Localization Theory of Module Categories

• Localization Theory of Module Categories is a framework that bridges triangulated or tensor categories with classical module categories through explicit localization and quotient constructions.
• It employs techniques like Gabriel–Zisman localization, calculus of fractions, and cotorsion pair methods to reconstruct and classify module categories effectively.
• The theory unifies approaches from cluster tilting, derived and tensor settings, providing deep insights into module substructures and support theory.

The localization theory of module categories provides a foundational bridge between triangulated or tensor categories and classical module categories via explicit localization or quotient constructions. It governs how abelian or triangulated module categories can be reconstructed, classified, or studied through localizing substructures, cotorsion pairs, and model-theoretic or homotopical methods.

1. Foundations: From Triangulated Categories to Module Categories

Localization theory typically begins with a triangulated category $\mathcal{C}$, assumed to be $k$-linear, Hom-finite, skeletally small, and Krull–Schmidt, and a choice of rigid object $T\in\mathcal{C}$ (satisfying $\operatorname{Ext}^1_{\mathcal{C}}(T,T)=0$). A central result is that the abelian category of finite-dimensional modules $\operatorname{mod}A$ over the opposite endomorphism algebra $$A=\operatorname{End}_{\mathcal{C}}(T)^{\mathrm{op}}$$ can be realized as a Gabriel–Zisman localization of $\mathcal{C}$ at an explicitly characterized class of morphisms $S$. The set $S$ consists of those morphisms $f$ which, when completed to a distinguished triangle,

$$\Sigma^{-1}Z \xrightarrow{g} X \xrightarrow{f} Y \xrightarrow{h} Z,$$

have both $g$ and $h$ factor through objects in the extension-closed subcategory $\Sigma T^\perp$. Equivalently, $f\in S$ if and only if $\operatorname{Hom}_{\mathcal{C}}(T,f)$ is an isomorphism of $A$-modules. The localization functor $L: \mathcal{C} \to S^{-1}\mathcal{C}$ inverts all maps in $S$ and is universal for this property. One then obtains an equivalence of categories $S^{-1}\mathcal{C} \simeq \operatorname{mod}A$ via the restriction of the hom-functor $H=\operatorname{Hom}_{\mathcal{C}}(T,-)$ (Buan et al., 2010, Palu, 2014).

This construction generalizes earlier results, such as the cluster-tilting theory in 2-Calabi–Yau categories of Keller–Reiten, by removing both the 2-Calabi–Yau and cluster-tilting hypotheses on $T$ and replacing the factor construction by a Gabriel–Zisman localization at the broader class $S$.

2. Calculus of Fractions and Abelian Localizations

A precise understanding of the Gabriel–Zisman localization relies on the interplay of regular morphisms and the calculus of fractions. The construction is technically grounded in the additive quotient $\mathcal{C}/\mathcal{I}$, where $$\mathcal{I} = \ker(\operatorname{Hom}_{\mathcal{C}}(T, -))$$, followed by localizing at the class $R$ of regular morphisms (those that are both monic and epic in $\mathcal{C}/\mathcal{I}$). In the setting where $\mathcal{C}$ is preabelian and integral, $R$ satisfies the axioms for both left and right calculus of fractions (Buan et al., 2011). This enables the explicit construction of the abelian category $(\mathcal{C}/\mathcal{I})[R^{-1}]$, which is shown to be equivalent to $\operatorname{mod}A$. In particular, kernels and cokernels in the localization lift from those in the quotient, and the functor induced by $\operatorname{Hom}_{\mathcal{C}}(T, -)$ is exact, full, faithful, and essentially surjective.

This approach gives a direct categorical route from the triangulated world to module categories and allows one to bypass the need for additional strong hypotheses or abstract homological algebra.

3. Extriangulated and Cotorsion-Pair Localizations

The localization framework extends to extriangulated categories—additive categories equipped with structures generalizing both exact and triangulated categories. Here, cotorsion pairs and Hovey twin cotorsion pairs (HTCP) play a central role. In this context, the Gabriel–Zisman localization is associated with a class of morphisms determined by deflations and inflations whose (co)cones belong to specified extension-closed subcategories (Ogawa, 2020).

The main theorem of Nakaoka–Palu states that for a given HTCP $P = ((S,T),(U,V))$, localization at the suitable class $W$ induces an equivalence

$Z/I \simeq \mathcal{C}[W^{-1}]$

where $Z = T \cap U$, $I = S \cap V$. This exact equivalence realizes both the Buan–Marsh localization and Iyama–Yoshino subfactor constructions as special cases, and the resulting heart of the cotorsion pair is an abelian category equivalent to $$\operatorname{mod}\operatorname{End}_{\mathcal{C}}(T)^{\mathrm{op}}$$ for rigid $T$ (Ogawa, 2020). The framework also supports recollements and gluing of abelian and triangulated subcategories, enhancing its impact on the structure theory of module and cluster categories.

