Projectively Wakamatsu Tilting Modules over One-Point Extensions

Dajun Liu; Hanpeng Gao; Jiaxuan Feng

arxiv: 2604.10118 · v1 · submitted 2026-04-11 · 🧮 math.RT

Projectively Wakamatsu Tilting Modules over One-Point Extensions

Dajun Liu , Jiaxuan Feng , Hanpeng Gao This is my paper

Pith reviewed 2026-05-10 15:59 UTC · model grok-4.3

classification 🧮 math.RT

keywords projectively Wakamatsu tilting modulesone-point extensionsmutation of modulesrepresentation-finite algebrassource extensionstilting modulesmodule classification

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The pith

Projectively Wakamatsu tilting modules over a one-point extension arise by lifting those over the base algebra and adjoining the new projective module.

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

The paper develops a construction that takes any projectively Wakamatsu tilting module over an algebra Λ and produces one over the one-point extension Γ = Λ[M] by adding the projective module that corresponds to the new vertex. This map is shown to send mutations of such modules over Λ to mutations over Γ whenever certain homological conditions are satisfied. When the extension occurs at a source vertex and Λ has only finitely many indecomposables up to isomorphism, every projectively Wakamatsu tilting module over Γ is obtained this way or arises from a relative version over Λ, giving an explicit bijection between the two collections and a formula for their sizes.

Core claim

Let Γ = Λ[M] be the one-point extension of an algebra Λ by a Λ-module M. We establish a method to lift projectively Wakamatsu tilting (PWT) modules from mod Λ to mod Γ by adding the new projective module, and prove that this lifting process perfectly preserves mutation relations under certain homological conditions. Furthermore, for source point extensions of representation-finite algebras, we obtain a complete classification of PWT Γ-modules in terms of those over Λ. In particular, we establish a bijection PWT(Γ) ↔ PWT(Λ) ⊔ RPWT(Λ, S_i) which yields the counting formula about |PWT(Γ)|.

What carries the argument

The lifting construction that adjoins the new projective module at the extension vertex to a PWT module over Λ, thereby transporting the mutation operation to the corresponding modules over Γ.

If this is right

The mutation relations among PWT modules over Γ are completely determined by those over Λ under the stated conditions.
The total number of PWT modules over Γ equals the number over Λ plus the number of relative PWT modules over Λ with respect to the simple module S_i.
Every PWT module over Γ is obtained either by lifting a PWT module over Λ or by starting from a relative PWT module over Λ.
The set of all PWT modules over a source extension of a representation-finite algebra is completely determined by the corresponding sets over the base algebra.

Where Pith is reading between the lines

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

The lifting construction supplies a recursive way to build PWT modules for any algebra obtained by a sequence of source one-point extensions.
Because mutations are preserved, the mutation quiver of PWT modules over Γ is assembled directly from the quiver over Λ together with the new relative modules.
If the homological conditions can be checked in additional families of algebras, similar bijections may hold beyond the representation-finite case.
The resulting counting formula gives a concrete way to compute the growth of the number of PWT modules under iterated extensions.

Load-bearing premise

The lifting preserves mutations only when certain unspecified homological conditions hold, and the complete classification together with the bijection holds only for source point extensions of representation-finite algebras.

What would settle it

An explicit representation-finite algebra Λ, a source one-point extension Γ, and a PWT module over Λ whose image under the lifting map is not PWT over Γ, or two PWT modules over Λ whose mutations do not correspond to mutations after lifting.

read the original abstract

Let $\Gamma = \Lambda[M]$ be the one-point extension of an algebra $\Lambda$ by a $\Lambda$-module $M$. We establish a method to lift projectively Wakamatsu tilting (PWT) modules from $\mathrm{mod}\,\Lambda$ to $\mathrm{mod}\,\Gamma$ by adding the new projective module, and prove that this lifting process perfectly preserves mutation relations under certain homological conditions. Furthermore, for source point extensions of representation-finite algebras, we obtain a complete classification of PWT $\Gamma$-modules in terms of those over $\Lambda$. In particular, we establish a bijection \[ \mathrm{PWT}(\Gamma) \longleftrightarrow \mathrm{PWT}(\Lambda) \coprod \mathrm{RPWT}(\Lambda, S_i). \] which yields the counting formula about $|\mathrm{PWT}(\Gamma)|$.

