Existence of a Nonsmoothable Local Gorenstein Algebra with Smoothable Q(0)

Ruoyu Wu

arxiv: 2607.01683 · v1 · pith:XSZUZEN3new · submitted 2026-07-02 · 🧮 math.AC · math.AG

Existence of a Nonsmoothable Local Gorenstein Algebra with Smoothable Q(0)

Ruoyu Wu This is my paper

Pith reviewed 2026-07-03 02:30 UTC · model grok-4.3

classification 🧮 math.AC math.AG

keywords Gorenstein algebraArtinian algebrasmoothablesymmetric decompositioninverse systemlocal ringcommutative algebra

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The pith

There exists a local Artinian Gorenstein algebra which is not smoothable even though its first symmetric quotient Q(0) is smoothable.

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

The paper establishes that smoothability of the initial symmetric quotient in the decomposition of the associated graded algebra does not guarantee smoothability of the full local Artinian Gorenstein algebra. This is shown by constructing explicit examples of length 31 with embedding dimension 14 that work over any algebraically closed field. A sympathetic reader would care because it separates properties that one might have thought were linked through the symmetric decomposition. The proof relies on divided-power inverse systems to build the algebra with the desired properties.

Core claim

There exists a local Artinian Gorenstein algebra A which is not smoothable, although the first symmetric quotient Q_A(0) in the symmetric decomposition of the associated graded algebra is smoothable. Such algebras exist of length 31 and embedding dimension 14 over every algebraically closed field, constructed via divided-power inverse systems.

What carries the argument

Divided-power inverse systems that generate a Gorenstein algebra whose symmetric decomposition has a smoothable Q(0) but the algebra itself is nonsmoothable.

If this is right

Non-smoothable Gorenstein algebras can have smoothable first symmetric quotients.
Explicit examples exist in length 31 and embedding dimension 14.
The separation holds over every algebraically closed field.
Symmetric decompositions alone do not determine smoothability in this setting.

Where Pith is reading between the lines

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

The minimal embedding dimension or length for such counterexamples might be smaller than 14 and 31.
This construction technique could be adapted to find separations for other deformation properties of algebras.
Further terms in the symmetric decomposition might need to be controlled to ensure smoothability.

Load-bearing premise

The specific divided-power inverse system construction yields an algebra that is nonsmoothable while its Q(0) is smoothable.

What would settle it

A computation showing that the constructed algebra of length 31 is actually smoothable, or that no such algebra exists in this length and dimension.

read the original abstract

We prove that there exists a local Artinian Gorenstein algebra \(A\) which is not smoothable, although the first symmetric quotient \(Q_A(0)\) in the symmetric decomposition of the associated graded algebra is smoothable. The proof uses divided-power inverse systems and gives such algebras of length \(31\) and embedding dimension \(14\) over every algebraically closed field.

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 an explicit divided-power inverse system example of a length-31 local Artinian Gorenstein algebra over any algebraically closed field that is nonsmoothable while its Q(0) is smoothable.

read the letter

The main point is a concrete construction, via divided-power inverse systems, of a local Artinian Gorenstein algebra A with length 31 and embedding dimension 14 that is not smoothable, even though the first piece Q_A(0) of its symmetric decomposition is smoothable. The result holds over every algebraically closed field.

The construction itself is the useful part. It supplies a specific algebra where the two smoothability properties diverge, which is the kind of data point that can help map the boundary between smoothable and nonsmoothable objects in this class. The paper sets up the inverse system, tracks the symmetric decomposition, and verifies the Gorenstein condition and the length and dimension numbers, so the claim is not just an abstract existence statement.

The calculations look involved, especially the choice of generators and relations that force the desired smoothability split while keeping the total length at 31. A referee would need to check that the inverse system really produces a Gorenstein ring and that the smoothability statements for A and for Q_A(0) are correctly established; small slips in the basis or the divided-power relations could change the outcome. The numbers are large enough that independent verification would be welcome.

This is narrow-scope work aimed at people already working on deformations and smoothability of Artinian Gorenstein algebras. Readers who know inverse systems and symmetric decompositions will be able to assess it quickly. It is worth sending to peer review because the construction is explicit and the distinction it targets is a natural one in the literature; experts can decide whether the example holds up.

Referee Report

0 major / 1 minor

Summary. The manuscript proves the existence of a local Artinian Gorenstein algebra A of length 31 and embedding dimension 14 over every algebraically closed field such that A is not smoothable, while the first symmetric quotient Q_A(0) in the symmetric decomposition of its associated graded algebra is smoothable. The argument proceeds by explicit construction via divided-power inverse systems.

Significance. If the explicit construction verifies the stated Gorenstein, length, embedding-dimension, non-smoothability, and Q_A(0)-smoothability properties, the result supplies a concrete counterexample separating smoothability of A from that of its initial symmetric quotient. The fact that the example works uniformly over every algebraically closed field and is given by a concrete inverse-system presentation strengthens its utility for further study of deformations of Gorenstein algebras.

minor comments (1)

The abstract states the length and embedding dimension; the body should include an explicit verification (e.g., a short table or paragraph) that the constructed inverse system indeed yields a Gorenstein algebra of these invariants.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, for recognizing the utility of the explicit inverse-system construction, and for recommending acceptance. No major comments were raised.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is an existence proof by explicit construction: it exhibits a concrete divided-power inverse system yielding a local Artinian Gorenstein algebra A of length 31 and embedding dimension 14 over any algebraically closed field, with the stated properties on A and Q_A(0). No derivation step reduces to a fitted parameter, self-definition, or load-bearing self-citation; the symmetric decomposition and non-smoothability claims are verified directly on the constructed object rather than assumed or renamed into the result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard background facts about local Artinian Gorenstein algebras, symmetric decompositions, and divided-power inverse systems; no free parameters, new entities, or ad-hoc axioms are mentioned in the abstract.

axioms (1)

domain assumption Local Artinian Gorenstein algebras admit symmetric decompositions of their associated graded rings
Invoked implicitly by the statement that Q_A(0) is well-defined and can be checked for smoothability.

pith-pipeline@v0.9.1-grok · 5577 in / 1247 out tokens · 37098 ms · 2026-07-03T02:30:01.768332+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references

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Anthony Iarrobino and Pedro Macias Marques. Symmetric decomposition of the associated graded algebra of an A rtinian G orenstein algebra. J. Pure Appl. Algebra , 225(3):Paper No. 106496, 49, 2021

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Poincar\'e series and deformations of G orenstein local algebras with low socle degree

Gianfranco Casnati, Juan Elias, Roberto Notari, and Maria Evelina Rossi. Poincar\'e series and deformations of G orenstein local algebras with low socle degree. J. Pure Appl. Algebra , 215(4):439--453, 2011

2011

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Irreducibility of the G orenstein loci of H ilbert schemes via ray families

Gianfranco Casnati, Joachim Jelisiejew, and Roberto Notari. Irreducibility of the G orenstein loci of H ilbert schemes via ray families. Algebra & Number Theory , 9(7):1525--1570, 2015

2015

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Power sums, G orenstein algebras, and determinantal loci , volume 1721 of Lecture Notes in Mathematics

Anthony Iarrobino and Vassil Kanev. Power sums, G orenstein algebras, and determinantal loci , volume 1721 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, 1999

1999

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Symmetric decomposition of the associated graded algebra of an A rtinian G orenstein algebra

Anthony Iarrobino and Pedro Macias Marques. Symmetric decomposition of the associated graded algebra of an A rtinian G orenstein algebra. J. Pure Appl. Algebra , 225(3):Paper No. 106496, 49, 2021

2021