Homological Invariants of Left and Right Serial Path Algebras

Ruoyu Guo

REVIEW 3 major objections 4 minor 23 references

Homological Invariants of Left and Right Serial Path Algebras

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Pith's one-line read For right serial path algebras, the right finitistic dimension equals the delooping level of the opposite algebra, and the derived delooping level agrees as well.

desk verdict New results on delooping levels for serial algebras, but the main lemma needs a fuller proof before I'd trust the equalities. read the letter →

arxiv 2506.02307 v1 pith:IUVHGDEX submitted 2025-06-02 math.RT math.RA

Ruoyu Guo This is my paper

classification math.RTmath.RA MSC 16E1016G2016E05

keywords deloopinglevelderivedfinitisticdimensionconjecturerightserialalgebrasleftmonomialpathsyzygiescompletionsequences

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that for a right serial path algebra $\Lambda$, the right little and big finitistic dimensions coincide and equal the delooping level and derived delooping level of the opposite algebra: $\operatorname{findim}\Lambda = \operatorname{Findim}\Lambda = \operatorname{dell}\Lambda^{\mathrm{op}} = \operatorname{ddell}\Lambda^{\mathrm{op}} < \infty$. This converts the finitistic dimension into a quantity computed by a finite algorithm over simple modules. For left serial algebras, the same equality is shown under a condition on where the extremal right-completion sequences live in the quiver, and a representation-finite example shows that without that condition the delooping level can overshoot while the derived delooping level still equals the finitistic dimension: $\operatorname{Findim}\Lambda = \operatorname{ddell}\Lambda^{\mathrm{op}} < \operatorname{dell}\Lambda$. The proof engine is the reversal of maximal right-completion sequences in the opposite quiver.

What carries the argument

The machinery is the right-completion sequence. For a monomial algebra $\Lambda = KQ/I$, a path $\beta$ is a right completion of a nonzero path $\alpha$ if $\alpha\beta$ is a minimal relation but no proper right factor of $\beta$ already kills $\alpha$; Lemma 3.3 identifies the syzygy of the right module $\alpha\Lambda$ with the direct sum of the modules $\beta\Lambda$ for all right completions $\beta$ of $\alpha$. The main proofs reverse a maximal sequence of right completions in the opposite quiver $Q^{\mathrm{op}}$ to obtain a path $p_n$ in $Q$, and Lemma 3.4 asserts that the reversed sequence carries at least $n$ minimal relations and forces $\operatorname{pd}_{\Lambda}(\Lambda/p_n\Lambda) \ge n+1$. In right serial algebras, every vertex has at most one outgoing arrow, so the relevant completion sequences are unique and the lower bound becomes an equality, which is how $\operatorname{dell}\Lambda^{\mathrm{op}} \le \operatorname{findim}\Lambda$ is obtained.

What would settle it

Pick a small right serial path algebra, enumerate every maximal right-completion sequence in $Q^{\mathrm{op}}$, reverse each one, and compute $\operatorname{pd}_{\Lambda}(\Lambda/p\Lambda)$ directly by resolving the principal right ideal; finding one reversed sequence of length $n$ whose quotient has projective dimension at most $n$ would refute Lemma 3.4 and collapse Theorems 4.2 and 4.6.

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Extended reading notes

Core claim

The central discovery is an exact equality chain for right serial path algebras: the right finitistic dimensions agree with the left delooping level of the opposite algebra, and the derived delooping level is trapped between them, hence also equal. In symbols, for every right serial path algebra $\Lambda$, $\operatorname{findim}\Lambda = \operatorname{Findim}\Lambda = \operatorname{dell}\Lambda^{\mathrm{op}} = \operatorname{ddell}\Lambda^{\mathrm{op}} < \infty$, and the common value is realized by the projective dimension of a module of the form $\Lambda/p\Lambda$ for some path $p$. For left serial path algebras, the paper proves the same equality whenever every simple $\Lambda^{\mathrm{op}}$-module attaining $\operatorname{dell}\Lambda^{\mathrm{op}}$ has its corresponding right-completion sequence supported entirely in the cyclic part or entirely in the tree part of the quiver. It also constructs a representation-finite left serial algebra where $\operatorname{dell}\Lambda$ strictly exceeds the finitistic dimension, while $\operatorname{ddell}\Lambda^{\mathrm{op}}$ does not, showing the derived delooping level is the sharper upper bound.

Load-bearing premise

The equality chain rests on Lemma 3.4, which asserts that reversing a right-completion sequence of length $n$ always produces at least $n$ minimal relations and a quotient module $\Lambda/p_n\Lambda$ of projective dimension at least $n+1$; the proof of that lemma, especially the syzygy formula for the higher syzygies, is only sketched.