4. Homotopical and Model-Theoretic Approaches

Localization theory of module categories admits a fundamental reinterpretation via homotopical algebra. The triangulated category with a rigid object is endowed with a model category structure in which "weak equivalences" are precisely the class $S$ of Buan–Marsh. The cofibrations and fibrations are specified such that the homotopy category $\operatorname{Ho}\mathcal{C}$ coincides with $S^{-1}\mathcal{C}$, and thus with the module category $\operatorname{mod}A$ (Palu, 2014).

Typically, this model structure is only left-weak, but suffices to recover all essential features for the localization process. The functor $\operatorname{Hom}_{\mathcal{C}}(T,-)$ becomes a left Quillen functor, and the passage to derived functors reproduces the module category. Challenges and new phenomena arise in hom-infinite or non-finitary settings, which may require further adaptations involving compactly generated or derived localization.

Model-theoretic localization frameworks, including right Bousfield localization and Eilenberg–Moore categories, also appear in the context of module categories over monads and tensor categories. They underpin the functorial and universal properties of (co)localization sequences among categories of modules and their derived categories (White et al., 2016).

5. Localization in Tensor and Derived Module Categories

In tensor-theoretic settings, especially for module categories over multiring or tensor categories, localization is mediated by Serre submodules and thick submodules. The Serre quotient $\mathcal{M}/\mathcal{N}$ of an abelian module category $\mathcal{M}$ by a stable Serre submodule $\mathcal{N}$ inherits a module structure compatible with the ambient tensor category. This yields exact localization sequences

$$0 \to \mathcal{N} \to \mathcal{M} \to \mathcal{M}/\mathcal{N} \to 0,$$

and compatibility at the derived level via the equivalence $$\mathbf{D}^b(\mathcal{M})/\mathbf{D}^b_{\mathcal{N}}(\mathcal{M}) \simeq \mathbf{D}^b(\mathcal{M}/\mathcal{N})$$, as established by Miyachi (Yu, 6 Jan 2026). These results connect abelian and derived localization frameworks and support reconstruction theorems: a sufficiently "full and dense" derived module functor between subcategories of tensor categories must arise from an underlying abelian module equivalence.

Consequences extend to Morita theory and the study of derived equivalences in settings such as Hopf algebra smash products, offering criteria for equivalences and invariance of module-theoretic data.

6. Stable Module Categories, Support Theory, and Tensor Triangulated Geometry

The global structure of module categories under localization is deeply informed by support theory and tensor triangulated geometry. In the stable module category $\operatorname{StMod}(kG)$ over a finite group scheme, tensor-ideal localizing subcategories correspond bijectively to subsets of the projective spectrum $\operatorname{Proj} H^*(G,k)$ of the cohomology ring (Benson et al., 2015, Benson et al., 2016). Tools such as $\pi$-points and $\pi$-support (corresponding to homological support and cosupport) provide concrete generators and classify localizing/colocalizing subcategories. Minimal localizing and colocalizing subcategories are generated or cogenerated by explicit endofinite modules canonically assigned to points of the spectrum.

In tensor triangulated frameworks, Balmer’s spectrum $\operatorname{Spc}$ of prime tensor ideals governs smashing localizations and classification of thick subcategories. The construction of Margolis Postnikov towers enables the description of localization sequences and $\infty$-stack structures on the collection of localized stable categories, leading to spectral sequences for invariants such as Picard groups (Ricka, 2016). These structures ensure that the associated local and global invariants, support data, and classification theorems of localizing subcategories mirror those of geometric and commutative algebraic contexts.

7. Broader Impact and Unification

The localization theory of module categories thus forms a unifying nexus in modern representation theory, homological algebra, and higher category theory. By means of explicit Gabriel–Zisman or Verdier localizations, cotorsion pair constructions, and model- or derived-category theoretic techniques, the theory recovers classical module categories, supports analysis of their substructure, and enables reconstruction theorems. It subsumes and extends prior results—such as Iyama–Yoshino subfactor theorems and cluster-tilting factor categories—to general settings, including extriangulated categories, stable module categories, and derived categories of tensor module categories. These developments tightly integrate module theoretic, categorical, and homotopical perspectives into a comprehensive localization framework for module categories (Buan et al., 2010, Buan et al., 2011, Palu, 2014, Benson et al., 2015, Benson et al., 2016, White et al., 2016, Ricka, 2016, Ogawa, 2020, Yu, 6 Jan 2026).