pith. sign in

arxiv: 2506.04899 · v5 · submitted 2025-06-05 · 🧮 math.AC

Canonical traces of graded fiber products: applications to disconnected Stanley--Reisner rings

Shinya Kumashiro , Sora Miyashita This is my paper

Pith reviewed 2026-05-19 11:33 UTC · model grok-4.3

classification 🧮 math.AC
keywords canonical tracegraded fiber productStanley-Reisner ringsimplicial complexGorenstein on the punctured spectrumCohen-Macaulay ringNoetherian ringcommutative algebra
0
0 comments X

The pith

An explicit formula for canonical traces of graded fiber products allows classification of Stanley-Reisner rings without the Cohen-Macaulay assumption.

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

This paper generalizes earlier classifications of canonical traces for Stanley-Reisner rings that are Gorenstein on the punctured spectrum. Previous results required the rings to satisfy the Cohen-Macaulay property, but the new work removes that restriction. The authors first derive an explicit formula for the canonical trace of graded fiber products of Noetherian rings. They apply the formula to Stanley-Reisner rings arising from disconnected simplicial complexes, which reduces the problem to the connected case. They then obtain a complete classification for connected complexes and combine the pieces to classify the disconnected case as well.

Core claim

The paper establishes an explicit formula for the canonical trace of graded fiber products of Noetherian rings. This formula is applied to Stanley-Reisner rings of disconnected simplicial complexes to reduce the classification problem to the connected case. For connected simplicial complexes, a complete classification of the canonical traces is given without assuming the Cohen-Macaulay property. The two parts are combined to produce the classification for disconnected complexes, extending the earlier results of Miyashita and Varbaro.

What carries the argument

The explicit formula for the canonical trace of graded fiber products of Noetherian rings, which reduces the disconnected simplicial complex case to the connected case.

Load-bearing premise

The explicit formula for the canonical trace of graded fiber products of Noetherian rings holds and can be applied to reduce the disconnected simplicial complex case to the connected case.

What would settle it

A graded fiber product of two Noetherian rings whose canonical trace does not match the explicit formula, or a disconnected simplicial complex whose Stanley-Reisner ring is Gorenstein on the punctured spectrum but whose canonical trace lies outside the listed classified forms.

read the original abstract

Recent work by Miyashita and Varbaro classified the canonical traces of Stanley--Reisner rings that are Gorenstein on the punctured spectrum, under the Cohen--Macaulay assumption. The purpose of this paper is to generalize the result to the non--Cohen--Macaulay case. First, we establish an explicit formula for the canonical trace of graded fiber products of Noetherian rings and apply it to Stanley--Reisner rings of disconnected simplicial complexes. This allows us to reduce the problem to the case of connected simplicial complexes. In that case, we succeeded in giving a complete classification without assuming the Cohen--Macaulay property. Finally, we combine these results to obtain a classification for disconnected simplicial complexes, complementing the work of Miyashita and Varbaro.

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

1 major / 2 minor

Summary. The paper establishes an explicit formula for the canonical trace of graded fiber products of Noetherian rings. It applies this formula to Stanley-Reisner rings of disconnected simplicial complexes, reducing the classification of canonical traces for rings that are Gorenstein on the punctured spectrum to the connected case. For connected complexes, a complete classification is obtained without assuming the Cohen-Macaulay property. These results are combined to classify the disconnected case, extending the work of Miyashita and Varbaro.

Significance. If the central results hold, the paper completes the classification of canonical traces for Stanley-Reisner rings Gorenstein on the punctured spectrum in the non-Cohen-Macaulay setting for both connected and disconnected complexes. The explicit formula for graded fiber products is a technical contribution with potential broader use in commutative algebra. Credit is due for removing the Cohen-Macaulay assumption and for the reduction via fiber products.

major comments (1)
  1. [applications to disconnected simplicial complexes] The reduction in the applications section relies on the claim that if R and S are Gorenstein on the punctured spectrum, then the graded fiber product R ×_k S is likewise Gorenstein on its punctured spectrum. This preservation is not immediate for Stanley-Reisner rings of disconnected complexes, where the components are identified only in degree zero; the argument should include an explicit verification that the property survives localization at non-maximal homogeneous primes (see the statement following the formula for the canonical trace and its application to k[Δ] when Δ is disconnected).
minor comments (2)
  1. [preliminaries] Clarify the notation for the irrelevant ideal and the punctured spectrum in the fiber-product setting to avoid ambiguity when passing from connected to disconnected cases.
  2. [introduction] The abstract refers to 'the purpose and method sections'; ensure these are explicitly labeled in the introduction for easier navigation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comment point by point below and will revise the manuscript to incorporate the suggested clarification.

