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On strongly G-regular rings

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Pith's one-line read For any field k, a commutative local artin algebra can be G-regular yet possess a non-projective Gorenstein projective module.

desk verdict Settles Chen's problems in the negative via an explicit tensor-product construction; the main counterexample is self-contained except for one cited classification that should be verified. read the letter →

arxiv 2608.08228 v1 pith:F5YXI3LV submitted 2026-08-08 math.AC math.RT

Kaito Kimura , Yuki Mifune , Yuya Otake , Ryo Takahashi This is my paper · ORCID
classification math.ACmath.RT MSC 13C6013D0913H1016E65
keywords stronglyG-regularringGorensteinprojectivemodulequasi-dominantCM-freealgebraweaklythicksubcategorytotallyacycliccomplexartin
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 strongly G-regular rings, defined by requiring every (including infinitely generated) Gorenstein projective module to be projective, and studies how this condition relates to quasi-dominance and to the homological finiteness of thick subcategories. Its main positive results characterize strong G-regularity in terms of quasi-dominant rings and, in the Cohen–Macaulay case, in terms of covariant or contravariant finiteness of a natural thick subcategory. Its main negative result constructs, over any field $k$, a commutative local artin algebra $R$ that is weakly Gorenstein and G-regular (CM-free) but not strongly G-regular; the explicit infinitely generated Gorenstein projective module built from a totally acyclic complex refutes the three open problems (A), (B), and (C) of the artin algebra literature. This matters because those problems asked whether the finiteness of Gorenstein projective modules would force them to decompose into finitely generated pieces, and the answer turns out to be no even in the commutative local artin case.

What carries the argument

The argument runs through two main mechanisms. On the structural side, the paper uses the thick subcategory $\operatorname{thick}_{D^b(R)}\{R, D\}$ generated by the ring and its dualizing complex: strong G-regularity is equivalent to this thick subcategory being the whole bounded derived category, and quasi-dominance is the local statement that the residue field can be built from $\{R, D, X\}$ for every $X$ of infinite projective dimension; the classification of such thick subcategories then flows from the correspondence between thick subcategories of $D^b(R)$ and of $\operatorname{mod} R$, restricted through the category $C(R)$. On the constructive side, Proposition 5.6 builds a one-periodic complex $X = (\cdots \to F \to F \to \cdots)$ with differential $c = \begin{pmatrix} a & b \\ b & -a \end{pmatrix}$ on $F = G \oplus G$; given endomorphisms $a, b$ of a free module $G$ with $a^2 = b^2 = 0$, $ab = ba$, and additive splittings $g$ and $h$ satisfying $ag + ga = \operatorname{id}_G$ and $b^*h + hb^* = \operatorname{id}$, the complex is totally acyclic, so $M = \operatorname{Im} c$ is Gorenstein projective. In Theorem 5.7, the tensor product $R = A \otimes_k B$ realizes such $a, b$ on a free module with basis indexed by $\mathbb{N}^2$ using bijections from $\mathbb{N} \times \{1,\dots,p\}$ and $\mathbb{N} \times \{1,\dots,q\}$ to $\mathbb{N}$; the two tensor factors ensure $ab = ba$, while the two acyclicity conditions are checked separately on $B$-bases and $A$-bases.

What would settle it

For $k = \mathbb{F}_2$ and $A = B = k[x,y]/(x^2, xy, y^2)$, compute explicitly with the definition in Theorem 5.7 whether $M = \operatorname{Im} c$ is Gorenstein projective and non-free. If $M$ is free, or if a finitely generated non-free Gorenstein projective module exists over $R = A \otimes_k B$, the counterexample fails.

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Extended reading notes

Core claim

The paper's central claim is that strong G-regularity is strictly stronger than G-regularity, even among commutative local artin algebras that are weakly Gorenstein. For any field $k$, take artinian local $k$-algebras $A$ and $B$ with square-zero maximal ideals of $k$-dimension at least 2 and form $R = A \otimes_k B$; the paper shows that $R$ is G-regular and satisfies (tr) (equivalently, is weakly Gorenstein), but the module $M = \operatorname{Im} c$, where $c$ is the one-periodic differential on $F = G \oplus G$ constructed in Section 5, is Gorenstein projective and not projective. Consequently the three problems (A), (B), and (C) all have negative answers: over the same ring $R$, CM-freeness does not force every Gorenstein projective module to be a direct sum of finitely generated ones, nor to be projective, and CM-finiteness does not imply virtual Gorensteinness. The paper also proves a classification theorem for thick subcategories over locally quasi-dominant rings and characterizes strong G-regularity by the equality of the singular and non-Gorenstein loci together with local quasi-dominance.