Editorial extensions

If this is right

For right serial path algebras, the finitistic dimension conjecture holds, and the finitistic dimension can be calculated by the finite algorithm for delooping levels of simple modules over the opposite algebra.
The common value is always realized by a module of the form $\Lambda/p\Lambda$, so a single principal right ideal quotient witnesses the finitistic dimension.
For Nakayama algebras, the equalities imply $\operatorname{dell}\Lambda = \operatorname{dell}\Lambda^{\mathrm{op}} = \operatorname{findim}\Lambda = \operatorname{Findim}\Lambda$ and the analogous equalities for the derived delooping level, recovering left-right symmetry of these invariants.
For left serial algebras satisfying the support condition, the same equality chain holds, giving a new sufficient condition for the finitistic dimension conjecture in this class.
Example 4.7 shows that even when the delooping level overshoots, the derived delooping level can still equal the big finitistic dimension, providing evidence that $\operatorname{ddell}\Lambda^{\mathrm{op}}$ is a tighter and more reliable upper bound.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The reversal technique may transfer to other monomial algebras with bounded outdegree, suggesting a quantitative bound on $\operatorname{dell}\Lambda^{\mathrm{op}} - \operatorname{Findim}\Lambda$ in terms of the maximum number of outgoing arrows per vertex, which would address Question 4.8.
The support condition in Theorem 4.6 is purely combinatorial and can be checked by inspecting only the finitely many extremal simple modules, so the theorem is directly usable as an algorithm for left serial examples.
One could search for left serial algebras whose quiver has extra branches entering the cycle to push $\operatorname{dell}\Lambda^{\mathrm{op}} - \operatorname{Findim}\Lambda$ beyond 1; a positive construction would answer Question 4.9 and simplify the known example of arbitrarily large gap.
If a counterexample to the finitistic dimension conjecture exists, the paper's perspective suggests it must admit sequences of right completions whose reversal produces modules of infinite projective dimension in a way that delooping levels cannot detect.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The headline result here is real and worth knowing: right serial path algebras join the list of algebras where Findim Λ = dell Λ^op, and Example 4.7 gives a clean left-serial case where ddell Λ^op succeeds and dell Λ^op does not. That is a genuine extension of the delooping-level program, and the paper deserves a close look.

What is good: the paper gives a uniform method (reverse maximal right-completions) that recovers known results for Nakayama and acyclic monomial algebras, and it isolates a specific structure—left serial quivers whose extremal sequences cross between cyclic and tree parts—that explains when the equality can fail. Example 4.7 is illuminating: a representation-finite algebra with Findim Λ = ddell Λ^op < dell Λ, and a clear reason (infinite path in the reversed completion) for the failure. The writing is honest about what is proved and what is deferred.

The main soft spot is exactly where the stress-test note lands: Lemma 3.4 is the engine for both Proposition 3.5 and Theorem 4.2, and its proof is a sketch. The claim that all additional minimal relations in the reversed sequence must start in y_{i+2} and end in y_i is non-obvious and is not proved for general monomial algebras, where vertices can have several incoming/outgoing arrows. The displayed syzygy formula Ω^j_Λ N = x_{n-j+2} y_{n-j+1}Λ for j=5,...,n+1 is asserted with no derivation. If there are extra right completions, the syzygy modules have extra summands and the conclusion pd Λ(Λ/p_nΛ) ≥ n+1 does not follow from the text as written. This is a load-bearing gap, not a cosmetic one.

Other concerns are smaller. Theorem 4.2 defers the dell Λ^op = 2 case to a remark; that is probably fine but should be in the proof or clearly justified. The proof also asserts that the four quantities are finite for right serial path algebras without a citation or argument. Example 4.7's display of a projective resolution is garbled: it shows a complex 0 → ... → ... → S1 → 0 and then says Findim Λ = 1, which does not parse as written; the intended statement is likely Findim Λ = 1 = Findim Λ^op? That needs fixing. The self-citations to [8,9] are fine because the cited results are formally argued elsewhere; the load-bearing inequality Findim Λ ≤ dell Λ^op is standard.

Overall, I think the results are probably correct and the gaps are addressable. The paper is not ready in its current form because the central lemma is not sufficiently verified, but it is exactly the kind of work a serious referee should spend time on.

Recommendation: send to a referee who can check the syzygy computation in Lemma 3.4; with that fixed, the paper would be a solid contribution.

