Mantese–Reiten Theorems Overview

• Mantese–Reiten Theorems are a set of classification and duality results in Auslander–Reiten theory that relate tilting modules with resolving subcategories.
• They extend classical tilting correspondences from Artin algebras to arbitrary rings and monomorphism categories using preenveloping and precovering subcategories.
• The framework provides concrete periodicity, homological duality, and AR-quiver classifications that unify representation theory, functorially finite subcategories, and extriangulated categories.

The Mantese–Reiten theorems provide a foundational classification and duality framework within Auslander–Reiten theory, generalizing structural correspondences between tilting modules and homological subcategories both in module categories of Artin algebras and, by recent advances, in arbitrary associative rings. These results characterize, via explicit bijections, the relationship between (equivalence classes of) Wakamatsu tilting or cotilting modules and functorially finite or preenveloping/precovering resolving subcategories. They underpin the mesh and tube combinatorics of Auslander–Reiten quivers and have been extended to monomorphism categories, functorially finite resolving subcategories, and the extriangulated categorical context.

1. Classical Mantese–Reiten Framework in Artin Algebra Module Categories

The classical Mantese–Reiten correspondences are established for module categories $A\text{-}\mathrm{mod}$ over Artin algebras. A Wakamatsu tilting module $T$ is a finitely generated module that is self-orthogonal ($\Ext^i_A(T,T)=0$ for $i>0$) and admits (co)resolutions satisfying specific vanishing conditions. The foundational theorems state:

• There is a bijection between isomorphism classes of basic Wakamatsu tilting left $A$-modules and covariantly finite coresolving subcategories of $A\text{-}\mathrm{mod}$ with finite coresolutions, via $T \mapsto T^\perp = \{M \mid \Ext^i_A(T, M) = 0, i > 0\}$.
• Dually, Wakamatsu tilting modules correspond to contravariantly finite resolving subcategories with finite resolutions using $T \mapsto {}^\perp T = \{M \mid \Ext^i_A(M, T) = 0, i > 0\}$.

These correspondences encapsulate the mechanisms by which tilting theory controls the structure and homological behavior of modules and their Auslander–Reiten (AR) sequences (Divaani-Aazar et al., 12 Dec 2025).

2. Generalization to Arbitrary Rings

Recent work has extended these correspondences beyond Artin algebras to arbitrary associative rings $R$. Two key upgrades are enacted:

• "Covariantly finite" subcategories are replaced by preenveloping subcategories, and "contravariantly finite" by precovering subcategories.
• The isomorphism condition among modules is generalized by introducing an equivalence relation $\sim$ via the action of invertible bimodules.

The main bijections are:

• Equivalence classes $[T]$ of Wakamatsu tilting right $R$-modules correspond to preenveloping, coresolving subcategories containing injectives and admitting an Ext-projective generator.
• Dually, $[T]$ are in bijection with resolving subcategories of $\mathrm{mod}$-$R$ with Ext-injective cogenerator.

The classification incorporates the subtleties of Morita theory—two tilting modules are equivalent if related through invertible bimodules—where all associated homological classes remain invariant (Divaani-Aazar et al., 12 Dec 2025).

3. Mantese–Reiten Theorems in Monomorphism Categories

The Mantese–Reiten phenomenon extends to monomorphism categories $\mathcal{S}_n(A)$, identified as subcategories of the length-$n$ morphism category $\mathrm{Mor}_n(A)$ whose objects are chains of injective maps between $A$-modules. Key results include:

• $\mathcal{S}_n(A)$ is functorially finite in $\mathrm{Mor}_n(A)$; hence, every indecomposable non-projective object of $\mathcal{S}_n(A)$ is at the right end of an almost-split (Auslander–Reiten) sequence.
• The AR translation $\tau_{\mathcal{S}_n}$ is given by

$$\tau_{\mathcal{S}_n} X \cong \mathrm{Mimo}(\tau(\mathrm{Cok}\, X)),$$

where $\mathrm{Mimo}$ and $\mathrm{Cok}$ respectively denote the minimal monomorphism approximation and cokernel computation.

