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Homological dimensions of complexes over coherent regular rings

James Gillespie, Alina Iacob

TL;DR

The paper extends Avramov–Foxby-type equivalences from Noetherian to coherent rings, showing that a ring is regular exactly when the ordinary and graded homological dimensions of all complexes coincide (e.g., $pd_R(X)=gr-pd_R(X)$ and $id_R(X)=gr-id_R(X)$). It develops an integrated framework based on FP-injective/projective modules, cotorsion pairs, and model structures, and leverages the coderived category via the FP-injective model to connect derived and coderived perspectives. Central to the results are characterizations in terms of FP-injective and FP-projective dimensions, DG- versus degreewise notions, and a triangulated quotient functor $ar{ extgamma}$ that identifies the coderived and derived categories for coherent regular rings. Collectively, the work unifies Noetherian results with coherent rings, shows that regularity can hold with infinite global dimension, and provides parallel injective, projective, and flat characterizations that illuminate the homological landscape of unbounded complexes over coherent rings.

Abstract

We show that Iacob-Iyengar's answer to a question of Avromov-Foxby extends from Noetherian to coherent rings. In particular, a coherent ring R is regular if and only if the injective (resp. projective) dimension of each complex X of R-modules agrees with its graded-injective (resp. graded-projective) dimension. The same is shown for the analogous dimensions based on FP-injective R-modules, and on flat R-modules.

Homological dimensions of complexes over coherent regular rings

TL;DR

The paper extends Avramov–Foxby-type equivalences from Noetherian to coherent rings, showing that a ring is regular exactly when the ordinary and graded homological dimensions of all complexes coincide (e.g., and ). It develops an integrated framework based on FP-injective/projective modules, cotorsion pairs, and model structures, and leverages the coderived category via the FP-injective model to connect derived and coderived perspectives. Central to the results are characterizations in terms of FP-injective and FP-projective dimensions, DG- versus degreewise notions, and a triangulated quotient functor that identifies the coderived and derived categories for coherent regular rings. Collectively, the work unifies Noetherian results with coherent rings, shows that regularity can hold with infinite global dimension, and provides parallel injective, projective, and flat characterizations that illuminate the homological landscape of unbounded complexes over coherent rings.

Abstract

We show that Iacob-Iyengar's answer to a question of Avromov-Foxby extends from Noetherian to coherent rings. In particular, a coherent ring R is regular if and only if the injective (resp. projective) dimension of each complex X of R-modules agrees with its graded-injective (resp. graded-projective) dimension. The same is shown for the analogous dimensions based on FP-injective R-modules, and on flat R-modules.
Paper Structure (7 sections, 15 theorems, 15 equations)

This paper contains 7 sections, 15 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Coherent regular rings
  3. Complexes of FP-injective and FP-projective modules
  4. Stovicek's FP-injective model structure for the coderived category
  5. Homological dimensions of complexes over coherent regular rings
  6. Injective and FP-injective characterizations of coherent regular rings
  7. Projective and Flat characterizations of coherent regular rings

Key Result

Theorem 2.2

The following are equivalent for a general ring $R$.

Theorems & Definitions (31)

  • Definition 2.1: Bertin bertin-coherent-regular
  • Theorem 2.2: Characterizations of coherent regular rings
  • proof
  • Example 2.3
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 21 more