Referee Report

3 major / 4 minor

Summary. The paper studies delooping-level style invariants for monomial path algebras, specializing to left and right serial path algebras. It recalls the delooping level, sub-derived delooping level, and derived delooping level, then introduces right-completion sequences and a reversal lemma (Lemma 3.4) to obtain lower bounds on projective dimensions. Theorem 4.2 claims that for right serial path algebras, findim Λ = Findim Λ = dell Λ^op = ddell Λ^op < ∞. Corollary 4.5 recovers the known symmetric equalities for Nakayama algebras, and Proposition 3.5 recovers the acyclic monomial case. Theorem 4.6 gives a sufficient condition for the same equality on left serial algebras, and Example 4.7 exhibits a representation-finite left serial case in which the equality fails and ddell still matches the finitistic dimension. The paper ends with open questions about quantifying the gaps between these invariants.

Significance. If the central proof is completed, the paper makes a solid contribution: it adds right serial path algebras to the list of algebras whose big finitistic dimension is computed by the delooping level of the opposite algebra, provides a finite algorithm for the invariant, and gives an instructive example where the derived delooping level is strictly better than the delooping level. The right-completion reversal technique is natural and potentially reusable. However, the main theorem currently rests on Lemma 3.4, whose proof is only sketched and whose syzygy computation is load-bearing, so the results are conditional on that lemma being proved in full.

major comments (3)

[Section 3, Lemma 3.4] The lemma is the engine of the paper, but its proof is a sketch. The equalities Ω_Λ N = p_nΛ, Ω_Λ^2 N = y_{n−1}x_{n−1}Λ, Ω_Λ^3 N = y_{n−2}Λ, and Ω_Λ^4 N = x_{n−2}y_{n−3}Λ are asserted by applying Lemma 3.3 without showing that the listed minimal relations are the only right completions that occur; the preceding paragraph only rules out relations starting in x_i, and the statement that all other relations have starting and terminal arrows in y_{i+2} and y_i is not proved. For n≥4, the formula Ω_Λ^j N = x_{n−j+2}y_{n−j+1}Λ for j=5,...,n+1 is stated without derivation. In a general monomial algebra, vertices can have several incoming or outgoing arrows, so additional right completions would add extra summands to these syzygies. Since the conclusion pd_Λ(Λ/p_nΛ)≥n+1 is used directly to prove dell Λ^op≤findim Λ in Theorems 4.2 and 4.6, the central equality is not established until this syzygy computation is supplied in full.
[Section 4, Theorem 4.2] The proof says it will prove the statement for dell Λ^op ≥ 3 and that the remaining dell Λ^op = 2 case will follow, but the case is never written. Remark 4.3 only states that the argument is essentially verbatim. Since the theorem is stated for all right serial algebras, the dell Λ^op = 2 case needs either a complete proof or an explicit reduction. In the same proof, after constructing the sequence (14), the claim that ~ x2 ~ y2 Λ^op is a summand of the fourth syzygy of M is another unproved syzygy calculation of the same type as Lemma 3.4; it should be justified or replaced by a reference to a completed Lemma 3.4.
[Section 4, Theorem 4.6] The proof asserts that when the vertices of the reversed sequence (17) lie in the cyclic part, all other right-completion sequences starting with p_n have finite length, but the parenthetical remark 'as they can only stay in the tree part' only addresses the tree-part case. For a sequence supported in the cyclic part, one needs an explicit argument that no infinite right-completion sequence arises from p_n; otherwise pd_Λ M could be infinite and the inequality n+1 ≤ pd_Λ M ≤ findim Λ would be unavailable.

minor comments (4)

[Introduction] There is a typo in 'Anslander-Reiten conjecture'; it should be 'Auslander-Reiten'. Also, 'useright modules' in the introduction should be 'use right modules'.
[Example 4.7] The displayed resolution (18) is typeset in a way that is hard to parse; it should be redrawn as a standard projective resolution with all maps and projective modules explicitly labeled.
[Example 4.7] The notation switches between Λ-modules and Λ^op-modules without consistently using tildes; please make explicit, for each displayed module or resolution, which algebra it is a module over.
[Section 4, Theorem 4.2] The sentence 'the remaining dell Λ^op = 2 case will follow' is not by itself a proof; if the case is not included in the main text, the sentence should be removed or replaced with a precise reference to an expanded Remark 4.3.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the central equality is proved by direct syzygy and right-completion computations, with cited prior bounds used as external general inequalities.

full rationale

The paper's main equality findim Λ = Findim Λ = dell Λ^op = ddell Λ^op is obtained by proving the missing inequality dell Λ^op ≤ findim Λ (Theorem 4.2) through a direct reversal of right-completion sequences and syzygy computations (Lemma 3.3, Lemma 3.4). The other inequalities used—findim ≤ Findim, Findim Λ ≤ ddell Λ^op, and ddell Λ^op ≤ dell Λ^op—are cited from prior literature [5] and [9]; these are general, parameter-free theorems whose statements do not include the right-serial equality, so the citations are external support rather than circular inputs. Lemma 3.4's syzygy formula is asserted with a sketched proof; that is a potential correctness gap, not a circularity, because the lemma is not defined in terms of the theorem's conclusion. No fitted parameter is renamed as a prediction, no quantity is defined in terms of the target result, and no load-bearing claim is justified solely by a self-citation whose own proof is unverified. The derivation is therefore self-contained apart from standard background inequalities, which does not constitute circularity under the stated hard rules.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new free parameters and no invented entities. It relies on established invariants and inequalities, plus a largely unproven finiteness assertion for serial algebras. The main proof engine is Lemma 3.4, which is an internal lemma rather than an external axiom, but its level of detail is a soundness concern.