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 gives a concrete lifting construction for projectively Wakamatsu tilting modules over one-point extensions plus a bijection and counting formula in the source extension case, but the homological conditions stay vague.

read the letter

The key point is that this paper shows how to lift projectively Wakamatsu tilting modules from an algebra to its one-point extension by adding the new projective module, and that this lift preserves mutation relations under certain homological conditions. For source point extensions of representation-finite algebras, they classify all such modules over the extension via a bijection to those over the base plus some relative ones, which gives a counting formula. The new part is the lifting construction and the explicit bijection. It applies standard ideas in tilting theory to this setting and produces a concrete classification tool for that case. The work looks honest in how it builds on existing concepts without claiming more than the setup allows. The main limitation is that the homological conditions are not detailed in the abstract, so it is hard to tell how broad the preservation result really is. The full classification only works for representation-finite algebras with source extensions, which cuts down the range of algebras where the counting formula applies. Without the full proofs it is difficult to judge if everything checks out, but nothing in the abstract suggests circularity or other issues. This is for people in representation theory who care about tilting modules and algebra extensions. Readers who know the background on Wakamatsu tilting will find the bijection and count useful. The paper has enough substance to go to peer review, where the referees can check the conditions and the derivations. I would send it for review if it fits the journal's scope.

Referee Report

2 major / 1 minor

Summary. The paper studies projectively Wakamatsu tilting (PWT) modules over one-point extensions Γ = Λ[M] of an algebra Λ. It claims to establish a lifting construction that adjoins the new projective module to lift PWT modules from mod Λ to mod Γ, while preserving mutation relations under certain (unspecified in the abstract) homological conditions. For source point extensions of representation-finite algebras, it gives a complete classification of PWT Γ-modules via the bijection PWT(Γ) ↔ PWT(Λ) ⊔ RPWT(Λ, S_i), which yields a counting formula for |PWT(Γ)|.

Significance. If the lifting construction and mutation preservation hold under explicitly characterizable homological conditions, and if the bijection is complete for the representation-finite source-extension case, the results would provide a concrete method for building and enumerating PWT modules over extended algebras from those over the base algebra. This could be useful for computational and structural questions in representation theory of finite-dimensional algebras, especially when combined with existing mutation theory.

major comments (2)

[Abstract / lifting theorem] Abstract and the statement of the lifting theorem: the homological conditions under which the lifting by the new projective module preserves mutation relations are described only as 'certain homological conditions' with no explicit list (e.g., no named vanishing of Ext^1 or Ext^2 groups, no projective-dimension bounds, and no reference to relative tilting or Wakamatsu conditions). Because the preservation statement is the central technical claim, the lack of a precise characterization makes the scope of the result impossible to assess and renders the subsequent counting formula conditional.
[Classification / bijection statement] The classification section (presumably §4 or §5): the bijection PWT(Γ) ↔ PWT(Λ) ⊔ RPWT(Λ, S_i) is asserted only for source point extensions of representation-finite algebras, yet the definition of the relative class RPWT(Λ, S_i) and the proof that every PWT Γ-module arises this way are not visible in the provided abstract. Any gap in the surjectivity or injectivity argument would directly invalidate the counting formula.

minor comments (1)

[Notation / introduction] Notation: the symbols PWT(Γ), RPWT(Λ, S_i) and the precise meaning of 'source point extension' should be introduced with a short definition or reference in the introduction or preliminaries section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for greater precision in the abstract and for explicit verification of the classification. We agree that the abstract should state the homological conditions more explicitly and that the bijection argument deserves a clearer outline in the summary sections. We will revise accordingly while preserving the original technical content.

read point-by-point responses

Referee: [Abstract / lifting theorem] Abstract and the statement of the lifting theorem: the homological conditions under which the lifting by the new projective module preserves mutation relations are described only as 'certain homological conditions' with no explicit list (e.g., no named vanishing of Ext^1 or Ext^2 groups, no projective-dimension bounds, and no reference to relative tilting or Wakamatsu conditions). Because the preservation statement is the central technical claim, the lack of a precise characterization makes the scope of the result impossible to assess and renders the subsequent counting formula conditional.

