 extends version 1.0 to every characteristic dividing \(n\) and to formal power-series bases in every dimension. It is a revision of the same manuscript, not a separate work. The broad noether-algebra question in OWR-12868-001 remains unresolved. The preprint is AI-assisted, self-audited and unrefereed; no independent review, proof-assistant certification, or absolute priority claim is asserted.