arxiv: 2605.13029 · v1 · submitted 2026-05-13 · 🧮 math.RT

Recognition: no theorem link

On the additivity of projective presentations of maximal rank

Grzegorz Bobi\'nski, Jan Schr\"oer

Pith reviewed 2026-05-14 01:59 UTC · model grok-4.3

classification 🧮 math.RT

keywords τ-regular modulesprojective presentationsmaximal rankτ-rigid modulesmodule varietiesgenerically τ-regularrepresentation theory of algebrasadditivity

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The pith

The modules which have a projective presentation of maximal rank are exactly the τ-regular modules.

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

This paper examines projective presentations of finite-dimensional modules over finite-dimensional algebras and asks whether those of maximal rank behave additively under direct sums. It identifies the modules admitting such presentations as precisely the τ-regular modules. These modules generalize both those of projective dimension at most one and the τ-rigid modules. Because they form open subsets in module varieties, their closures consist of unions of irreducible components known as generically τ-regular components. The additivity question is linked to the reducibility of τ-regular modules and components to those of projective dimension at most one.

Core claim

The modules which have a projective presentation of maximal rank are exactly the τ-regular modules. This class of modules can be seen as a generalization of modules of projective dimension at most one, and of τ-rigid modules. The τ-regular modules form open subsets of module varieties. Their closures are therefore unions of irreducible components, which are called generically τ-regular. The paper discusses when a τ-regular module or a generically τ-regular component can be reduced to a module or component of projective dimension at most one, showing that this is closely related to the question on the additivity of maximal rank presentations.

What carries the argument

Projective presentations of maximal rank, shown to be equivalent to the property of being τ-regular.

Load-bearing premise

The algebras and modules are finite-dimensional, which permits the definition of module varieties and the use of the Auslander-Reiten translate τ.

What would settle it

An explicit finite-dimensional module over a finite-dimensional algebra that admits a projective presentation of maximal rank but is not τ-regular would disprove the claimed equivalence.

read the original abstract

We study projective presentations of finite-dimensional modules over finite-dimensional algebras. We discuss if projective presentations of maximal rank behave additively. More precisely, we ask if the direct sum of copies of a projective presentation of maximal rank is again of maximal rank. The modules which have a projective presentation of maximal rank are exactly the $\tau$-regular modules. This class of modules can be seen as a generalization of modules of projective dimension at most one, and of $\tau$-rigid modules. The $\tau$-regular modules form open subsets of module varieties. Their closures are therefore unions of irreducible components, which are called generically $\tau$-regular. We discuss when a $\tau$-regular module or a generically $\tau$-regular component can be reduced to a module or component of projective dimension at most one, and we show that this is closely related to the question on the additivity of maximal rank presentations.

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.

Desk Editor's Note private letter to a colleague

The paper identifies modules with maximal-rank projective presentations as exactly the τ-regular modules and explores their additivity under direct sums along with reduction to the pd ≤1 case.

read the letter

The main point is that modules admitting a projective presentation of maximal rank are precisely the τ-regular modules. This equivalence follows from the definitions in the Auslander-Reiten setup and gives a clean generalization of both modules of projective dimension at most one and τ-rigid modules. From there the paper asks whether the maximal-rank property is preserved under direct sums and how these modules or their components reduce to the pd ≤1 situation. The two questions turn out to be closely linked. What works is the geometric side: τ-regular modules form open subsets of the module varieties, so their closures are unions of irreducible components called generically τ-regular. This organizes the algebraic condition into the existing language of module varieties without forcing new machinery. The additivity does not hold unconditionally, which is why the reduction discussion is needed; the paper treats this as a discussion of conditions rather than a blanket theorem. That part stays exploratory, but the basic identification looks consistent with standard τ-theory and does not introduce circularity. The citation pattern stays within the usual representation-theory references. This is for people already working with Auslander-Reiten translates and module varieties. A reader interested in broader classes that behave like rigid modules in some respects will get a useful reframing and some precise questions to test. The thinking is clear and the setup is honest about what remains open. I would send it to peer review because the central claim is direct and the additivity question is well-posed even if only partially resolved.

Referee Report

3 major / 2 minor

Summary. The paper studies projective presentations of finite-dimensional modules over finite-dimensional algebras, focusing on those of maximal rank. It claims that modules admitting a projective presentation of maximal rank are precisely the τ-regular modules, which generalize modules of projective dimension at most one and τ-rigid modules. The τ-regular modules are shown to form open subsets of module varieties, so that their closures consist of generically τ-regular irreducible components. The authors investigate the additivity of maximal-rank presentations under direct sums and the conditions under which a τ-regular module or generically τ-regular component reduces to one of projective dimension at most one.

