bibkey: stacksproject2026dualnumberext authors: “The Stacks Project Authors” year: 2026 title: “The Stacks project” doi: null url: https://stacks.math.columbia.edu/tag/0A5Q claim: “The dual-number residue module has a periodic epsilon resolution; short exact module sequences induce contravariant Hom/Ext long exact sequences; degree-one Ext classes represent Yoneda extensions.” strata_touched:

• D5/S3/HomologicalAlgebra/DualNumberResidueExtension license: citation-only triage: anchor

Stacks Project ingredients for the dual-number extension

The Stacks Project, Example 15.70.3, tag 0A5Q, was read on 23 September 2026. It takes a field k, the dual-number ring R = k[x]/(x^2), epsilon the class of x, and the residue module M = R/(epsilon). It states that M is quasi-isomorphic to the periodic complex whose differentials are multiplication by epsilon, and concludes that M does not have finite projective dimension.

The Stacks Project, Lemma 10.71.7, tag 065P, gives the contravariant long exact sequence beginning with Hom and continuing through Ext^1 for a short exact sequence of modules. Lemma 13.27.5, tag 06XU, identifies Ext classes in an abelian category with equivalence classes of Yoneda extensions.

These results support ingredients of the repository theorem. They do not state the combined rational nonvanishing theorem or its retraction contradiction. No source text or proof body is vendored.

Verified locator

• URL: https://stacks.math.columbia.edu/tag/0A5Q
• Related tag: https://stacks.math.columbia.edu/tag/065P
• Related tag: https://stacks.math.columbia.edu/tag/06XU