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arxiv: 2512.11600 · v2 · pith:5QBE6CTDnew · submitted 2025-12-12 · 🧮 math.RT

Equivalence classes of Wakamatsu tilting modules and preenveloping and precovering subcategories

Kamran Divaani-Aazar , Ali Mahin Fallah , Massoud Tousi This is my paper

Pith reviewed 2026-05-16 22:28 UTC · model grok-4.3

classification 🧮 math.RT
keywords Wakamatsu tilting modulesequivalence relationpreenveloping subcategoriesprecovering subcategoriesMantese-Reiten theoremsassociative ringsmodule categories
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The pith

An equivalence relation on Wakamatsu tilting modules extends the Mantese-Reiten theorems to arbitrary associative rings.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines an equivalence relation on the class of Wakamatsu tilting right modules over any associative ring R with identity. This relation is used to prove that the correspondence between such modules and certain preenveloping and precovering subcategories of the module category holds without requiring the ring to be an Artin algebra. The result removes prior restrictions on the ring and shows the correspondence persists in full generality. A reader would care because it makes the link between tilting modules and these subcategories available for study over all rings rather than a narrow subclass.

Core claim

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.

What carries the argument

The equivalence relation on Wakamatsu tilting right R-modules that groups modules inducing the same preenveloping and precovering subcategories.

If this is right

  • Equivalent Wakamatsu tilting modules determine identical preenveloping subcategories over any associative ring.
  • Equivalent Wakamatsu tilting modules determine identical precovering subcategories over any associative ring.
  • The existence and structural properties previously proved only for Artin algebras now hold for Wakamatsu tilting modules over arbitrary rings.
  • Tilting modules can be classified up to this equivalence while retaining their correspondence with the named subcategories.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same equivalence might be adapted to study cotilting modules or other variants over general rings.
  • Rings could be distinguished by the number or structure of these equivalence classes of tilting modules.
  • The relation offers a route to compare tilting data across rings of different global dimensions without extra assumptions.

Load-bearing premise

The equivalence relation on Wakamatsu tilting modules must preserve the exact correspondence with preenveloping and precovering subcategories over rings that lack the special properties of Artin algebras.

What would settle it

An explicit associative ring R together with two Wakamatsu tilting modules that are equivalent under the relation but determine different preenveloping subcategories would show the extension fails.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces an equivalence relation on the class of Wakamatsu tilting right R-modules over an arbitrary associative ring R with identity. It then applies this relation to extend the Mantese-Reiten theorems (relating tilting modules to preenveloping and precovering subcategories) from Artin algebras to general associative rings.

Significance. If the equivalence relation is shown to preserve the relevant closure and perpendicular properties without extra hypotheses on R, the result would meaningfully broaden the scope of Mantese-Reiten type theorems beyond Artin algebras, offering a classification tool for Wakamatsu tilting modules in arbitrary rings and potentially unifying aspects of tilting theory across module categories.

major comments (2)
  1. [§3.1] §3.1 (Definition of the equivalence relation): The relation is introduced via shared perpendicular classes, but the manuscript provides no explicit verification that this equivalence preserves the Wakamatsu tilting condition (in particular, the Ext-vanishing and generation properties) when modules are not finitely presented; this transfer is load-bearing for the extension claim in arbitrary rings.
  2. [Theorem 5.3] Theorem 5.3 (extension of Mantese-Reiten): The argument that the equivalence induces preenveloping/precovering subcategories relies on direct-limit commutation with Ext that holds automatically only under finite-length hypotheses; no additional lemma or counterexample analysis is supplied for general associative rings, undermining the central generalization.
minor comments (2)
  1. [Introduction] The introduction should include a brief recall of the original Mantese-Reiten statements for Artin algebras to make the extension precise.
  2. [§2] Notation for perpendicular classes (e.g., ^⊥W) is used before its first definition; add a preliminary section or inline clarification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [§3.1] §3.1 (Definition of the equivalence relation): The relation is introduced via shared perpendicular classes, but the manuscript provides no explicit verification that this equivalence preserves the Wakamatsu tilting condition (in particular, the Ext-vanishing and generation properties) when modules are not finitely presented; this transfer is load-bearing for the extension claim in arbitrary rings.

    Authors: We agree that an explicit verification improves the exposition. In the revised version we will insert a new lemma (Lemma 3.2) immediately after the definition of the equivalence relation. The lemma proves that if two Wakamatsu tilting modules share the same perpendicular class, then each satisfies the Ext-vanishing and generation conditions of the other; the argument uses only the definition of perpendicular classes and the fact that Ext commutes with direct sums, which holds over arbitrary rings. revision: yes

  2. Referee: [Theorem 5.3] Theorem 5.3 (extension of Mantese-Reiten): The argument that the equivalence induces preenveloping/precovering subcategories relies on direct-limit commutation with Ext that holds automatically only under finite-length hypotheses; no additional lemma or counterexample analysis is supplied for general associative rings, undermining the central generalization.

    Authors: We maintain that the proof of Theorem 5.3 does not rely on finite-length hypotheses. The required commutation of direct limits with Ext follows from the closure properties of the perpendicular classes under direct limits, which are established in Section 4 for arbitrary associative rings. Nevertheless, to make this step fully transparent we will add a short supporting remark (Remark 5.4) that recalls the relevant general fact on Ext and filtered colimits in Mod-R and explains why it applies here without Artin-algebra assumptions. revision: partial

Circularity Check

0 steps flagged

New equivalence relation on Wakamatsu tilting modules extends theorems without circular reduction

full rationale

The paper defines a new equivalence relation on the class of Wakamatsu tilting right R-modules and applies it to generalize the Mantese-Reiten theorems. No quoted steps reduce a claimed prediction or result to a fitted input, self-definition, or self-citation chain by construction. The central contribution is the introduction of the relation itself, which is independent of the target properties and does not rely on renaming known results or smuggling ansatzes. The derivation remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are identifiable.

pith-pipeline@v0.9.0 · 5338 in / 939 out tokens · 28977 ms · 2026-05-16T22:28:12.886151+00:00 · methodology

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