read point-by-point responses

  1. Referee: [applications to disconnected simplicial complexes] The reduction in the applications section relies on the claim that if R and S are Gorenstein on the punctured spectrum, then the graded fiber product R ×_k S is likewise Gorenstein on its punctured spectrum. This preservation is not immediate for Stanley-Reisner rings of disconnected complexes, where the components are identified only in degree zero; the argument should include an explicit verification that the property survives localization at non-maximal homogeneous primes (see the statement following the formula for the canonical trace and its application to k[Δ] when Δ is disconnected).

    Authors: We appreciate the referee highlighting this point. The manuscript derives an explicit formula for the canonical trace of graded fiber products and applies it to reduce the disconnected case to the connected case for Stanley-Reisner rings. While the reduction implicitly relies on the preservation of the Gorenstein-on-the-punctured-spectrum property, we agree that an explicit verification of this preservation under localization at non-maximal homogeneous primes is not immediate in the disconnected setting (where components meet only in degree zero) and would strengthen the argument. In the revised manuscript we will add a short lemma or paragraph immediately following the formula, verifying that if R and S are Gorenstein on their punctured spectra then so is the graded fiber product R ×_k S, by relating localizations at relevant homogeneous primes to the corresponding localizations in R and S. revision: yes

Circularity Check

1 steps flagged

Minor self-citation to co-author prior work; new formula and non-CM classification provide independent content

specific steps
  1. self citation load bearing [Abstract]
    "Recent work by Miyashita and Varbaro classified the canonical traces of Stanley--Reisner rings that are Gorenstein on the punctured spectrum, under the Cohen--Macaulay assumption."

    The cited classification by co-author Miyashita serves as the base case that the new fiber-product formula reduces the disconnected problem to, before the paper extends to non-CM; the overall disconnected classification therefore depends on the prior result without re-deriving its core in this manuscript.

full rationale

The paper cites recent work by Miyashita and Varbaro (co-author overlap) for the CM case of canonical traces on Gorenstein-on-punctured-spectrum SR rings, then introduces an explicit formula for graded fiber products to reduce the disconnected case and claims a new complete classification for the connected non-CM case. This self-citation is minor and not load-bearing for the central new results. No self-definitional reductions, fitted inputs renamed as predictions, ansatz smuggling, or renaming of known results appear in the abstract or described derivation chain. The explicit formula and non-CM extension constitute independent mathematical content, keeping the paper self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard commutative algebra but introduces no free parameters, new axioms beyond background ring theory, or invented entities; the central contribution is the derived formula and classification.

axioms (1)
  • standard math Standard properties of Noetherian rings, graded modules, and fiber products in commutative algebra.
    Invoked to establish the explicit formula for canonical traces.

pith-pipeline@v0.9.0 · 5670 in / 1185 out tokens · 44337 ms · 2026-05-19T11:33:19.447884+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Trace ideals of exterior powers of the module of differentials

    math.AC 2026-05 unverdicted novelty 7.0

    Trace ideals of exterior powers of differentials characterize polynomial and formal power series ranks of rings and define the singular locus via the top differential trace for reduced equidimensional rings.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · cited by 1 Pith paper

  1. [1]

    Aoyama,Some basic results on canonical modules, Journal of Mathematics of Kyoto University23 (1983), no

    Y . Aoyama,Some basic results on canonical modules, Journal of Mathematics of Kyoto University23 (1983), no. 1, 85–94

    work page 1983

  2. [2]

    Aoyama, S

    Y . Aoyama, S. Goto, and Others, On the endomorphism ring of the canonical module , Journal of Mathematics of Kyoto University 25 (1985), no. 1, 21–30

    work page 1985

  3. [3]

    F Atiyah and I

    M. F Atiyah and I. G. Macdonald, Introduction to commutative algebra, CRC Press, 2018

    work page 2018

  4. [4]

