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arxiv: 2409.15081 · v2 · submitted 2024-09-23 · 🧮 math.AC · math.AG· math.RA

Finite-dimensional monomial algebras are determined by their automorphism group

Roberto D\'iaz , Giancarlo Lucchini Arteche This is my paper

Pith reviewed 2026-05-23 21:16 UTC · model grok-4.3

classification 🧮 math.AC math.AGmath.RA
keywords monomial algebrasautomorphism groupsfinite-dimensional algebraslocal algebrasmonomial idealscotangent space
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The pith

Finite-dimensional monomial algebras are characterized by their automorphism groups among local algebras with fixed cotangent space dimension.

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

The paper shows that any finite-dimensional monomial algebra can be recovered from its automorphism group when the comparison is limited to finite-dimensional local algebras whose cotangent spaces all have one fixed dimension. This means the monomial ideal itself can be reconstructed directly from the group of algebra automorphisms. A reader would care because the result turns an algebraic symmetry into a complete invariant for this class, so isomorphism questions reduce to group questions inside the restricted family. The claim is not made for arbitrary algebras, only those sharing the cotangent-space dimension. If correct, the automorphism group therefore encodes the entire monomial structure without needing the ideal explicitly.

Core claim

Finite-dimensional monomial algebras are characterized by their automorphism group among finite-dimensional, local algebras with cotangent space of fixed dimension. In particular, the monomial ideal can be recovered from the automorphism group of the corresponding monomial algebra.

What carries the argument

The automorphism group of the algebra, which encodes and determines the monomial ideal uniquely inside the fixed cotangent-dimension class.

If this is right

  • The monomial ideal is recoverable directly from the automorphism group.
  • Distinct monomial algebras in the class must have distinct automorphism groups.
  • Any algebra in the class that is not monomial must have an automorphism group different from every monomial algebra.
  • Isomorphism of such monomial algebras reduces to isomorphism of their automorphism groups.

Where Pith is reading between the lines

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

  • The result supplies an explicit reconstruction procedure that turns group data into the ideal generators.
  • It may simplify computational checks for whether a given local algebra is monomial when the automorphism group is known.
  • The same technique could be tested on other restricted families, such as graded algebras with fixed Hilbert function.

Load-bearing premise

The comparison class is restricted to finite-dimensional local algebras whose cotangent space has a fixed dimension.

What would settle it

Exhibit either two non-isomorphic monomial algebras with identical automorphism groups or one monomial algebra and one non-monomial algebra sharing the same automorphism group, all with the same cotangent-space dimension.

read the original abstract

A monomial algebra is the quotient of a polynomial algebra by an ideal generated by monomials. We prove that finite-dimensional monomial algebras are characterized by their automorphism group among finite-dimensional, local algebras with cotangent space of fixed dimension. In particular, we show how to recover a monomial ideal given the automorphism group of the corresponding monomial algebra.

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

0 major / 0 minor

Summary. The paper proves that finite-dimensional monomial algebras are characterized by their automorphism groups among finite-dimensional local algebras with cotangent space of fixed dimension. It also shows how to recover the monomial ideal from the automorphism group of the corresponding monomial algebra.

Significance. If the result holds, the characterization and explicit recovery procedure would constitute a useful contribution to commutative algebra by linking algebraic structure to automorphism data within a clearly delimited class of algebras.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes a characterization theorem showing that finite-dimensional monomial algebras are uniquely determined by their automorphism groups inside the explicitly restricted class of finite-dimensional local algebras with fixed cotangent space dimension. The abstract and claim describe a direct algebraic recovery of the monomial ideal from the group, with no equations, fitted parameters, predictions, or self-citations invoked as load-bearing steps. The derivation is presented as a self-contained proof within the stated scope, without reduction to inputs by construction or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract relies on standard definitions of monomial algebras, local algebras, cotangent spaces, and automorphism groups in commutative algebra; no free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)
  • standard math Standard definitions and basic properties of monomial algebras, local rings, cotangent spaces, and automorphism groups in commutative algebra.
    The paper invokes these established notions to state the characterization.

pith-pipeline@v0.9.0 · 5577 in / 1236 out tokens · 50515 ms · 2026-05-23T21:16:20.893220+00:00 · methodology

discussion (0)

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Reference graph

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