Relative Derived Torsion in Categories of Reduced Modules: Comparison and Non-Noetherian Examples

Let $I$ be an ideal of a commutative ring $R$. We use relative derived torsion in exact full abelian subcategories of $I$-reduced modules to study ambient derived ideal-power torsion. Exact inclusion forces $IM=I^2M$ whenever $I^2$ is finitely generated. Over a Noetherian ring, every finite object consequently decomposes into its torsion part and $IM$, and its positive-degree ambient cohomology vanishes. For $R=S[t]$ and $I=JS[t]+(t)$ with $J^2=J$, we compute every stage of the canonical Ext system and its transition maps, obtaining a comparison with $\Ext_S^q(S/J,-)$. Valuation-domain and product-of-fields examples yield nonzero cyclic first cohomology. A further construction has cyclic coefficients and cyclic ambient cohomology in every positive even degree, while its positive-degree relative cohomology vanishes. Applications determine kernels of canonical Ext maps for these coefficient modules and identify the obstruction to ring-valued specialization in the questions of Eisenbud, Mustaţă, and Stillman.