    Bagherpoor and A

    M. Bagherpoor and A. Taherizadeh, Trace ideals of semidualizing modules and two generalizations of nearly Gorenstein rings, Communications in Algebra 51 (2023), no. 2, 446–463

    work page 2023

  5. [5]

    A Bj ¨orner, Topological methods, in: Handbook of combinatorics

    work page

  6. [6]

    Bruns and J

    W. Bruns and J. Herzog, Cohen–Macaulay rings, Cambridge university press, 1998

    work page 1998

  7. [7]

    Caminata and F

    A. Caminata and F. Strazzanti, Nearly Gorenstein cyclic quotient singularities , Beitr ¨age zur Algebra und Geome- trie/Contributions to Algebra and Geometry 62 (2021), no. 4, 857–870

    work page 2021

  8. [8]

    Celikbas, J

    E. Celikbas, J. Herzog, and S. Kumashiro, Traces of semi-invariants, arXiv preprint arXiv:2312.00983 (2023)

    work page arXiv 2023

  9. [9]

    H. Dao, T. Kobayashi, and R. Takahashi, Trace ideals of canonical modules, annihilators of Ext modules, and classes of rings close to being Gorenstein, Journal of Pure and Applied Algebra 225 (2021), no. 9, 106655

    work page 2021

  10. [10]

    N. Endo, S. Goto, and R. Isobe, Almost Gorenstein rings arising from fiber products, Canadian Mathematical Bulletin 64 (2021), no. 2, 383–400

    work page 2021

  11. [11]

    Ficarra, The canonical trace of Cohen–Macaulay algebras of codimension 2, arXiv preprint arXiv:2406.07517 (2024)

    A. Ficarra, The canonical trace of Cohen–Macaulay algebras of codimension 2, arXiv preprint arXiv:2406.07517 (2024)

    work page arXiv 2024

  12. [12]

    Ficarra, J

    A. Ficarra, J. Herzog, D. I Stamate, and V . Trivedi,The canonical trace of determinantal rings, Archiv der Mathematik 123 (2024), no. 5, 487–497

    work page 2024

  13. [13]

    Gasanova, J

    O. Gasanova, J. Herzog, T. Hibi, and S. Moradi, Rings of Teter type, Nagoya Mathematical Journal 248 (2022), 1005–1033

    work page 2022

  14. [14]

    Goto and K

    S. Goto and K. Watanabe, On graded rings, I, Journal of the Mathematical Society of Japan 30 (1978), no. 2, 179–213

    work page 1978

  15. [15]

    T. Hall, M. K ¨olbl, K. Matsushita, and S. Miyashita, Nearly Gorenstein polytopes , Electronic Journal of Combinatorics (2023)

    work page 2023

  16. [16]

    Hashimoto and Y

    M. Hashimoto and Y . Yang, Indecomposability of graded modules over a graded ring , arXiv preprint arXiv:2306.14523 (2023)

    work page arXiv 2023

  17. [17]

    Herzog, T

    J. Herzog, T. Hibi, and D. I. Stamate,The trace of the canonical module, Israel Journal of Mathematics233 (2019), 133–165

    work page 2019

  18. [18]

    Herzog and E

    J. Herzog and E. Kunz, Die werthalbgruppe eines lokalen rings der dimension 1, Berichte der Heidelberger Akademie der Wissenschaften (1971)

    work page 1971

  19. [19]

    Jafari, F

    R. Jafari, F. Strazzanti, and S. Z. Armengou, On nearly Gorenstein affine semigroups , arXiv preprint arXiv:2411.12081 (2024). CANONICAL TRACES OF GRADED FIBER PRODUCTS: APPLICATIONS TO STANLEY–REISNER RINGS 25

    work page arXiv 2024

  20. [20]

    Kimura, Trace ideals, conductors, and ideals of finite (phantom) projective dimension, arXiv preprint arXiv:2501.03442 (2025)

    K. Kimura, Trace ideals, conductors, and ideals of finite (phantom) projective dimension, arXiv preprint arXiv:2501.03442 (2025)

    work page arXiv 2025

  21. [21]

    Kumashiro, When are trace ideals finite?, Mediterranean Journal of Mathematics 20 (2023), no

