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Homological and homotopical aspects of Gorenstein flat modules and complexes relative to duality pairs

Víctor Becerril, Marco A. Pérez

Abstract

We study homological and homotopical aspects of Gorenstein flat modules over a ring with respect to a duality pair $(\mathcal{L,A})$. These modules are defined as cycles of exact chain complexes with components in $\mathcal{L}$ which remain exact after tensoring by objects in $\mathcal{A} \cap {}^\perp\mathcal{A} = \mathcal{A} \cap \Big( \bigcap_{i \in \mathbb{Z}_{> 0}} {\rm Ker}({\rm Ext}^i_{R^{\rm o}}(-,\mathcal{A})) \Big)$. In the case where $(\mathcal{L,A})$ is product closed and bicomplete (meaning in addition that $\mathcal{L}$ is closed under extensions, (co)products, $R \in \mathcal{L}$, $(\mathcal{A,L})$ is also a duality pair, and $\mathcal{A}$ is the right half of a hereditary complete cotorsion pair) we prove that these relative Gorenstein flat modules are closed under extensions, and that the corresponding Gorenstein flat dimension is well behaved in the sense that it recovers many of the properties and characterizations of its (absolute) Gorenstein flat counterpart (for instance, it can be described in terms of torsion functors). The latter in turn is a consequence of a Pontryagin duality relation that we show between these relative Gorenstein flat modules and certain Gorenstein injective modules relative to $\mathcal{A}$. We also find several hereditary and cofibrantly generated abelian model structures from these Gorenstein flat modules and complexes relative to $(\mathcal{L,A})$. At the level of chain complexes, we find three recollements between the homotopy categories of these model structures, along with several derived adjunctions connecting these recollements.

Homological and homotopical aspects of Gorenstein flat modules and complexes relative to duality pairs

Abstract

We study homological and homotopical aspects of Gorenstein flat modules over a ring with respect to a duality pair . These modules are defined as cycles of exact chain complexes with components in which remain exact after tensoring by objects in . In the case where is product closed and bicomplete (meaning in addition that is closed under extensions, (co)products, , is also a duality pair, and is the right half of a hereditary complete cotorsion pair) we prove that these relative Gorenstein flat modules are closed under extensions, and that the corresponding Gorenstein flat dimension is well behaved in the sense that it recovers many of the properties and characterizations of its (absolute) Gorenstein flat counterpart (for instance, it can be described in terms of torsion functors). The latter in turn is a consequence of a Pontryagin duality relation that we show between these relative Gorenstein flat modules and certain Gorenstein injective modules relative to . We also find several hereditary and cofibrantly generated abelian model structures from these Gorenstein flat modules and complexes relative to . At the level of chain complexes, we find three recollements between the homotopy categories of these model structures, along with several derived adjunctions connecting these recollements.
Paper Structure (6 sections, 34 theorems, 69 equations, 5 figures)

This paper contains 6 sections, 34 theorems, 69 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Relative Gorenstein flat modules
  4. Gorenstein flat dimensions relative to duality pairs
  5. Model structures from relative Gorenstein flat modules and dimensions
  6. Relative Gorenstein injective and flat complexes

Key Result

Proposition 2.2

Let $\mathcal{X} \subseteq \mathcal{G}$ such that $\mathrm{resdim}_{\mathcal{X}}(-)$ is stable. For every short exact sequence $A \rightarrowtail X \twoheadrightarrow C$ with $X \in \mathcal{X}$ and $A, C \in \mathcal{X}^\wedge$ and every $n \in \mathbb{Z}_{> 0}$, one has that $\mathrm{resdim}_{\mat

Figures (5)

  • Figure 1: Implications in Proposition \ref{['Bennis']}.
  • Figure 2: At each implication, the containment $\mathcal{L}^+ \subseteq \mathcal{A}$ is assumed.
  • Figure 3: At each implication, it is assumed that $\mathcal{L}$ is a relative generator in $\mathsf{Mod}(R)$.
  • Figure 4: A display of the implications studied in Theorem \ref{['theo:finiteness_GFdim']} for pairs $(\mathcal{L,A})$ with $\mathcal{L}^+ \subseteq \mathcal{A}$.
  • Figure 5: Display of all model structures on $\mathsf{Ch}(R)$, their homotopy categories, recollements and derived adjunctions between them.

Theorems & Definitions (72)

  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • Proposition 2.6
  • Remark 2.7
  • Definition 3.1
  • Remark 3.2
  • ...and 62 more