Quillen equivalence for chain homotopy categories induced by balanced pairs

1. Problem Statement

The paper addresses the problem of determining sufficient conditions under which the chain homotopy categories $K(\mathcal{X})$ and $K(\mathcal{Y})$ of two additive subcategories $\mathcal{X}$ and $\mathcal{Y}$ in an abelian category $\mathcal{A}$ are triangulated equivalent.

• Context: Previous work (notably by Chen [5]) established that an admissible balanced pair $(\mathcal{X}, \mathcal{Y})$ of finite dimension induces a triangle-equivalence $K(\mathcal{X}) \simeq K(\mathcal{Y})$.
• Limitation: The finite dimension hypothesis restricts the applicability of these results. For instance, it excludes cases involving infinite global dimensions, such as pure projective/injective modules over certain rings (e.g., products of fields).
• Goal: The authors seek broader sufficient conditions for the equivalence $K(\mathcal{X}) \simeq K(\mathcal{Y})$ that do not rely on finite dimensionality, specifically by utilizing model category theory and Quillen equivalences.

2. Methodology

The authors employ a model-categorical approach, realizing chain homotopy categories as homotopy categories of specific model structures.

• Balanced Pairs: They utilize the concept of a balanced pair $(\mathcal{X}, \mathcal{Y})$, where $\mathcal{X}$ is contravariantly finite and $\mathcal{Y}$ is covariantly finite. The pair is admissible if right $\mathcal{X}$-approximations are epimorphisms and left $\mathcal{Y}$-approximations are monomorphisms.
• Exact Category Setup:
  • Let $\mathcal{E}$ be the class of short exact sequences in $\mathcal{A}$ that remain exact under $\text{Hom}_{\mathcal{A}}(\mathcal{X}, -)$ (equivalently, under $\text{Hom}_{\mathcal{A}}(-, \mathcal{Y})$).
  • This forms an exact category $(\mathcal{A}, \mathcal{E})$, and consequently, the category of complexes $\text{Ch}(\mathcal{A}, \mathcal{E})$ is also exact.
• Model Structures via Hovey Triples:
  • The authors construct model structures on $\text{Ch}(\mathcal{A}, \mathcal{E})$ using Hovey triples (correspondence between model structures and complete cotorsion pairs).
  • They define specific classes of complexes:
    • $\mathcal{E}\text{-dw}\tilde{\mathcal{X}}$: Complexes with components in $\mathcal{X}$.
    • $(\mathcal{E}\text{-dw}\tilde{\mathcal{X}})^\perp$: The right orthogonal with respect to $\text{Ext}^1_{\text{Ch}(\mathcal{E})}$.
  • Key Construction: They prove that if $(\mathcal{E}\text{-dw}\tilde{\mathcal{X}})^\perp$ is closed under direct sums, the triple $(\mathcal{E}\text{-dw}\tilde{\mathcal{X}}, (\mathcal{E}\text{-dw}\tilde{\mathcal{X}})^\perp, \text{Ch}(\mathcal{A}))$ forms a model structure $M_{\text{dw}\mathcal{X}}$ whose homotopy category is equivalent to $K(\mathcal{X})$.
  • Dually, they construct $M_{\text{dw}\mathcal{Y}}$ for $K(\mathcal{Y})$.

3. Key Contributions

A. Realization of Chain Homotopy Categories as Model Categories

The paper provides a rigorous construction showing that $K(\mathcal{X})$ and $K(\mathcal{Y})$ can be realized as the homotopy categories of model structures on $\text{Ch}(\mathcal{A}, \mathcal{E})$.

• Theorem 4.9: Establishes that under mild closure conditions (direct sums for $\mathcal{X}$, direct products for $\mathcal{Y}$), the model structures $M_{\text{dw}\mathcal{X}}$ and $M_{\text{dw}\mathcal{Y}}$ exist and satisfy:
  $$\text{Ho}(M_{\text{dw}\mathcal{X}}) \simeq K(\mathcal{X}) \quad \text{and} \quad \text{Ho}(M_{\text{dw}\mathcal{Y}}) \simeq K(\mathcal{Y})$$
  These are relative analogues of Positselski's contraderived and coderived model structures.

