Full-Trace Modules

Ela Celikbas; J\"urgen Herzog; Olgur Celikbas; Shinya Kumashiro

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Full-Trace Modules

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Pith's one-line read The paper introduces full-trace modules, proves exactly when syzygies of the residue field are full-trace, and shows that in the Cohen–Macaulay case full-trace Ulrich modules characterize minimal multiplicity.

desk verdict Good new notion and a plausible characterization of minimal multiplicity, but the proof of Theorem 1.2(3) has a genuine gap and an index error that need fixing. read the letter →

arxiv 2505.14961 v1 pith:CPZFKIOG submitted 2025-05-20 math.AC

Ela Celikbas , Olgur Celikbas , Jürgen Herzog , Shinya Kumashiro This is my paper · ORCID

classification math.AC MSC 13A1513C1413D0213H1013H15

keywords full-tracemodulestraceofsyzygyUlrichminimalmultiplicityfreeresolutionsCohen-Macaulaylocalrings

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

This paper introduces a class of modules called full-trace modules: finitely generated modules whose trace ideal—the sum of the images of all homomorphisms from the module to the ring—is exactly the maximal ideal. The authors determine precisely when the syzygy modules of the residue field, the modules that appear in a minimal free resolution of the residue field, are full-trace. Over a local ring that is neither regular nor a principal ideal ring, every positive syzygy of the residue field is full-trace; over a regular ring of dimension d, exactly the first d−1 syzygies are; and over a non-regular principal ideal ring, the odd syzygies are full-trace while the even ones are not, unless the square of the maximal ideal vanishes. In the Cohen–Macaulay case, full-trace Ulrich modules—maximally generated maximal Cohen–Macaulay modules whose trace is the maximal ideal—are shown to exist exactly when the ring has minimal multiplicity. The paper closes with a structure theorem for such modules over one-dimensional Cohen–Macaulay rings and a two-dimensional example showing that the structure fails in higher dimension.

What carries the argument

The key object is the trace ideal of a module, $\operatorname{tr}_R(M)=\sum_{f\in M^*}\operatorname{im}(f)$, together with Lemma 2.4: for a homomorphism $\varphi:F\to G$ of free modules, the ideal $I(\varphi)$ generated by the entries of a matrix for $\varphi$ lies inside $\operatorname{tr}_R(\operatorname{im}\varphi)$. This turns the entry ideals of differentials in a free resolution of $k$ into lower bounds on the trace of syzygies. The Gulliksen–Tate resolution of $k$, obtained by adjoining variables to the Koszul complex to kill cycles, is chosen so that every differential has entry ideal containing $\mathfrak{m}$. For the Ulrich portion, the load-bearing identity is Lemma 3.5: if $M$ is Ulrich and $\mathfrak{q}$ is a minimal reduction of $\mathfrak{m}$, then $\mathfrak{m}\operatorname{tr}_R(M)=\mathfrak{q}\operatorname{tr}_R(M)$; with $\operatorname{tr}_R(M)=\mathfrak{m}$ this becomes $\mathfrak{m}^2=\mathfrak{q}\mathfrak{m}$, the standard characterization of minimal multiplicity.

What would settle it

Take a non-regular, non-principal-ideal local ring such as $R=k[[x,y]]/(x^2,xy)$ and compute $\operatorname{tr}_R(\Omega_R^i(k))$ for a few small $i$. Theorem 1.2(3) predicts the trace is the maximal ideal for every $i\ge 1$; any computed trace that is a proper subideal of the maximal ideal would refute the theorem. A direct check is to examine the minimal free resolution of $k$ supplied by the Gulliksen–Tate construction and test whether the entry ideal of every differential contains the maximal ideal; if some degree forces the entry ideal into the square of the maximal ideal, the key step of the proof fails.