Load-bearing premise

The counterexample inherits from a cited classification the claim that the tensor product ring $A \otimes_k B$ (with both maximal ideals of dimension at least two and square zero) is G-regular and satisfies (tr); if that classification fails, the example no longer separates the two notions.

Editorial extensions

If this is right

  • Problems (A), (B), and (C) all have negative answers: CM-freeness does not force every Gorenstein projective module to be a direct sum of finitely generated ones, nor to be projective, and CM-finiteness does not imply virtual Gorensteinness.
  • Strong G-regularity is strictly stronger than G-regularity even in the artinian local setting: there is a CM-free ring with a non-projective Gorenstein projective module.
  • Under mild assumptions, strong G-regularity is a local property and is characterized by local quasi-dominance together with equality of the singular and non-Gorenstein loci.
  • For a henselian Cohen–Macaulay non-Gorenstein local ring, strong G-regularity is equivalent to the covariant (equivalently contravariant) finiteness of the thick subcategory generated by the ring and its canonical module.
  • New families of strongly G-regular rings are obtained: quotients of regular rings by powers of complete-intersection ideals and certain products of parameter ideals are locally dominant and strongly G-regular when the number of generators exceeds one.

Reading between the lines

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

  • The explicit construction of a totally acyclic complex from commuting square-zero endomorphisms with additive splittings could be adapted to other tensor-product settings, potentially yielding non-projective Gorenstein projective modules over graded or noncommutative algebras.
  • The counterexample is G-regular and weakly Gorenstein but not strongly G-regular; a known theorem for artin algebras then implies it is not virtually Gorenstein, so weak Gorensteinness does not imply virtual Gorensteinness for commutative artin algebras.
  • The classification theorem suggests that quasi-dominance may be the right unifying hypothesis for subcategory classification beyond Cohen–Macaulay rings; one could test whether the correspondence extends to categories larger than $C(R)$ or to rings without dualizing complexes.
  • The non-projective Gorenstein projective module $M = \operatorname{Im} c$ is a one-periodic image, giving a concrete infinitely generated Gorenstein projective module over an artin algebra; computing its endomorphism ring or deciding whether it decomposes into finitely generated modules would be a natural next step.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper introduces and studies strongly G-regular rings, i.e., commutative noetherian rings over which every Gorenstein projective module, not necessarily finitely generated, is projective. It establishes a classification of thick subcategories over locally quasi-dominant rings (Theorem 2.11 and Corollary 2.13), characterizes strong G-regularity in terms of the derived category, quasi-dominance, and the singular/Gorenstein loci (Theorem 3.8 and Corollary 3.11), and proves a higher-dimensional analogue of the Beligiannis–Krause characterization via covariantly/contravariantly finite thick subcategories (Theorem 4.5). The central new result is Theorem 5.7: for any field k there is a commutative local finite-dimensional k-algebra R that is weakly Gorenstein and G-regular (CM-free) but not strongly G-regular; an explicit infinitely generated Gorenstein projective module is constructed. Corollary 5.9 then gives negative answers to Chen's Problems A, B, and C. The only point of fragility is that the G-regularity and the condition (tr) of the constructed ring are imported verbatim from Atkins–Vraciu [2, Example 2.5(a)], and no independent verification or precise statement of that example is provided.

