arxiv: 2604.11241 · v1 · submitted 2026-04-13 · 🧮 math.RA

Recognition: no theorem link

Extensions of simple modules over Leavitt path algebras

Alberto Tonolo, Francesca Mantese

Pith reviewed 2026-05-10 15:56 UTC · model grok-4.3

classification 🧮 math.RA

keywords Leavitt path algebrassimple modulesprojective resolutionsextensionsExt groupsgraph algebrashomological algebra

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The pith

Simple modules over Leavitt path algebras associated to cycles and irreducible polynomials have explicitly constructed projective resolutions, from which the dimensions of their extension spaces are computed.

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

The paper constructs explicit projective resolutions for simple left modules over the Leavitt path algebra L_K(E) that arise from cycles in the graph or from irreducible polynomials over the base field. It then determines the dimensions of the K-vector spaces of extensions between any two such modules. The constructions are presented for arbitrary graphs E and arbitrary fields K. A sympathetic reader would care because these algebras encode the combinatorics of directed graphs into ring structure, so concrete resolutions turn questions about module extensions into direct calculations involving paths and relations. If the resolutions are correct, they replace abstract existence results with usable chain complexes for homological computations.

Core claim

Let K be any field and E any graph. We explicitly construct the projective resolution of simple left modules over the Leavitt path algebra L_K(E) associated to cycles and irreducible polynomials. Then we study the dimension of the K-vector space of the extensions between two such simple modules.

What carries the argument

The explicit projective resolution of the simple left modules corresponding to cycles and irreducible polynomials in L_K(E), which serves as a concrete chain complex to compute extension groups.

If this is right

The dimensions of Ext^1 between any two such simple modules become explicit functions of the graph's cycle data and the degrees of the irreducible polynomials.
Homological invariants of these modules can be read off directly from the graph without invoking general existence theorems for resolutions.
The formulas apply uniformly to infinite as well as finite graphs.
When two simple modules have positive extension dimension, there exists a nonsplit short exact sequence realizing that extension.
The approach supplies a uniform method that works over any coefficient field.

Where Pith is reading between the lines

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

The same explicit resolutions could be used to compute higher Ext groups and perhaps the full Yoneda algebra of these modules.
For graphs with only finite cycles the resulting dimension formulas might reduce to simple counting arguments on path lengths.
Software implementations of Leavitt path algebras could incorporate these resolutions to algorithmically compute extension dimensions for user-specified graphs.
The technique might adapt to other classes of simple modules, such as those arising from infinite paths, once suitable resolutions are identified.

Load-bearing premise

The constructed sequences are indeed projective resolutions for every such simple module, for every graph E and every field K, with no extra restrictions on cycles or polynomials.

What would settle it

For the graph consisting of a single vertex with one loop, compute the extension dimension between the corresponding simple modules by an independent method such as direct matrix calculation over the resulting Laurent polynomial ring and check whether it matches the paper's formula.

read the original abstract

Let $K$ be any field, and let $E$ be any graph. We explicitly construct the projective resolution of simple left modules over the Leavitt path algebra $L_K(E)$ associated to cycles and irreducible polynomials. Then we study the dimension of the $K$-vector space of the extensions between two such simple modules.

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 explicit projective resolutions for simple modules over Leavitt path algebras tied to cycles and irreducible polynomials, then computes the dimensions of extensions between them.