B. The Main Equivalence Theorem

The central result provides a condition for the equivalence of the two homotopy categories without assuming finite dimension.

• Theorem 4.10: If the orthogonal classes coincide, i.e., $(\mathcal{E}\text{-dw}\tilde{\mathcal{X}})^\perp = {}^\perp(\mathcal{E}\text{-dw}\tilde{\mathcal{Y}})$, then there exists a Quillen equivalence between $M_{\text{dw}\mathcal{X}}$ and $M_{\text{dw}\mathcal{Y}}$. Consequently:
  $$K(\mathcal{X}) \simeq K(\mathcal{Y})$$
  This condition is shown to be less restrictive than the finite dimension hypothesis used in previous literature.

C. Recovery and Extension of Previous Results

The authors demonstrate that their framework recovers known theorems (like Chen's [5, Theorem A]) and extends them to infinite-dimensional settings.

• Corollary 4.11: If the global resolution dimension is finite, the condition of Theorem 4.10 holds, recovering Chen's result.
• New Applications: The framework successfully handles cases where dimensions are infinite, provided the orthogonality condition holds.

4. Specific Results and Applications

The paper applies the main theorem to three significant areas:

1. Cotorsion Triples:

   • If $(\mathcal{X}, \mathcal{Z}, \mathcal{Y})$ is a complete and hereditary cotorsion triple, the paper proves that if specific "dg" (derived) and "dw" (degree-wise) classes coincide, the equivalence restricts to the subcategories of projective and injective objects: $K(\mathcal{P}) \simeq K(\mathcal{I})$ (Corollary 5.3).

2. Gorenstein Homological Algebra:

   • Virtually Gorenstein Rings: For a left virtually Gorenstein ring $R$, if the equality between derived and degree-wise classes holds for Gorenstein projective ($\mathcal{GP}$) and Gorenstein injective ($\mathcal{GI}$) modules, then $K(\mathcal{GP}) \simeq K(\mathcal{GI})$ (Corollary 5.4).
   • Finite Gorenstein Weak Global Dimension: The authors recover and extend Chen's result [5] and Wang-Estrada's result [33] for rings with finite Gorenstein weak global dimension, showing $K(\mathcal{GP}) \simeq K(\mathcal{GI})$ restricts to $K(\mathcal{P}) \simeq K(\mathcal{I})$ (Corollary 5.8).

3. Pure Homological Algebra:

   • Pure Projective/Injective Objects: In a locally finitely presented Grothendieck category, the pair of pure projective ($\mathcal{PP}$) and pure injective ($\mathcal{PI}$) objects forms an admissible balanced pair.
   • Infinite Dimension Case: The paper proves $K(\mathcal{PP}) \simeq K(\mathcal{PI})$ even when the pure global dimension is infinite (e.g., for von Neumann regular rings with infinite pure global dimension). This refines a result by Št'ovíček [32] and improves upon Chen's [5] requirement of finite pure global dimension (Corollary 5.11).

5. Significance

• Methodological Shift: The paper shifts the paradigm from comparing relative derived categories (as done by Chen) to establishing Quillen equivalences between model categories. This provides a more robust structural framework for proving triangulated equivalences.
• Removal of Finite Dimension Constraints: By focusing on the equality of orthogonal classes $(\mathcal{E}\text{-dw}\tilde{\mathcal{X}})^\perp = {}^\perp(\mathcal{E}\text{-dw}\tilde{\mathcal{Y}})$ rather than finite resolution dimensions, the authors significantly broaden the scope of rings and categories for which these equivalences hold.
• Unification: The approach unifies various known equivalences (Gorenstein, pure, classical projective/injective) under a single model-theoretic umbrella.
• Refinement: It offers a refined understanding of the relationship between Gorenstein and pure homological properties, particularly in cases where traditional dimension-based criteria fail.

In summary, this work establishes a powerful general criterion for the triangulated equivalence of chain homotopy categories induced by balanced pairs, leveraging model category theory to overcome limitations imposed by finite dimensionality in classical homological algebra.