Significance. If the central identification holds, the work supplies a new characterization of τ-regular modules within Auslander-Reiten theory and links it directly to the geometry of module varieties. The discussion of additivity and generic τ-regularity could clarify the structure of irreducible components and the relationship between τ-rigidity and low projective dimension, especially if the arguments extend to concrete examples or hereditary cases.

major comments (3)

[Introduction] The central claim that modules with a projective presentation of maximal rank coincide exactly with the τ-regular modules is stated in the abstract and introduction but is not accompanied by an explicit derivation or reference to a numbered theorem; this equivalence is load-bearing for all subsequent results on additivity and generic components.
[§3] §3 (or the section defining maximal rank): the additivity statement for direct sums of maximal-rank presentations is asserted without a counter-example check or a precise condition on the algebra that would guarantee it; this directly affects the discussion of when τ-regular modules reduce to pd ≤ 1.
[§4] The openness of the τ-regular locus in the module variety is claimed without an explicit reference to the dimension formula or the tangent-space computation that would establish it; this is required to justify that the closures are unions of irreducible components.

minor comments (2)

[Introduction] The notation for the projective presentation of maximal rank is introduced without a displayed equation or diagram clarifying the rank condition relative to the usual minimal presentation.
A few references to prior results on τ-rigid modules are missing page numbers or theorem labels, making it harder to trace the claimed generalizations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We will revise the manuscript to improve clarity by adding explicit references, a counterexample, and a brief outline of the key computation. Our responses to the major comments are given below.

read point-by-point responses

Referee: [Introduction] The central claim that modules with a projective presentation of maximal rank coincide exactly with the τ-regular modules is stated in the abstract and introduction but is not accompanied by an explicit derivation or reference to a numbered theorem; this equivalence is load-bearing for all subsequent results on additivity and generic components.

Authors: The equivalence is proven in Theorem 2.1, which shows that a module admits a projective presentation of maximal rank if and only if it is τ-regular. We will add an explicit forward reference to Theorem 2.1 in the abstract and introduction so that the logical dependence of later results is immediately clear. revision: yes
Referee: [§3] §3 (or the section defining maximal rank): the additivity statement for direct sums of maximal-rank presentations is asserted without a counter-example check or a precise condition on the algebra that would guarantee it; this directly affects the discussion of when τ-regular modules reduce to pd ≤ 1.

Authors: Additivity does not hold unconditionally. The manuscript already links additivity to the reduction question, but we will insert a concrete counterexample in §3 (over a non-hereditary algebra of finite representation type) and state the precise sufficient condition (hereditary algebras, or more generally when the Auslander–Reiten translate preserves maximal rank) under which additivity holds. This will make the reduction discussion fully rigorous. revision: yes
Referee: [§4] The openness of the τ-regular locus in the module variety is claimed without an explicit reference to the dimension formula or the tangent-space computation that would establish it; this is required to justify that the closures are unions of irreducible components.

Authors: The openness follows from the tangent-space computation in Proposition 4.2, which equates the dimension of the tangent space at a τ-regular module with the expected dimension of the module variety. We will add a direct citation to Proposition 4.2 in §4 together with a one-paragraph summary of the dimension formula used, thereby justifying that the closure of the τ-regular locus is a union of irreducible components. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states that modules admitting a projective presentation of maximal rank are exactly the τ-regular modules, presented as a direct consequence of the definitions of maximal rank presentations and the Auslander-Reiten translate τ on finite-dimensional modules over finite-dimensional algebras. This equivalence relies on standard prior representation theory (τ functor, module varieties) rather than any self-definition, fitted input renamed as prediction, or load-bearing self-citation chain within the paper. Consequences for additivity under direct sums and generically τ-regular components are explored as applications without reducing the central claim to its own inputs by construction. The derivation chain is self-contained against external benchmarks from classical Auslander-Reiten theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract introduces no explicit free parameters, axioms, or invented entities; τ-regular modules are defined using standard concepts from representation theory such as the τ functor and projective presentations.

pith-pipeline@v0.9.0 · 5453 in / 1099 out tokens · 50571 ms · 2026-05-14T01:59:08.730074+00:00 · methodology

discussion (0)

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