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$(n,d)$-Coherent Rings

Rafael Parra

Abstract

We investigate finiteness conditions on modules of bounded projective dimension and their connection with generalized notions of coherence. For a ring $R$, we consider the class $\mathsf{FP}_n^{\le d}(R)$ of finitely $n$-presented modules of projective dimension at most $d$ and develop the corresponding relative homological theory. We establish several characterizations of left $(n,d)$-coherent rings in the sense of Mao and Ding [43], in terms of $\mathsf{FP}_n^{\le d}(R)$ and the associated classes of $\mathsf{FP}_n^{\le d}$-injective, $\mathsf{FP}_n^{\le d}$-projective, $\mathsf{FP}_n^{\le d}$-flat, and $\mathsf{FP}_n^{\le d}$-cotorsion modules. As a consequence, when $d \ge \gD(R)$ or $d=\infty$, we recover Costa's $n$-coherence [17] and obtain new characterizations of regularly coherent rings.

$(n,d)$-Coherent Rings

Abstract

We investigate finiteness conditions on modules of bounded projective dimension and their connection with generalized notions of coherence. For a ring , we consider the class of finitely -presented modules of projective dimension at most and develop the corresponding relative homological theory. We establish several characterizations of left -coherent rings in the sense of Mao and Ding [43], in terms of and the associated classes of -injective, -projective, -flat, and -cotorsion modules. As a consequence, when or , we recover Costa's -coherence [17] and obtain new characterizations of regularly coherent rings.

Paper Structure

This paper contains 10 sections, 72 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Finitely $n$-presented modules of bounded projective dimension and $(n,d)$-coherent modules
  3. $(n,d)$-coherent modules
  4. $(n,d)$-coherent rings
  5. Regularity and coherence
  6. $\mathsf{FP}_n^{\le d}$-injective and $\mathsf{FP}_n^{\le d}$-projective modules
  7. $\mathsf{FP}_n^{\le d}$-injective and $\mathsf{FP}_n^{\le d}$-projective modules over $(n,d)$-coherent rings
  8. $\mathsf{FP}_n^{\le d}$-flat and $\mathsf{FP}_n^{\le d}$-cotorsion modules
  9. $\mathsf{FP}_n^{\le d}$-flat and $\mathsf{FP}_n^{\le d}$-cotorsion modules over $(n,d)$-coherent rings
  10. Acknowledgements

Key Result

Proposition 2.1

Let $n \in \mathbb{N}$ and $d \in \mathbb{N}$. For any ring $R$, the following statements hold:

Theorems & Definitions (143)

  • Proposition 2.1
  • proof
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 133 more