The brick chain complexity of an artin algebra

Claus Michael Ringel

arxiv: 2606.27879 · v1 · pith:RQF2V72O · submitted 2026-06-26 · math.RT

The brick chain complexity of an artin algebra

Claus Michael Ringel This is my paper

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classification math.RT

keywords Artin algebrabrick chain filtrationbrick chain complexityindecomposable modulemodule filtrationrepresentation theory of algebras

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The pith

Artin algebras exist with arbitrarily large brick chain complexity.

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

Modules over Artin algebras always admit brick chain filtrations. The brick chain complexity of an algebra is the largest minimal filtration length needed for any of its indecomposable modules. The paper shows how to build algebras where this number can be made larger than any given bound. This establishes that the complexity is unbounded in the class of Artin algebras.

Core claim

We consider the category of finitely generated modules over an Artin algebra A. It is known that any module M has a brick chain filtration. We say that M has brick chain complexity at most t provided M has a brick chain filtration of length at most t. The brick chain complexity of A is by definition the supremum of the brick chain complexity of the indecomposable A-modules. The aim of this note is to calculate the brick chain complexity for some algebras. We will exhibit algebras with arbitrarily large brick chain complexity.

What carries the argument

The brick chain filtration of a module, a chain of submodules whose successive quotients are bricks, whose shortest length defines the module complexity and whose supremum over indecomposables defines the algebra complexity.

Load-bearing premise

Every finitely generated module over an Artin algebra admits a brick chain filtration.

What would settle it

A proof that brick chain complexity is bounded above by some fixed number for every Artin algebra.

read the original abstract

We consider the category of finitely generated modules over an artin algebra $A$. It is known that any module $M$ has a brick chain filtration. We say that M has brick chain complexity at most $t$ provided $M$ has a brick chain filtration of length at most $t$. The brick chain complexity of A is by definition the supremum of the brick chain complexity of the indecomposable $A$-modules. The aim of this note is to calculate the brick chain complexity for some algebras. We will exhibit algebras with arbitrarily large brick chain complexity.

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

Ringel defines brick chain complexity via filtrations on indecomposables and claims it is unbounded for some Artin algebras, but the note rests on an un-referenced existence statement for the filtrations themselves.

read the letter

The core of this note is a new definition: brick chain complexity of an Artin algebra A is the supremum, over indecomposable modules, of the shortest brick chain filtration length for that module. The author states that every module admits such a filtration as background and then claims to exhibit algebras where the resulting number is arbitrarily large.

What the paper does is introduce this numerical invariant cleanly and set a concrete existence goal. That is the actual novelty at the level of the text. The constructions, if they are explicit in the body, would give the first examples where this particular measure grows without bound.

The soft spot is exactly the one flagged in the stress-test. The definition only makes sense if the filtrations exist for every finitely generated module, yet the abstract presents this as known without a citation or short argument. If the full note supplies a reference to prior work on brick chains, the issue is minor and easily fixed. If it does not, readers have to take the premise on trust, which weakens the claim that the complexity can be made arbitrarily large. No concrete algebras or filtration lengths appear in the abstract, so the unboundedness result cannot be checked from the given text alone.

This is a short, focused note aimed at people already working inside the representation theory of Artin algebras, especially those tracking numerical invariants on module categories. A specialist who already knows the background on bricks and filtrations will see the definition immediately and can evaluate the examples once they are written down.

The work shows clear engagement with the literature and a well-posed question, so it deserves a serious referee even though the note is brief and the central premise needs a reference. I would send it to peer review.

Referee Report

1 major / 0 minor

Summary. The manuscript defines the brick chain complexity of an Artin algebra A as the supremum, over all indecomposable finitely generated A-modules M, of the minimal length of a brick chain filtration of M. It states as background that every such module admits at least one brick chain filtration, and announces the intention to compute the invariant for certain algebras while exhibiting examples where the complexity is arbitrarily large.

