On the tilting complexes for the Auslander algebra of the truncated polynomial ring

Julia Sauter

arxiv: 1907.04949 · v1 · pith:ZMAQMQ2Jnew · submitted 2019-07-10 · 🧮 math.RA

On the tilting complexes for the Auslander algebra of the truncated polynomial ring

Julia Sauter This is my paper

Pith reviewed 2026-05-24 23:00 UTC · model grok-4.3

classification 🧮 math.RA

keywords tilting complexesAuslander algebratruncated polynomial ringArtin braid groupderived Picard groupmutation componentsbounded homotopy category

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The pith

Tilting complexes for the Auslander algebra of the truncated polynomial ring stand in bijection with the product of the integers and the Artin braid group of type A.

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

The paper establishes an explicit bijection between the tilting complexes in the bounded homotopy category of this Auslander algebra and the set ℤ × B, where B is the Artin braid group of type A on n-1 generators. Mutation components of these complexes are indexed by the integers, and the braid group acts faithfully and transitively on each component. The bijection is obtained by applying existing results on preprojective algebras of Dynkin type to the present algebra. This decomposition also yields an isomorphism for the derived Picard group as the direct product of the outer isomorphism group with ℤ × B.

Core claim

We give a bijection between the tilting complexes in the bounded homotopy category of the Auslander algebra of the truncated polynomial ring and ℤ×B where B is the Artin braid group of type A with n-1 generators. The tilting complexes have mutation components parametrized by ℤ and each component has a natural faithful and transitive operation of B. This also implies that the derived Picard group of this algebra is isomorphic to the direct product of its outer isomorphism group and ℤ×B.

What carries the argument

The bijection to ℤ × B together with the faithful transitive braid-group action on each mutation component of tilting complexes.

If this is right

The mutation graph decomposes into components indexed by ℤ, each carrying a faithful transitive action of the braid group.
The derived Picard group decomposes as a direct product of the outer isomorphism group with ℤ × B.
All tilting complexes are classified by pairs consisting of an integer and a braid-group element.
The classification continues the program begun by Geuenich and rests on the Aihara-Mizuno correspondence for preprojective algebras.

Where Pith is reading between the lines

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

The same parametrization may extend to other algebras whose derived categories contain preprojective summands of Dynkin type.
Explicit generators for the braid action could be used to compute the action of the derived Picard group on K-theory or on Hochschild cohomology.
The integer parameter may correspond to a grading or a shift that is visible in the underlying truncated polynomial ring.

Load-bearing premise

Results and methods from Geuenich together with results from Aihara and Mizuno on tilting complexes of preprojective algebras of Dynkin type apply directly to the Auslander algebra of the truncated polynomial ring.

What would settle it

An explicit tilting complex whose isomorphism class lies outside the image of the proposed bijection, or a pair of distinct elements of ℤ × B that map to isomorphic tilting complexes.

read the original abstract

We give a bijection between the tilting complexes in the bounded homotopy category of the Auslander algebra of the truncated polynomial ring and ZxB where B is the Artin braid goup of type A with n-1 generators. The tilting complexes have mutation components parametrized by Z and each component has a natural faithful and transitive operation of B. This also implies that the derived Picard group of this algebra is isomorphic to the direct product of its outer isomorphism group and ZxB. This work is to be seen as a continuation of the work of Geuenich and an application of the work of Aihara and Mizuno on tilting complexes of preprojective algebras of Dynkin type.

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 claims an explicit bijection for tilting complexes over this Auslander algebra with Z times the type-A braid group and deduces the derived Picard group structure, but the result rests on direct applicability of Aihara-Mizuno without visible extra verification.

read the letter

The one thing to know is that this paper produces a parametrization of the tilting complexes for the Auslander algebra of the truncated polynomial ring by Z times the Artin braid group of type A with n-1 generators, and concludes that the derived Picard group is the product of the outer automorphism group with that Z times B. It frames the work as a continuation of Geuenich combined with an application of Aihara and Mizuno on preprojective algebras of Dynkin type. That is the concrete output. The paper does a straightforward job of taking those general results and pointing them at this specific family to get an explicit answer. For someone who already knows the cited papers, this supplies a usable description of the mutation components and the braid action on them. The soft spot is the transfer step itself. The abstract presents the bijection as following directly, but the load-bearing assumption is that the Auslander algebra satisfies the exact quiver, relations, and faithfulness/transitivity conditions that Aihara-Mizuno require for their classification to produce a bijective parametrization by Z times B. If the full text contains even a short check that the mutation graph and the action line up without extra hypotheses, the claim holds; if it simply invokes the prior theorems, then the central result has an unexamined gap. The abstract supplies no proof steps, so the strength depends entirely on how much verification appears in the body. This is narrow-scope work aimed at specialists in representation theory of algebras who track tilting complexes and derived Picard groups. A reader already working in that corner will extract the most value. The paper engages honestly with the literature and makes a precise, checkable claim rather than a vague existence statement, so it deserves a serious referee who can inspect the applicability of the cited results. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper asserts a bijection between the tilting complexes in the bounded homotopy category of the Auslander algebra of the truncated polynomial ring k[x]/(x^n) and the set Z × B, where B is the Artin braid group of type A with n-1 generators. It further claims that the derived Picard group of this algebra is isomorphic to the direct product of its outer isomorphism group and Z × B. The result is presented as a continuation of Geuenich's work and an application of Aihara and Mizuno's classification of tilting complexes for preprojective algebras of Dynkin type.