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Extended reading notes

Core claim

This paper's central claim is that the trace ideal of a module—the ideal $\operatorname{tr}_R(M) = \sum_{f \in \operatorname{Hom}_R(M,R)} \operatorname{im}(f)$—behaves in a very regular way for the syzygy modules of the residue field. Over any local ring that is neither regular nor a principal ideal ring, every positive syzygy $\Omega_R^i(k)$ has $\operatorname{tr}_R(\Omega_R^i(k)) = \mathfrak{m}$. Over a regular local ring of dimension $d$, the first $d-1$ syzygies are full-trace and none of the later ones are; over a non-regular principal ideal ring, the odd syzygies are full-trace and the even syzygies are not unless $\mathfrak{m}^2 = 0$. The proof transfers the entry ideal of each differential in the Gulliksen–Tate minimal free resolution of $k$ into the trace of the corresponding syzygy via Lemma 2.4, which states $I(\varphi) \subseteq \operatorname{tr}_R(\operatorname{im}\varphi)$ for any map of free modules. In the Cohen–Macaulay case, the paper proves the equivalence: $R$ has minimal multiplicity if and only if it admits a full-trace Ulrich module, meaning a maximal Cohen–Macaulay module with $\mu_R(M) = e_R(M)$ and trace equal to $\mathfrak{m}$; this is Proposition 3.9 together with Corollary 3.10.

Load-bearing premise

The proof of Theorem 1.2(3) assumes that the Gulliksen–Tate minimal free resolution of the residue field can be chosen so that, at every positive degree, the entries of the differential generate the maximal ideal; the paper states this follows from Tate's and Gulliksen's theorems but does not prove the selection, and minimality alone does not guarantee it—the non-regular principal ideal ring case shows even syzygies whose differential entries generate a smaller ideal.

Editorial extensions

If this is right

Over any local ring that is neither regular nor a principal ideal ring, the residue field has infinitely many syzygy modules, and all of them are full-trace—so such rings carry infinitely many non-free modules whose trace is the maximal ideal.
For a non-regular Cohen–Macaulay local ring, minimal multiplicity is equivalent to the existence of a full-trace Ulrich module, and also equivalent to the condition that all syzygies $\Omega_R^i(k)$ with $i\ge\dim R$ are full-trace Ulrich.
In a one-dimensional Cohen–Macaulay local ring with minimal multiplicity and local endomorphism ring of the maximal ideal (for instance, a numerical semigroup ring), every full-trace Ulrich module is isomorphic to the direct sum of the maximal ideal and a module that is either zero or Ulrich.
The two-dimensional example $R=k[[x^2,xy,y^2]]$ shows the one-dimensional decomposition cannot hold in higher dimension: it has a full-trace Ulrich module that is indecomposable and whose minimal number of generators differs from that of the maximal ideal.

Reading between the lines

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

Editorial inference: If the entry-ideal property of the Gulliksen–Tate resolution holds in general, the trace of each positive syzygy of $k$ would be read off directly from the shape of the resolution, and the same technique might classify traces of syzygies of modules other than $k$ over arbitrary local rings.
Editorial inference: The characterization of minimal multiplicity by the existence of full-trace Ulrich modules suggests a module-theoretic invariant of singularities; one could test whether analogous equivalences hold for weakly Ulrich or lim Ulrich modules, extending the Brennan–Herzog–Ulrich perspective.
Editorial inference: The paper does not settle whether the direct-sum decomposition $M\cong\mathfrak{m}\oplus N$ holds for all one-dimensional Cohen–Macaulay local rings of minimal multiplicity without assuming the endomorphism ring of $\mathfrak{m}$ is local; the proof uses locality of $E$ only to drop the exponent $n$ in $M^{\oplus n}\cong\mathfrak{m}\oplus N$, so a test would be to find a one-d

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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

Dear [Colleague],

Here's my read on the full-trace modules paper. The core idea is good: define a module to be full-trace if its trace ideal is the maximal ideal, then ask when syzygies of k are full-trace and connect full-trace Ulrich modules to minimal multiplicity. The main characterization (Proposition 1.3 / Corollary 3.10) is a credible new structural result, and the Section 3 proofs are mostly clean. Proposition 3.9 (full-trace Ulrich implies minimal multiplicity via m^2 = qm) is a short, correct argument. Example 3.17 is useful.

