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Differential modules: a perspective on Bass' question

David Nkansah

TL;DR

This paper develops a differential-module analogue of Bass’s finite-injective-dimension criterion within the $Q$-shaped derived category framework, showing that over a local noetherian ring a finitely generated module $M$ has finite injective dimension exactly when the minimal semi-injective resolution of $(M,0)$ has finitely many copies of the residue-field injective envelope. It introduces a robust adjunction between differential modules and complexes via expansion, compression, and cocompression, and defines differential tensor/Hom operations that respect this structure. A key corollary is a differential-module characterization of local Cohen–Macaulay rings, tying Bass-type invariants to the differential-homological structure. The work provides both theoretical machinery for non-graded homological algebra and concrete criteria linking differential-module data to classical ring-theoretic properties, with illustrative examples showing when minimal semi-injectives are preserved under the adjunctions.

Abstract

Guided by the $Q$-shaped derived category framework introduced by Holm and Jorgensen, we provide a differential module analogue of a classical result that characterises when a finitely generated module over a local commutative noetherian ring has finite injective dimension. As an application, we characterise local Cohen-Macaulay rings using the homological algebra of differential modules.

Differential modules: a perspective on Bass' question

TL;DR

This paper develops a differential-module analogue of Bass’s finite-injective-dimension criterion within the -shaped derived category framework, showing that over a local noetherian ring a finitely generated module has finite injective dimension exactly when the minimal semi-injective resolution of has finitely many copies of the residue-field injective envelope. It introduces a robust adjunction between differential modules and complexes via expansion, compression, and cocompression, and defines differential tensor/Hom operations that respect this structure. A key corollary is a differential-module characterization of local Cohen–Macaulay rings, tying Bass-type invariants to the differential-homological structure. The work provides both theoretical machinery for non-graded homological algebra and concrete criteria linking differential-module data to classical ring-theoretic properties, with illustrative examples showing when minimal semi-injectives are preserved under the adjunctions.

Abstract

Guided by the -shaped derived category framework introduced by Holm and Jorgensen, we provide a differential module analogue of a classical result that characterises when a finitely generated module over a local commutative noetherian ring has finite injective dimension. As an application, we characterise local Cohen-Macaulay rings using the homological algebra of differential modules.

Paper Structure

This paper contains 27 sections, 20 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. For question one:
  3. For question two:
  4. For question three:
  5. The main theorem
  6. Structure of the paper and the global setup
  7. Structure of the paper.
  8. Global Setup.
  9. A lemma
  10. The expansion and (co)compression functors
  11. Expansion functor.
  12. Compression functor.
  13. Cocompression functor.
  14. Stalk functors.
  15. Evaluation functors and their left adjoint.
  16. ...and 12 more sections

Key Result

Theorem A

Let $(R,\mathfrak{m},k)$ be a local commutative noetherian ring and let $M$ be finitely generated $R$-module. Consider the differential $R$-module $(M,0)$ with zero differential and choose a minimal semi-injective resolution $(M,0)\xrightarrow[]{}(I,d_I)$. Then the following statements are equivalen Furthermore, $R$ is Cohen-Macaulay if and only if there exists a finitely generated $R$-module $M$

Theorems & Definitions (49)

  • Theorem A: Theorem \ref{['thm: finite injective dimension IFF bass number finite']} and Corollary \ref{["thm: Bass' question"]}
  • Lemma 1.1
  • proof
  • Remark 2.1
  • Remark 2.2
  • Example 2.3: Projective flags
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 39 more