Equivalence classes of Wakamatsu tilting modules and preenveloping and precovering subcategories

Abstract: Let R be an associative ring with identity. We introduce an equivalence relation on the class of Wakamatsu tilting right R modules. By using this equivalence relation, we extend the Mantese Reiten theorems from the setting of Artin algebras to that of arbitrary associative rings.

• The paper establishes equivalence classes of Wakamatsu tilting modules using invertible bimodules to achieve a classification up to Morita equivalence.
• It constructs explicit bijections between these module classes and maximal preenveloping coresolving as well as precovering resolving subcategories in module categories.
• The results extend classical dualities to arbitrary associative rings, enabling a unified framework in tilting theory and homological algebra.

Equivalence Classes of Wakamatsu Tilting Modules and (Pre)enveloping/(Pre)covering Subcategories

Introduction

The paper "Equivalence classes of Wakamatsu tilting modules and preenveloping and precovering subcategories" (2512.11600) generalizes foundational results in tilting theory beyond the restriction to Artin algebras, providing a framework for Wakamatsu tilting modules over arbitrary associative rings. It formalizes equivalence classes of such modules via invertible (Morita) bimodule transformations, and establishes bijections between these module classes and certain homologically well-behaved subcategories, such as preenveloping coresolving and precovering resolving subcategories, both in the module category and its finitely generated portion. These equivalence relations and correspondences extend and unify known results by Auslander-Reiten and Mantese-Reiten.

Summary of Main Results

Equivalence Relations and Morita Theory

The authors define an equivalence relation $\sim$ on Wakamatsu tilting right $R$-modules, where $T \sim T'$ if $T$ and $T'$ are related via invertible bimodules (projective bimodules of rank one inducing Morita equivalence of module categories). They show that, modulo such equivalence, the classification of tilting objects is insensitive to Morita transformations, reflecting categorical invariance tied to the Picard group of rank-one projective bimodules.

Notably, when $R$ is a Noetherian algebra over a Noetherian commutative semi-local complete ring, the equivalence classes under $\sim$ reduce to isomorphism classes, establishing that Morita-equivalent Wakamatsu tilting modules are in fact isomorphic in this restricted setting.

Correspondences with Homological Subcategories

Via explicit constructions and categorical arguments, the paper proves bijections between the equivalence classes $[T]$ of Wakamatsu tilting right $R$-modules and:

• Preenveloping coresolving subcategories of $\mathrm{Mod}{-}R$ admitting an Ext-projective generator maximal relative to that generator.
• Resolving subcategories of $\mathrm{mod}{-}R$ (the category of finitely generated right $R$-modules) possessing an Ext-injective cogenerator maximal among those with the same cogenerator.
• Precovering resolving subcategories admitting a (possibly product-complete) Ext-injective cogenerator, with the results specialized for product-complete cotilting modules.

These correspondences extend the dualities first established by Auslander-Reiten in the classical (finite-dimensional) context, and Mantese-Reiten for Wakamatsu tilting modules over Artin algebras, to a general non-commutative and non-Noetherian regime.

Wakamatsu Cotilting Modules and Product-Complete Case

For rings satisfying strong finiteness and completeness assumptions, the authors show that for finite length modules the distinction between Wakamatsu tilting and cotilting collapses—they coincide—generalizing the previously Artin algebra-constrained statements.

In addition, product-completeness (closure under arbitrary direct products) is characterized as an essential property for certain Ext-injective cogenerators, and yields refined surjectivity and injectivity statements about the correspondences with precovering subcategories.

Technical Contributions

• Invertible Bimodules and Morita Equivalence: A rigorous equivalence relation is introduced on tilting and cotilting modules using the categorical machinery of invertible (Morita) bimodules and progenerators, connecting module-theoretic invariants with structural properties of underlying rings.
• Explicit Construction of Bijections: The bijections are constructed via homological algebra, leveraging cotorsion pair arguments, closure properties (coresolving, resolving), and canonical functorial isomorphisms, ensuring that the subcategories classified correspond precisely to maximal families determined by the tilting or cotilting object up to equivalence.
• Extension to Noetherian Complete Algebras: The classification is shown to be sharp in the context of Noetherian algebras over semi-local commutative complete rings, yielding concrete invariance up to isomorphism, rather than merely up to Morita equivalence.

Numerical and Strong Claims

• One-to-one correspondences are established between:
  • Equivalence classes of Wakamatsu tilting (and product-complete cotilting) right $R$-modules under $\sim$
  • Maximal preenveloping coresolving subcategories of $\mathrm{Mod}{-}R$ (Theorem 1.1)
  • Maximal resolving subcategories of $\mathrm{mod}{-}R$ (Theorem 1.2)
  • Maximal precovering resolving subcategories of $\mathrm{Mod}{-}R$ with product-complete cogenerator (Theorem 1.3)
• Equivalence under $\sim$ collapses to isomorphism for basic finite length tilting modules over Noetherian algebras over semi-local complete rings.

Practical and Theoretical Implications

The abstraction of tilting and cotilting theory to arbitrary associative rings, respecting equivalence under Morita invariance, profoundly impacts homological classification in representation theory, noncommutative algebraic geometry, and singularity categories. Especially in the study of module approximations (precovers, preenvelopes) and derived categories, these results facilitate categorical resolutions and relative homological constructions, unrestricted by traditional finiteness conditions or specific algebra classes.

The explicit connection to the Picard group and rank-one projective bimodules further informs the study of endomorphism rings, equivalence classes of noncommutative spaces, and categorical derived equivalences.

Future Directions

Prospective research lines include:

• Extension to broader settings (e.g., differential graded rings, triangulated categories).
• Analysis of dualities for infinite-dimensional contexts, or for categories with more exotic closure properties.
• Investigation of derived and stable equivalence induced by Wakamatsu tilting objects in singularity categories.
• Connections with semidualizing modules and their role in relative derived functor vanishing and Gorenstein homological dimensions.
• Computational algorithms for classifying tilting/cotilting objects up to equivalence via Picard group invariants.

Conclusion

This work (2512.11600) substantially advances the structural and categorical understanding of Wakamatsu tilting and cotilting modules by rigorously defining equivalence classes via invertible bimodules and establishing bijective correspondences with well-behaved homological subcategories. The results provide foundational classifications applicable to arbitrary associative rings, aligning the study of module categories with categorical invariants, and offering a robust framework for future advances in relative homological algebra and representation theory.