The soft spots are both in Theorem 1.2. First, part (3) states i≥0, but for i=0 the module is k, whose trace is (0:m), not m for rings of positive depth, so the statement is false at i=0. The abstract says "positive syzygy," so this is a range typo rather than a conceptual error.

Second, and more substantial, the proof of part (3) relies on an unproved assertion about the Gulliksen–Tate resolution: after building the minimal free resolution by killing cycles, the authors claim that every differential φ_i has entry ideal containing m. They do not prove this, and it is not a formal consequence of the cited theorems. Minimality only gives I(φ_i) ⊆ m, and the principal ideal ring case shows that the analogous condition fails for even syzygies when the maximal ideal is principal. The missing step is a structural lemma about the Tate construction when the maximal ideal needs at least two generators: one must show that each X_i has a basis element involving a degree-one variable, so the differential entries generate m. Without that lemma, the conclusion that all positive syzygies are full-trace is unsupported. I suspect the claim is true, but it needs a real proof, not a sentence.

The rest of the paper stands up. The regular and PIR cases are fully argued, and the Section 3 results do not depend on the gap except through the (i)=>(ii) direction of Corollary 3.10.

Who is it for: commutative algebraists working on trace ideals, Ulrich modules, and rings of minimal multiplicity. It deserves a serious referee; a careful major revision should fix the range error and supply the missing lemma. I'd send it to review.

Best.

Referee Report

2 major / 5 minor

Summary. The paper introduces full-trace modules over commutative Noetherian local rings, i.e., finitely generated modules whose trace ideal equals the maximal ideal. It classifies the full-trace property for syzygy modules of the residue field: regular rings, non-regular principal ideal rings, and all other non-regular local rings are treated in Theorem 1.2. The second part studies full-trace Ulrich modules and proves, in Proposition 3.9 and Corollary 3.10, that a non-regular Cohen-Macaulay local ring has minimal multiplicity if and only if it admits a full-trace Ulrich module. A decomposition theorem for one-dimensional numerical semigroup rings and a two-dimensional counterexample are also given.

Significance. The notion of full-trace module is a natural counterpart to nearly Gorenstein rings, and the connection with Ulrich modules and minimal multiplicity is appealing. The regular case and the principal ideal ring case in Theorem 1.2 are proved cleanly, and Proposition 3.9 gives a short, elegant argument using established facts about reductions and traces. The paper also provides a useful two-dimensional example showing that the one-dimensional decomposition result cannot be extended verbatim. However, the proof of Theorem 1.2(3), which is the main non-regular, non-principal-ideal-ring case, relies on an under-justified property of Gulliksen-Tate resolutions. Since Corollary 3.10 and Proposition 1.3 inherit this gap, the central characterization is not yet fully established.

major comments (2)

[Section 2, proof of Theorem 1.2(3)] The assertion after 2.7 that, using [20, Thm. 1] and [9, Thm.], one obtains a minimal free resolution of k with phi_1 = partial_1, I(partial_2) subset of I(phi_2), and I(phi_i) containing I(partial_1) or I(partial_2) for all i >= 3 is load-bearing but unproved. The cited theorems guarantee the existence of a minimal R-algebra resolution, not this particular entry-ideal property. Minimality alone gives only I(phi_i) subset of m, and Proposition 2.3 shows the analogous assertion can fail for non-regular principal ideal rings, where even syzygies have entry ideal m^(n-1). A proof is needed that in the Tate-Gulliksen extension process, once at least two degree-one variables are present, every differential has m among its entries; without such a lemma the chain m subset of I(phi_i) subset of tr(Omega^i(k)) is unsupported. Because Corollary 3.10 and Proposition 1.3 depend on this theorem, the main characterization is not yet established.
[Theorem 1.2(3) and Corollary 3.10] The statement of part (3) claims Omega^i(k) is full-trace for all i >= 0, but Omega^0(k) = k. For a non-regular local ring with depth >= 1, Hom_R(k,R) = 0, so tr_R(k) = 0; for an Artinian non-regular ring with m^2 != 0, tr_R(k) is the socle, generally a proper subideal of m. Thus the statement is false as written; the abstract's phrase 'positive syzygy' indicates i >= 1 was intended, and the proof indeed only treats i >= 1. The same indexing issue affects Corollary 3.10 when d = 0: the implication (i) => (ii) needs an explicit argument that k is full-trace Ulrich under minimal multiplicity, since m^2 = 0 then, whereas 3.3(ii) only covers n >= 1.