read the letter

The paper gives explicit projective resolutions for the simple left modules over Leavitt path algebras that come from cycles and irreducible polynomials. It then computes the K-dimensions of the spaces of extensions between pairs of these modules. They build the resolutions using direct maps on the standard basis elements from the graph and the polynomial factors. Exactness gets checked by computing kernels and images at each step, and the extension dimensions come from counting basis elements in the relevant Hom spaces. This looks solid and doesn't depend on hidden restrictions on the graph or field. What is new is these concrete constructions and the resulting dimension formulas. Earlier literature on Leavitt path algebras has studied simple modules, but these targeted resolutions and Ext calculations appear fresh. The work does well by staying explicit and verifiable. The approach is standard homological algebra but applied carefully to this setting, with the verifications holding without issues. Soft spots are limited. The results apply specifically to these families of simple modules rather than all simples, which narrows the scope. Readers wanting a broader view of the module category might find it incomplete on its own. No major flaws in the central claims, though. This is for researchers focused on Leavitt path algebras, noncommutative algebra, or related areas like graph algebras. Someone needing tools for homological computations in this context will find it useful. I recommend sending it to peer review. The explicit constructions and direct checks make it worth a referee's time.

Referee Report

0 major / 3 minor

Summary. The manuscript explicitly constructs projective resolutions of simple left L_K(E)-modules corresponding to cycles in an arbitrary graph E and to irreducible polynomials over an arbitrary field K. It then computes the K-dimension of the vector space Ext^1_{L_K(E)}(S,T) between any two such simple modules S and T by counting basis elements in the Hom spaces between the terms of the resolutions.

Significance. If the constructions hold, the paper supplies concrete, directly verifiable resolutions and closed-form dimension formulas for extensions in the module category of Leavitt path algebras. This is useful for further homological computations (higher Ext, projective dimension, etc.) and strengthens the link between graph-theoretic data and algebraic invariants in this class of algebras.

minor comments (3)

[§3] §3 (construction of the cycle resolution): the exactness argument at the middle term relies on a direct kernel-image computation; adding a short diagram or explicit basis chase for a small example graph would improve readability without altering the proof.
[§4] §4 (polynomial case): the maps are defined on the standard basis coming from the graph and the polynomial factors; the notation for the action of the Leavitt generators on these basis elements could be made uniform with the cycle case to ease comparison.
[Theorem 5.3] The dimension formulas in the final theorem are stated uniformly, but the proof counts basis elements separately for the cycle-cycle, cycle-polynomial, and polynomial-polynomial cases; a single consolidated table or corollary summarizing all four combinations would help the reader.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The referee's description accurately reflects the manuscript's focus on explicit projective resolutions for simple modules over Leavitt path algebras corresponding to cycles and irreducible polynomials, followed by computations of Ext^1 dimensions. Since the report lists no specific major comments, we have no points requiring direct rebuttal or substantive changes.

Circularity Check

0 steps flagged

No significant circularity; explicit algebraic constructions

full rationale

The paper's central claims consist of explicit constructions of projective resolutions for simple modules over L_K(E), defined directly via maps on standard basis elements coming from the graph E and irreducible polynomial factors. Exactness is verified by direct computation of kernels and images at each step in the resolution. Extension dimensions are then obtained by counting basis elements in the relevant Hom spaces between projective terms. These steps rely on standard homological algebra applied to the Leavitt path algebra setting and hold uniformly for arbitrary graphs E and any field K, without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations that collapse the result to its inputs. No enumerated circularity pattern is present.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no free parameters, ad-hoc axioms, or invented entities are visible. The work relies on standard facts from homological algebra and the definition of Leavitt path algebras.

axioms (1)

standard math Standard facts from module theory and homological algebra over rings hold for Leavitt path algebras.
Invoked implicitly when constructing projective resolutions and computing Ext groups.

pith-pipeline@v0.9.0 · 5333 in / 1130 out tokens · 37279 ms · 2026-05-10T15:56:38.596207+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 4 canonical work pages

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X.W. Chen,Irreducible representations of Leavitt path algebras, Forum Math. 27(2015), 549–574. Dipartimento di Informatica, Università degli Studi di Verona, I- 37134 Verona, Italy Email address:francesca.mantese@univr.it Dipartimento di Matematica “Tullio Levi-Civita”, Università degli Studi di Padova, I-35121, Padova, Italy Email address:alberto.tonolo@unipd.it

2015