Significance. If the announced constructions are carried out correctly, the result would establish that brick chain complexity is an unbounded invariant on the class of Artin algebras. This could furnish a new numerical measure of the complexity of module categories and might be useful for distinguishing representation types or for studying filtrations by bricks.

major comments (1)

[Abstract] Abstract: the definition of brick chain complexity presupposes that every finitely generated module possesses a brick chain filtration, yet the manuscript states this only as 'it is known' without supplying a reference, a citation, or a self-contained argument. This premise is load-bearing for the entire note, because the supremum is undefined if even one indecomposable module lacks such a filtration; without it the subsequent claim of exhibiting algebras with arbitrarily large complexity cannot be interpreted.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We address the major comment below and will make the necessary revisions to improve the paper.

read point-by-point responses

Referee: [Abstract] Abstract: the definition of brick chain complexity presupposes that every finitely generated module possesses a brick chain filtration, yet the manuscript states this only as 'it is known' without supplying a reference, a citation, or a self-contained argument. This premise is load-bearing for the entire note, because the supremum is undefined if even one indecomposable module lacks such a filtration; without it the subsequent claim of exhibiting algebras with arbitrarily large complexity cannot be interpreted.

Authors: We acknowledge that citing or proving the existence of brick chain filtrations for all finitely generated modules is essential for the definition to be rigorous. Upon revision, we will either provide a suitable reference from the literature on Artin algebras and module filtrations or include a brief self-contained argument establishing this fact. This will resolve the concern and ensure the subsequent results are well-defined. revision: yes

Circularity Check

0 steps flagged

No circularity; definition uses external background fact, constructions are independent

full rationale

The paper states as background that every module admits a brick chain filtration and defines complexity as the sup of minimal filtration lengths over indecomposables. The central result is an explicit construction of algebras realizing arbitrarily large values of this sup. No equation or claim reduces a derived quantity to a fitted input, self-citation, or definitional tautology; the existence premise is external to the paper's derivations and the exhibited examples stand on their own constructions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The definition rests on the background statement that every module admits a brick chain filtration; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption Any module M over an Artin algebra has a brick chain filtration.
Explicitly stated as known in the abstract.

pith-pipeline@v0.9.1-grok · 5607 in / 1048 out tokens · 43124 ms · 2026-06-29T02:21:30.395443+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 2 canonical work pages · 2 internal anchors

[1]

The brick chain complexity of an artin algebra

Brick chain ﬁltrations. 1.1.We deal with an artin algebraA; the modules to be considered are the left A-modules of ﬁnite length. Given a setXof modules, letE(X) be the class of modules which have a ﬁltration with all factors inX. We recall that abrickis a module whose endomorphism ring is a division ring. IfBis a brick, the modules inE(B) will be said to ...

work page internal anchor Pith review Pith/arXiv arXiv 2026
[2]

foundation bricks

Brick chain complexity. Deﬁnition and some examples. 2.1.As we have mentioned, a moduleMis said to havebrick chain complexityat mostt provided there is a brick chain ﬁltration with at mosttfactors. Thebrick chain complexity of an algebraAis the supremum of the brick chain complexity of the indecomposable A-modules (it is a natural number or∞). 2.2. Algebr...
[3]

A brickBis said to be afoundation brickfor the moduleMprovided there is a brick chain ﬁltration ofMof type (B 1,B 2,...,B m ) withB 1 =B

Foundation bricks. A brickBis said to be afoundation brickfor the moduleMprovided there is a brick chain ﬁltration ofMof type (B 1,B 2,...,B m ) withB 1 =B. 3.1. Lemma.LetBbe a foundation brick forM. Then the sumt BMof the images B→Mis a direct sum of copies ofB. We recall that the sumt B (M) of the images of the mapsB→Mis called thetrace ofBinM. Proof: S...
[4]

4.1.For anyn≥3, there is a directed gentle algebras withnsimple modules, with brick chain complexity equal ton

Some algebras with ﬁnite brick chain complexity. 4.1.For anyn≥3, there is a directed gentle algebras withnsimple modules, with brick chain complexity equal ton. Proof. We consider the algebraAwith vertices 1,2,...,nand two arrowsi⇔i+1 for 1≤i<n(always labeledαandβ). As relations, we take all the wordsαβandβα, thus Ais gentle and directed. 1 2 3 ... n−1 n ...
[5]

As a preparation for section 6, we include some observations about Ktronecker mod- ules; these are the modules over the Kronecker algebra

The brick chain ﬁltrations of a regular Kronecker module. As a preparation for section 6, we include some observations about Ktronecker mod- ules; these are the modules over the Kronecker algebra. The Kronecker algebraAis the 6 path algebra of the quiver with two vertices 1,2 and two arrows 1⇔2. There is the well- known trisection of the indecomposableA-m...
[6]