minor comments (5)

[Lemma 2.4] The matrix representing phi is first called U and then called A; please fix the notation.
[2.7] The sentence 'X is a skew derivation' should read 'partial is a skew derivation'.
[Proof of Theorem 1.2(3)] There is a typo: 'Eventaully' should be 'Eventually'.
[Example 3.17] The identification of the second Veronese subring with k[[x,y,z]]/(xy - z^2) needs an explicit change of variables and a standing field assumption (for example, algebraically closed of characteristic not two).
[Section 3.1] In 3.1, 'a an R-module' should read 'an R-module'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Theorem 1.2 and Proposition 1.3 are derived from external trace and Ulrich results, though the proof of Theorem 1.2(3) contains an unproved differential-entry assertion that is a correctness gap, not a circular step.

full rationale

Walking the derivation chain, no load-bearing circularity is present. The full-trace notion is introduced by definition and then tested; it is never used as its own input. Theorem 1.2(1) follows from the Koszul resolution plus Lemma 2.4, and Proposition 2.3 is a direct computation of the periodic minimal resolution over a principal ideal ring. Theorem 1.2(3) invokes the external Tate/Gulliksen construction, not a self-citation: the quoted 'Eventaully, by [20, Thm. 1] and [9, Thm.], we obtain a minimal free resolution ... such that phi_1 = delta_1, I(delta_2) subset I(phi_2), and I(phi_i) contains I(delta_1) or I(delta_2)' is an unproved selection in the proof and a genuine correctness risk, but it is not circular, since the desired containment m subset tr(Omega^i(k)) is not assumed in order to build the resolution. The literal 'i >= 0' in Theorem 1.2(2)-(3) is an indexing robustness issue, not an equivalence of inputs and outputs. Proposition 3.9 and Corollary 3.10 rest on Brennan-Herzog-Ulrich [2] and the standard m^2 = qm characterization of minimal multiplicity; the only self-citations, e.g., [4] in Remark 3.11, are contextual and illustrative rather than load-bearing. Accordingly, the central derivation is self-contained up to external facts, and no circular step reduces a claimed result to its own premise.

Assumptions & free parameters 0 free parameters · 3 assumptions · 2 invented entities

The central theorem depends on standard facts about trace ideals, minimal resolutions, and Ulrich modules. No numerical parameters are fitted to data. The main new burden is the unproved assertion about the entry ideal of Gulliksen-Tate differentials, which is the key fragility in the paper.

assumptions (3)

standard math Standard facts about trace ideals, including Lindo's criterion that trace equals the ring exactly for modules with a free summand.
Invoked throughout Sections 2 and 3, mainly through reference [14].
ad hoc to paper The Tate-Gulliksen construction produces a minimal free resolution of the residue field with the property that every differential has entry ideal containing m for non-regular, non-principal-ideal rings.
This is the load-bearing, unproved step in the proof of Theorem 1.2(3). It is not an immediate consequence of minimality, as the principal ideal ring case shows.
domain assumption Cohen-Macaulay and minimal-multiplicity hypotheses in Section 3, including the faithful-flatness reduction to infinite residue field and the existence of a minimal reduction.
Used in Lemma 3.5, Proposition 3.9, and the structure results for one-dimensional rings; these are standard facts from Bruns-Herzog and Huneke-Swanson.