    S. Kumashiro, When are trace ideals finite?, Mediterranean Journal of Mathematics 20 (2023), no. 5, 278

    work page 2023

  22. [22]

    Kumashiro, N

    S. Kumashiro, N. Matsuoka, and T. Nakashima, Nearly Gorenstein local rings defined by maximal minors of a 2× n matrix, Semigroup forum, 2025, pp. 1–27

    work page 2025

  23. [23]

    Lindo, Trace ideals and centers of endomorphism rings of modules over commutative rings , Journal of Algebra 482 (2017), 102–130

    H. Lindo, Trace ideals and centers of endomorphism rings of modules over commutative rings , Journal of Algebra 482 (2017), 102–130

    work page 2017

  24. [24]

    Lu, The chain algebra of a pure poset, arXiv preprint arXiv:2410.05024 (2024)

    D. Lu, The chain algebra of a pure poset, arXiv preprint arXiv:2410.05024 (2024)

    work page arXiv 2024

  25. [25]

    Lyle and S

    J. Lyle and S. Maitra, Annihilators of (co) homology and their influence on the trace ideal, arXiv preprint arXiv:2409.04686 (2024)

    work page arXiv 2024

  26. [26]

    Matsumura, Commutative ring theory, Cambridge university press, 1989

    H. Matsumura, Commutative ring theory, Cambridge university press, 1989

    work page 1989

  27. [27]

    Miyashita, Levelness versus nearly Gorensteinness of homogeneous rings , Journal of Pure and Applied Algebra 228 (2024), no

    S. Miyashita, Levelness versus nearly Gorensteinness of homogeneous rings , Journal of Pure and Applied Algebra 228 (2024), no. 4, 107553

    work page 2024

  28. [28]

    , A linear variant of the nearly Gorenstein property, arXiv preprint arXiv:2407.05629 (2024)

    work page arXiv 2024

  29. [29]

    , When do pseudo-Gorenstein rings become Gorenstein?, arXiv preprint arXiv:2502.01133 (2025)

    work page arXiv 2025

  30. [30]

    Miyashita and M

    S. Miyashita and M. Varbaro, The canonical trace of Stanley–Reisner rings that are Gorenstein on the punctured spectrum, International Mathematics Research Notices 2025 (June 2025), no. 12

    work page 2025

  31. [31]

    Miyazaki, On the Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph, International Electronic Journal of Algebra 30 (2021), no

    M. Miyazaki, On the Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph, International Electronic Journal of Algebra 30 (2021), no. 30, 269–284

    work page 2021

  32. [32]

    Miyazaki and J

    M. Miyazaki and J. Page, Non-Gorenstein loci of Ehrhart rings of chain and order polytopes , Journal of Algebra 643 (2024), 241–283

    work page 2024

  33. [33]

    Moscariello and F

    A. Moscariello and F. Strazzanti, Nearly Gorenstein numerical semigroups with five generators have bounded type, arXiv preprint arXiv:2501.08260 (2025)

    work page arXiv 2025

  34. [34]

    Ogoma, Existence of dualizing complexes, Journal of Mathematics of Kyoto University 24 (1984), no

    T. Ogoma, Existence of dualizing complexes, Journal of Mathematics of Kyoto University 24 (1984), no. 1, 27–48

    work page 1984

  35. [35]

    P Stanley, Combinatorics and commutative algebra, V ol

    R. P Stanley, Combinatorics and commutative algebra, V ol. 41, Springer Science & Business Media, 2007

    work page 2007

  36. [36]

    Varbaro and H

    M. Varbaro and H. Yu, Lefschetz duality for local cohomology, Journal of Algebra 639 (2024), 498–515

    work page 2024

  37. [37]

    H Villarreal, Monomial algebras, V ol

    R. H Villarreal, Monomial algebras, V ol. 238, Marcel Dekker New York, 2001. (Kumashiro) D EPARTMENT OF MATHEMATICS , O SAKA INSTITUTE OF TECHNOLOGY , 5-16-1 O MIYA, ASAHI -KU, O S- AKA , 535-8585, J APAN Email address: shinya.kumashiro@oit.ac.jp Email address: shinyakumashiro@gmail.com (Miyashita) D EPARTMENT OF PURE AND APPLIED MATHEMATICS , G RADUATE S...

    work page 2001