An algebra with inﬁnite brick chain complexity. 6.1. The algebra.Let us present an algebraAwith brick chain complexity∞. Take the path algebra with two simple modules 1,2, with two arrows 1⇔2, two arrows 2⇔1 and a loop at 1; we assume that 1 is a node (that means: all pathsx←1←yare zero relations; there are ﬁve such paths), and that there is no other rela...
[7]

References. [R1] C. M. Ringel. Brick chain ﬁltrations. A report. arXiv:2411.18427 [R2] C. M. Ringel. In preparation. Claus Michael Ringel Fakult¨ at f¨ ur Mathematik, Universit¨ at Bielefeld POBox 100131, D-33501 Bielefeld, Germany ringel@math.uni-bielefeld.de 9

work page internal anchor Pith review Pith/arXiv arXiv

[1] [1]

The brick chain complexity of an artin algebra

Brick chain ﬁltrations. 1.1.We deal with an artin algebraA; the modules to be considered are the left A-modules of ﬁnite length. Given a setXof modules, letE(X) be the class of modules which have a ﬁltration with all factors inX. We recall that abrickis a module whose endomorphism ring is a division ring. IfBis a brick, the modules inE(B) will be said to ...

work page internal anchor Pith review Pith/arXiv arXiv 2026

[2] [2]

foundation bricks

Brick chain complexity. Deﬁnition and some examples. 2.1.As we have mentioned, a moduleMis said to havebrick chain complexityat mostt provided there is a brick chain ﬁltration with at mosttfactors. Thebrick chain complexity of an algebraAis the supremum of the brick chain complexity of the indecomposable A-modules (it is a natural number or∞). 2.2. Algebr...

[3] [3]

A brickBis said to be afoundation brickfor the moduleMprovided there is a brick chain ﬁltration ofMof type (B 1,B 2,...,B m ) withB 1 =B

Foundation bricks. A brickBis said to be afoundation brickfor the moduleMprovided there is a brick chain ﬁltration ofMof type (B 1,B 2,...,B m ) withB 1 =B. 3.1. Lemma.LetBbe a foundation brick forM. Then the sumt BMof the images B→Mis a direct sum of copies ofB. We recall that the sumt B (M) of the images of the mapsB→Mis called thetrace ofBinM. Proof: S...

[4] [4]

4.1.For anyn≥3, there is a directed gentle algebras withnsimple modules, with brick chain complexity equal ton

Some algebras with ﬁnite brick chain complexity. 4.1.For anyn≥3, there is a directed gentle algebras withnsimple modules, with brick chain complexity equal ton. Proof. We consider the algebraAwith vertices 1,2,...,nand two arrowsi⇔i+1 for 1≤i<n(always labeledαandβ). As relations, we take all the wordsαβandβα, thus Ais gentle and directed. 1 2 3 ... n−1 n ...

[5] [5]

As a preparation for section 6, we include some observations about Ktronecker mod- ules; these are the modules over the Kronecker algebra

The brick chain ﬁltrations of a regular Kronecker module. As a preparation for section 6, we include some observations about Ktronecker mod- ules; these are the modules over the Kronecker algebra. The Kronecker algebraAis the 6 path algebra of the quiver with two vertices 1,2 and two arrows 1⇔2. There is the well- known trisection of the indecomposableA-m...

[6] [6]

An algebra with inﬁnite brick chain complexity. 6.1. The algebra.Let us present an algebraAwith brick chain complexity∞. Take the path algebra with two simple modules 1,2, with two arrows 1⇔2, two arrows 2⇔1 and a loop at 1; we assume that 1 is a node (that means: all pathsx←1←yare zero relations; there are ﬁve such paths), and that there is no other rela...

[7] [7]

References. [R1] C. M. Ringel. Brick chain ﬁltrations. A report. arXiv:2411.18427 [R2] C. M. Ringel. In preparation. Claus Michael Ringel Fakult¨ at f¨ ur Mathematik, Universit¨ at Bielefeld POBox 100131, D-33501 Bielefeld, Germany ringel@math.uni-bielefeld.de 9

work page internal anchor Pith review Pith/arXiv arXiv