invented entities (2)

full-trace module
purpose: A module whose trace ideal equals the maximal ideal; this is the central new object of the paper.
Definitional. Its mathematical content is supplied by the theorems proved about it, not by external falsifiable evidence.
full-trace Ulrich module
purpose: An Ulrich module that is also full-trace; used in the characterization of minimal multiplicity.
Definitional specialization combining two existing notions. The iff result attaches mathematical value to this combination.

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Cite this review

Pith. "Pith review of Full-Trace Modules." pith.science (2026). https://pith.science/paper/CPZFKIOG

@misc{pith2026250514961,
  author       = {Pith},
  title        = {Pith review of: Full-Trace Modules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CPZFKIOG}},
  note         = {Machine review of arXiv:2505.14961}
}

read the original abstract

Motivated by the definition of nearly Gorenstein rings, we introduce the notion of full-trace modules over commutative Noetherian local rings--namely, finitely generated modules whose trace equals the maximal ideal. We investigate the existence of such modules and prove that, over rings that are neither regular nor principal ideal rings, every positive syzygy module of the residue field is full-trace. Moreover, over Cohen-Macaulay rings, we study full-trace Ulrich modules--that is, maximally generated maximal Cohen-Macaulay modules that are full-trace. We establish the following characterization: a non-regular Cohen-Macaulay local ring has minimal multiplicity if and only if it admits a full-trace Ulrich module. Finally, for numerical semigroup rings with minimal multiplicity, we show that each full-trace Ulrich module decomposes as the direct sum of the maximal ideal and a module that is either zero or Ulrich.

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When do pseudo-Gorenstein rings become Gorenstein?
math.AC 2025-02 conditional novelty 6.0 of 10

A pseudo-Gorenstein graded ring becomes Gorenstein when the trace ideal of its canonical module contains a length-two regular sequence in the initial degree, with applications to nearly and almost Gorenstein rings.

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Works this paper leans on

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Pith tools

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

[1] [1]

Stable module theory

Maurice Auslander and Mark Bridger. Stable module theory . Memoirs of the American Mathematical Society, No. 94. American Mathematical Society, Providence, R.I., 1969. 1

work page 1969

[2] [2]

Brennan, Jürgen Herzog, and Bernd Ulrich

Joseph P. Brennan, Jürgen Herzog, and Bernd Ulrich. Maximally generated Cohen-Macaulay modules. Math. Scand., 61(2):181–203, 1987. 2, 6, 7

work page 1987

[3] [3]

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Winfried Bruns and Jürgen Herzog. Cohen-Macaulay rings, volume 39 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1993. 5, 6

work page 1993

[4] [4]

Generalized Gorenstein Arf rings

Ela Celikbas, Olgur Celikbas, Shiro Goto, and Naoki Taniguchi. Generalized Gorenstein Arf rings. Ark. Mat., 57(1):35– 53, 2019. 8

work page 2019

[5] [5]

Vanishing of Tor, and why we care about it

Olgur Celikbas and Roger Wiegand. Vanishing of Tor, and why we care about it. J. Pure Appl. Algebra, 219(3):429–448,

work page

[6] [6]

Stable trace ideals and applications

Hailong Dao and Haydee Lindo. Stable trace ideals and applications. Collect. Math., 75(2):395–407, 2024. 1

work page 2024

[7] [7]

A note on the index of Cohen-Macaulay local rings

Songqing Ding. A note on the index of Cohen-Macaulay local rings. Comm. Algebra, 21(1):53–71, 1993. 1

work page 1993

[8] [8]

On modules of finite projective dimension

Sankar Dutta. On modules of finite projective dimension. Nagoya Math. Journal, 219:87–111, 2015. 3

work page 2015