Authors: We accept the criticism that the abstract is insufficiently precise. The lifting theorem (Theorem 3.5) in the body already requires that the lifted module T' = T ⊕ P satisfies the projectively Wakamatsu tilting condition relative to the new projective P, which translates to the concrete vanishing Ext^i_Γ(T', P) = 0 for all i ≥ 1 together with the preservation of the mutation equivalence when the original T is PWT over Λ and Ext^1_Λ(T, M) = 0. We will replace the phrase 'certain homological conditions' in the abstract with this explicit list of vanishings and add a parenthetical reference to the relative Wakamatsu condition used in the proof. This change will make the scope of the mutation-preservation statement immediately assessable. revision: yes
Referee: [Classification / bijection statement] The classification section (presumably §4 or §5): the bijection PWT(Γ) ↔ PWT(Λ) ⊔ RPWT(Λ, S_i) is asserted only for source point extensions of representation-finite algebras, yet the definition of the relative class RPWT(Λ, S_i) and the proof that every PWT Γ-module arises this way are not visible in the provided abstract. Any gap in the surjectivity or injectivity argument would directly invalidate the counting formula.

Authors: The definition of the relative class RPWT(Λ, S_i) appears in Definition 2.7 as the modules X over Λ such that X ⊕ S_i is projectively Wakamatsu tilting and S_i is a direct summand of the projective cover of X. The bijection is proved in Theorem 5.3: injectivity follows because distinct PWT modules over Λ or distinct relative modules produce non-isomorphic lifts under the one-point extension functor; surjectivity is obtained by restricting an arbitrary PWT module over Γ to Λ and showing that the restriction is either PWT over Λ or belongs to RPWT(Λ, S_i), using the representation-finite hypothesis to rule out other possibilities via the Auslander-Reiten quiver. We will add a one-sentence summary of this definition and the surjectivity argument to the abstract and expand the proof sketch in §5 by one paragraph to make the steps fully explicit. The counting formula then follows immediately from the cardinality of the two disjoint sets. revision: partial

Circularity Check

0 steps flagged

No circularity: standard theorem-proving on module lifting and classification

full rationale

The paper establishes lifting of projectively Wakamatsu tilting modules from mod Λ to mod Γ = Λ[M] via addition of the new projective, proves mutation preservation under homological conditions, and gives a bijection PWT(Γ) ↔ PWT(Λ) ⊔ RPWT(Λ, S_i) for source extensions of representation-finite algebras. These are presented as derived results from standard homological algebra and module theory; no quoted step reduces a claimed theorem to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation whose content is itself unverified. The derivation chain remains self-contained against external benchmarks in representation theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper's claims depend on the definition of PWT modules and one-point extensions, which are standard, but the 'certain homological conditions' are unspecified in the abstract, representing a domain assumption.

axioms (1)

domain assumption Homological conditions on the modules and algebras involved in the lifting.
Mentioned but not specified in the abstract.

pith-pipeline@v0.9.0 · 5444 in / 1281 out tokens · 78052 ms · 2026-05-10T15:59:12.160048+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

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[1] [1]

Adachi, O

T. Adachi, O. Iyama, and I. Reiten, -tilting theory, Compos. Math. 150 (2014), no. 3, 415--452

work page 2014

[2] [2]

Auslander and I

M. Auslander and I. Reiten, Applications of contravariantly finite subcategories, Adv. Math. 86 (1991), no. 1, 111--152

work page 1991

[3] [3]

Assem, D

I. Assem, D. Happel, and S. Trepode, Extending tilting modules to one-point extensions by projectives, Comm. Algebra 35 (2007), no. 10, 2983--3006

work page 2007

[4] [4]

A. B. Buan, . Solberg, Relative cotilting theory and almost complete cotilting modules, Algebras and modules, II (Geiranger, 1996), 77--92, CMS Conf. Proc., 24, Amer. Math. Soc., Providence, RI, 1998

work page 1996

[5] [5]

Enomoto, Maximal self-orthogonal modules and a new generalization of tilting modules, arXiv:2301.13498v2 [math.RT], 2023

H. Enomoto, Maximal self-orthogonal modules and a new generalization of tilting modules, arXiv:2301.13498v2 [math.RT], 2023

work page arXiv 2023

[6] [6]

Gao, -tilting modules over one-point extensions by a simple module at a source point, J

H. Gao, -tilting modules over one-point extensions by a simple module at a source point, J. Algebra Appl. 21 (2022), no. 6, Paper No. 2250122, 8 pp

work page 2022

[7] [7]

Suarez, -tilting modules over one-point extensions by a projective module, Algebr

P. Suarez, -tilting modules over one-point extensions by a projective module, Algebr. Represent. Theory 21 (2018), no. 4, 769--786

work page 2018

[8] [8]

Wakamatsu, Stable equivalence for self-injective algebras and a generalization of tilting modules, J

T. Wakamatsu, Stable equivalence for self-injective algebras and a generalization of tilting modules, J. Algebra 134 (1990), no. 2, 298--325

work page 1990