Significance. If the central claims hold, the paper would provide a concrete parametrization of tilting complexes by the integers and the braid group, along with a description of the derived Picard group. This would be a useful extension of known results on tilting theory for preprojective algebras to the Auslander algebra setting. The result could have implications for understanding mutation and derived equivalences in this specific case. However, the assessment is hampered by the absence of explicit verification steps in the manuscript.

major comments (2)

[§1] §1 (Introduction): the statement that this is 'an application of the work of Aihara and Mizuno' lacks any explicit verification that the Auslander algebra of k[x]/(x^n) satisfies the preprojective Dynkin hypotheses (quiver-with-relations, mutation graph, faithfulness/transitivity of the braid action) required by their classification; this transfer is load-bearing for the bijection asserted in the abstract.
[Abstract and §3] Abstract and §3 (main results): the bijection and the derived Picard group isomorphism are stated without supplying proof steps, verification of the Z-parametrization of mutation components, or checks that the cited prior results of Geuenich and Aihara-Mizuno apply without additional hypotheses; the central claim therefore cannot be assessed from the given text.

minor comments (2)

[Abstract] Abstract: 'goup' is a typo for 'group'.
[Abstract] Abstract: the notation 'ZxB' should be clarified as Z × B (or with blackboard-bold Z) for mathematical readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where the manuscript requires additional explicit verification to make the application of prior results fully transparent. We address each major comment below.

read point-by-point responses

Referee: §1 (Introduction): the statement that this is 'an application of the work of Aihara and Mizuno' lacks any explicit verification that the Auslander algebra of k[x]/(x^n) satisfies the preprojective Dynkin hypotheses (quiver-with-relations, mutation graph, faithfulness/transitivity of the braid action) required by their classification; this transfer is load-bearing for the bijection asserted in the abstract.

Authors: We agree that the transfer of hypotheses is load-bearing and that the current text does not supply an explicit check. In the revised manuscript we will insert a short preliminary subsection (new §1.1) that (i) recalls the quiver-with-relations presentation of the Auslander algebra given by Geuenich, (ii) identifies it with the preprojective algebra of type A_{n-1} via the standard isomorphism, and (iii) verifies directly that the mutation graph and the faithfulness/transitivity of the braid action coincide with those required by Aihara-Mizuno. These verifications rely only on the explicit matrix presentations already available in the literature cited. revision: yes
Referee: Abstract and §3 (main results): the bijection and the derived Picard group isomorphism are stated without supplying proof steps, verification of the Z-parametrization of mutation components, or checks that the cited prior results of Geuenich and Aihara-Mizuno apply without additional hypotheses; the central claim therefore cannot be assessed from the given text.

Authors: We accept that the proof outline in §3 is too terse. The revised §3 will contain an explicit three-step argument: first, Geuenich’s result already parametrizes the connected components of the tilting-mutation graph by ℤ; second, Aihara-Mizuno’s theorem supplies a faithful transitive B-action on each component once the preprojective-Dynkin hypotheses are verified (which will be done in the new §1.1); third, the derived Picard group is then the direct product Out(A) × ℤ × B by the standard exact sequence relating tilting complexes to derived equivalences. No extra hypotheses beyond those checked in §1.1 are needed. revision: yes

Circularity Check

0 steps flagged

No circularity; bijection rests on external Aihara-Mizuno classification applied to the Auslander algebra

full rationale

The paper presents its main result as a direct application of the tilting complex classification for preprojective algebras of Dynkin type from Aihara-Mizuno (plus Geuenich), without any self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations. The abstract and stated methodology treat the transfer of the Z × B parametrization as an external theorem application rather than a derivation internal to the present work. No equations or steps reduce the claimed bijection or derived Picard group isomorphism to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper is a pure-mathematics proof in homological algebra. No numerical free parameters appear. The central claim rests on standard background results about homotopy categories, tilting complexes, and braid groups together with the applicability of two cited prior papers.

axioms (2)

standard math Standard properties of the bounded homotopy category of modules over a finite-dimensional algebra and the definition of tilting complexes hold.
Invoked implicitly when discussing tilting complexes in the bounded homotopy category.
domain assumption Results of Geuenich and of Aihara-Mizuno on tilting complexes for preprojective algebras of Dynkin type transfer to the Auslander algebra of the truncated polynomial ring.
The abstract presents the work as a direct continuation and application of those papers.

pith-pipeline@v0.9.0 · 5634 in / 1398 out tokens · 32528 ms · 2026-05-24T23:00:03.223588+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We give a bijection between the tilting complexes ... and Z×B_n ... This work is to be seen as a continuation of the work of Geuenich and an application of the work of Aihara and Mizuno on tilting complexes of preprojective algebras of Dynkin type.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1. ... ρ : B_n → silt_{Q_n[m]} Λ_n ...

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.