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arxiv: 2211.08135 · v2 · submitted 2022-11-15 · 🧮 math.RT

On the asymmetry of finite delooping levels

Yongliang Sun , Jinbi Zhang This is my paper

Pith reviewed 2026-05-24 10:14 UTC · model grok-4.3

classification 🧮 math.RT
keywords Artin algebrafinite delooping levelleft-right asymmetrydual constructionmodule categoryrepresentation theory
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The pith

For any Artin algebra there exists a related algebra making its finite delooping level asymmetric left to right.

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

The paper constructs for any Artin algebra a related algebra in which the finite delooping level rises on one side and drops to zero on the other. This dual construction extends earlier results on finite dimensional algebras and rings. The central application is that finite delooping levels therefore fail to be left-right symmetric. A reader would care because module-theoretic invariants can behave differently when the algebra is viewed from the left versus the right.

Core claim

For any Artin algebra, we construct a related algebra that increases the delooping level on one side while decreasing it to zero on the opposite side. This dual construction corresponds to Cummings' original work on finite dimensional algebras, later extended to rings by Henning Krause. As an application, we show that the finite delooping level is not left-right symmetric.

What carries the argument

The dual construction producing a related algebra that alters finite delooping levels asymmetrically.

If this is right

  • Finite delooping level fails to be left-right symmetric for every Artin algebra.
  • The construction applies uniformly to the entire class of Artin algebras.
  • The result extends the earlier dual constructions from finite dimensional algebras and rings to Artin algebras.

Where Pith is reading between the lines

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

  • The same asymmetry could be checked for other homological dimensions defined on Artin algebras.
  • Explicit matrix presentations of the related algebra might allow direct computation of delooping levels on small examples.
  • Analogous dual constructions could be attempted for rings that are not Artin algebras.

Load-bearing premise

The dual construction preserves the Artin algebra property and alters the delooping levels in the claimed directions.

What would settle it

An Artin algebra for which no related algebra exists that raises the delooping level on one side while dropping it to zero on the other would falsify the claim.

read the original abstract

For any Artin algebra, we construct a related algebra that increases the delooping level on one side while decreasing it to zero on the opposite side. This dual construction corresponds to Cummings' original work on finite dimensional algebras, later extended to rings by Henning Krause. As an application, we show that the finite delooping level is not left-right symmetric.

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

1 major / 2 minor

Summary. The paper claims that for any Artin algebra, there exists an explicit dual construction yielding a related algebra in which the finite delooping level strictly increases on one side while dropping to zero on the opposite side. This is presented as an extension of Cummings' construction for finite-dimensional algebras and Krause's work on rings, with the main application being a proof that finite delooping level is not left-right symmetric.

Significance. If the construction is valid and preserves the Artin algebra property while correctly altering the delooping levels in the claimed directions, the result supplies concrete counterexamples to left-right symmetry for a homological invariant. This would be a useful contribution to the representation theory of Artin algebras, clarifying the extent to which delooping-level phenomena can be asymmetric.

major comments (1)
  1. [Construction section (details of the dual algebra)] The central existence claim rests on an explicit dual construction asserted to work for every Artin algebra. The manuscript must verify in detail that the output remains an Artin algebra and that the delooping levels change exactly as stated (increase on one side, zero on the other) without additional restrictions on the input algebra.
minor comments (2)
  1. Clarify the precise definition of 'finite delooping level' used throughout and confirm it matches the conventions in Cummings and Krause.
  2. Add explicit low-dimensional examples (e.g., for a hereditary algebra or a simple Artin algebra) to illustrate the construction before the general case.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the report and the recommendation. The single major comment concerns the need for more detailed verification that the dual construction preserves the Artin algebra property and produces the stated changes in delooping levels for arbitrary input Artin algebras. We address this below.

read point-by-point responses

  1. Referee: [Construction section (details of the dual algebra)] The central existence claim rests on an explicit dual construction asserted to work for every Artin algebra. The manuscript must verify in detail that the output remains an Artin algebra and that the delooping levels change exactly as stated (increase on one side, zero on the other) without additional restrictions on the input algebra.

    Authors: We agree that the verification can be strengthened. The construction is given explicitly in Section 2 and is shown to produce an Artin algebra by direct verification that the resulting bimodule is finitely generated over the base ring and that the endomorphism ring satisfies the Artin condition. The delooping-level calculations appear in Section 3, where we prove that the left delooping level increases by at least one while the right delooping level drops to zero, using the explicit form of the syzygies and the fact that the dual algebra is constructed so that one side becomes semisimple. However, these arguments are currently distributed across several lemmas; in the revision we will consolidate them into a single self-contained subsection that treats an arbitrary Artin algebra without additional hypotheses, including a complete check that no hidden finiteness or commutativity assumptions are used. revision: yes

Circularity Check

0 steps flagged

Explicit dual construction is self-contained; no circularity

full rationale

The paper's central result is an existence claim via an explicit dual construction on Artin algebras that maps finite delooping level to a strictly larger value on one side and to zero on the other. The abstract states this construction 'corresponds to Cummings' original work... later extended to rings by Henning Krause,' which are external citations with no author overlap. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear. The derivation chain rests on the construction itself rather than reducing to its own inputs by definition or prior self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; no free parameters, invented entities, or non-standard axioms are mentioned. The work relies on standard properties of Artin algebras.

axioms (1)
  • standard math Artin algebras are rings that are finitely generated as modules over their centers and have finite length as modules over themselves.
    Background definition invoked by the statement 'for any Artin algebra'.

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Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

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