assumptions (4)

domain assumption The inequalities Findim Λ^op ≤ sub-ddell Λ ≤ dell Λ and Findim Λ^op ≤ ddell Λ ≤ dell Λ are taken as background from [5] and [9].
Used in Theorems 2.3, 2.6, and in the chain of inequalities in the proofs of Section 4; not reproved here.
standard math Every finite dimensional algebra over an algebraically closed field is Morita equivalent to a basic quiver path algebra, and finitistic dimensions are invariant under Morita equivalence and field extensions.
Invoked in Section 1 to justify restricting to path algebras KQ/I.
domain assumption Right serial path algebras are monomial and have finite delooping level; the text asserts this without a proof or explicit citation at that point.
The proof of Theorem 4.2 opens with 'Since right serial path algebras are monomial, the four quantities are all finite', and the Introduction states that 2-syzygy finite algebras have finite delooping level.
domain assumption The syzygy of μΛ over a monomial algebra is the direct sum of (ωr)Λ over right completions ωr of μ, as stated in Lemma 3.3.
This is the main structural description used throughout Section 3 and Section 4, and it relies on the known behavior of syzygies over monomial algebras.

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Cite this review

Pith. "Pith review of Homological Invariants of Left and Right Serial Path Algebras." pith.science (2026). https://pith.science/paper/IUVHGDEX

@misc{pith2026250602307,
  author       = {Pith},
  title        = {Pith review of: Homological Invariants of Left and Right Serial Path Algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IUVHGDEX}},
  note         = {Machine review of arXiv:2506.02307}
}

read the original abstract

We investigate the relationship between the delooping level (dell) and the finitistic dimension of left and right serial path algebras. These 2-syzygy finite algebras have finite delooping level, and it can be calculated with an easy and finite algorithm. When the algebra is right serial, its right finitistic dimension is equal to its left delooping level. When the algebra is left serial, the above equality only holds under certain conditions. We provide examples to demonstrate this and include discussions on the sub-derived (subddell) and derived delooping level (ddell). Both subddell and ddell are improvements of the delooping level. We motivate their definitions and showcase how they can behave better than the delooping level in certain situations throughout the paper.

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Cambridge University Press, 2006

Ibrahim Assem, Daniel Simson, and Andrzej Skowro´ nski.Elements of the representation theory of as- sociative algebras: Techniques of representation theory. Cambridge University Press, 2006

work page 2006

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work page Pith review arXiv 2024

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Generalised igusa-todorov func- tions and lat-igusa-todorov algebras.Journal of Algebra, 580:63–83, 2021

Diego Bravo, Marcelo Lanzilotta, Octavio Mendoza, and Jos´ e Vivero. Generalised igusa-todorov func- tions and lat-igusa-todorov algebras.Journal of Algebra, 580:63–83, 2021

work page 2021

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The finitistic dimension of an artin algebra with radical square zero.Proceedings of the American Mathematical Society, 149(12):5001–5012, 2021

Vincent G´ elinas. The finitistic dimension of an artin algebra with radical square zero.Proceedings of the American Mathematical Society, 149(12):5001–5012, 2021

work page 2021

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The depth, the delooping level and the finitistic dimension.Advances in Mathematics, 394:108052, 2022

Vincent G´ elinas. The depth, the delooping level and the finitistic dimension.Advances in Mathematics, 394:108052, 2022

work page 2022

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Projective resolutions over artin algebras with zero relations.Illinois Journal of Mathematics, 29(1):180–190, 1985

Edward L Green, Dieter Happel, and Dan Zacharia. Projective resolutions over artin algebras with zero relations.Illinois Journal of Mathematics, 29(1):180–190, 1985

work page 1985

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Finitistic dimensions of finite dimensional monomial algebras.Journal of algebra, 136(1):37–50, 1991

Edward L Green, Ellen Kirkman, and James Kuzmanovich. Finitistic dimensions of finite dimensional monomial algebras.Journal of algebra, 136(1):37–50, 1991

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Symmetry of Derived Delooping Level

Ruoyu Guo. Symmetry of derived delooping level.arXiv preprint arXiv:2406.00253, 2024

work page Pith review arXiv 2024