• For selfinjective Nakayama algebras $A(m, t)$, the periodicity formula

$\tau_{\mathcal{S}_n}^{2m(n+1)} X \cong X$

holds, implying the AR translation in monomorphism categories possesses uniform $\mathbb{Z}/(2m(n+1))$-action (Xiong et al., 2011).

These results generalize the classical AR theory from submodule categories $\mathcal{S}_2(A)$, as developed by Ringel–Schmidmeier.

4. Functorially Finite Resolving Subcategories and AR-Quiver Classification

The Mantese–Reiten framework provides generalizations of Brauer–Thrall I and 1½, the Happel–Preiser–Ringel theorem, and Dynkin classification in functorially finite resolving subcategories $\mathcal{X} \subseteq A\text{-}\mathrm{mod}$ (Krebs, 2015):

• A functorially finite resolving subcategory contains only finitely many indecomposable objects if and only if their Jordan–Hölder length is bounded.
• If there exist infinitely many modules of fixed length, then this finitude propagates, guaranteeing infinite representation type.
• Infinite, connected $\tau_\mathcal{X}$-stable AR components with periodic elements are necessarily stable tubes (with mesh type $\mathbb{Z}A_m/\langle\phi\rangle$).
• The left (or right) subgraph type of the AR quiver equals a Dynkin diagram if and only if the subcategory is representation-finite.

Sectional paths, degrees of irreducible morphisms, and helical/coray-tube structures are essential invariants for decomposing morphisms and classifying AR-components, ensuring the extension of classical mesh and tree-theoretic results (Krebs, 2015).

5. Extriangulated Categories and Mantese–Reiten Duality

The notion of extriangulated categories unifies exact, abelian, and triangulated categories by axiomatizing extension classes and realizing conflations. Within this context, Mantese–Reiten-type results hold:

• An almost-split $\mathcal{E}$-extension exists for every indecomposable non-projective object, manifesting via sequences that are characterized by minimality and endo-locality conditions.
• Auslander–Reiten–Serre duality holds: $(\tau, n)$ provides an equivalence between the stable and costable categories $\underline{\mathscr{C}}$ and $\overline{\mathscr{C}}$, with natural isomorphisms relating radical layers and mesh combinatorics.
• The AR theory is preserved under relative extriangulated categories, ideal quotients, and extension-closed functorially finite subcategories.
• The stable category inherits the structure of a $\tau$-category if sufficiently many projectives/injectives and sink/source morphisms exist, enabling import of classical ladder and hammock arguments and explicit reconstruction via mesh category equivalence (Iyama et al., 2018).

6. Technical Features: Periodicity, Dualizations, and Equivalence

A critical technical aspect is the quantification and classification of functors and translations:

• The Serre functor $F_{\mathcal{S}}$ in stable monomorphism categories satisfies $F_{\mathcal{S}}^{m(n+1)} \cong \mathrm{id}$ for Nakayama algebras $A(m, t)$, generalizing the periodicity observed for submodule categories.
• The equivalence relation on tilting modules via invertible bimodules grades the classification results in the general non-Artin context.
• All classical Mantese–Reiten results (existence of AR sequences, duality, mesh, and Dynkin classification) extend to extriangulated categories and their stable/costable categories with virtually unchanged statements and mesh combinatorics.

7. Applications and Advanced Generalizations

The Mantese–Reiten theorems unify disparate strands in tilting theory, representation type classification, Auslander–Reiten quiver geometry, and homological algebra. The most recent extensions encompass:

• Bijective classification of preenveloping/precovering subcategories of $\mathrm{Mod}$-$R$ associated to Wakamatsu tilting and cotilting modules up to equivalence by invertible bimodules (Divaani-Aazar et al., 12 Dec 2025).
• Explicit periodicity and rotation formulas for Auslander–Reiten translations in monomorphism categories (Xiong et al., 2011).
• Mesh-theoretic and combinatorial control of AR-components in functorially finite subcategories (Krebs, 2015).
• Full extriangulated generality, including the inverse problem: realizability of any locally finite symmetrizable $\tau$-quiver as the AR quiver of an extriangulated category (Iyama et al., 2018).

This synthesis establishes the Mantese–Reiten framework as a universal scaffold for module and category classifications in modern representation theory.