Significance. If the cited result is correct, the paper resolves a cluster of open problems due to Chen and Ringel–Zhang in the negative, and it provides a substantial new theory connecting strong G-regularity with quasi-dominance and thick subcategory finiteness. The construction of a totally acyclic complex in Proposition 5.6 is explicit and the verification of total acyclicity in Theorem 5.7 is detailed and appears correct; there is no post-hoc data selection. The paper also contains useful new families of dominant and strongly G-regular rings. The main caveat is that the decisive G-regularity and (tr) properties of the counterexample are not proved in the text; they rest on a single citation. This makes the central claim conditionally sound rather than fully self-contained.

major comments (2)
  1. [§5, Theorem 5.7] The final sentence of Theorem 5.7, 'By [2, Example 2.5(a)] and Remark 3.3(2), R is G-regular and satisfies (tr)', is the sole justification for the properties that make R a counterexample to Chen's Problems A, B, and C in Corollary 5.9. The cited example is not stated, and the paper does not verify that its hypotheses hold for an arbitrary field k and for all p,q ⩾ 2. If the cited example requires additional assumptions (for instance, an algebraically closed field, or p,q > 2), or if it is misquoted, then the ring R may fail to be G-regular or to satisfy (tr), and the negative answers to Chen's problems would not follow. Since this is a load-bearing step, please either state [2, Example 2.5(a)] precisely and confirm that the present setting satisfies its hypotheses, or give a self-contained proof of the G-regularity and (tr) of R.
  2. [§5, Remark 5.8(2) and Corollary 5.9] The paper concludes that the constructed rings are weakly Gorenstein in the sense of Ringel–Zhang by combining (tr) with the equivalence asserted in Remark 3.3(2). This is an additional advertised property in Theorem 1.4 and the abstract, but the connection between (tr) and weak Gorensteinness for artin algebras is only referenced indirectly. Please provide a precise reference to [45] for the equivalence and, if needed, a one-sentence explanation showing that the artinian local algebra R satisfies the definition of weak Gorensteinness directly from (tr).
minor comments (5)
  1. [§5, Theorem 5.7] The statement 'The ring R is local with maximal ideal I=mR+nR' is used without proof; it follows from the standard fact that the tensor product over a field of two local k-algebras is local, but the authors should either justify it or supply a citation.
  2. [§2, Proposition 2.7(2)] The isomorphism m⊗^L_R R/xR ≅ m⊗_R R/xR is used without comment; it holds because x is R-regular, but a brief justification would improve clarity.
  3. [Introduction, Theorem 1.1] The displayed statement of Theorem 1.1 contains a garbled typesetting of the correspondence; the final version should ensure that the bijection between thick subcategories and generalization-closed subsets is rendered correctly.
  4. [§3, Remark 3.3(2)] The phrase 'strongly G-regular' in the sense of Atkins–Vraciu is defined using Tor-condition; the paper should explicitly note that for artinian local rings this condition is equivalent to the conjunction of G-regularity and the condition (tr), and cite the relevant theorem of [45] for the equivalence with weak Gorensteinness.
  5. [§5, Corollary 5.2] The notation 'dim k(0 :R m)' in the proof should be typeset as 'dim_k(0:_R m)' for readability; this is a purely typographical issue.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central counterexample is constructed from scratch, with external citations (including [2]) serving as genuine support rather than as restatements of the target conclusion.

full rationale

The paper's main counterexample (Theorem 5.7) is proved by explicitly constructing a one-periodic complex X with differential c on F = G ⊕ G and verifying total acyclicity in Proposition 5.6. The proof does not re-use Chen's Problems A/B/C or the paper's own definition of strong G-regularity as an input; the non-projective Gorenstein projective module M = Im c is built directly from the data of the tensor factors. The only imported assertion in the central chain is the G-regularity and (tr) of R = A ⊗_k B with dim_k m ≥ 2 and dim_k n ≥ 2, which is taken from [2, Example 2.5(a)]; that is an external peer-reviewed theorem by non-overlapping authors (Atkins and Vraciu), not a fitted parameter, not a re-labeling of Chen's questions, and not a self-citation. A failure of that cited theorem would create a correctness risk for the negative answers to Chen's problems, but external citation is not circularity. The self-citations that do occur ([50] for thick-subcategory classification, [49] for the proof of Theorem 4.5, [26] for finite CM-representation type) are to prior published theorems with independent proofs; they are used as lemmas rather than as the conclusion being derived. No circular step can be exhibited with the required specificity: no equation is equal to its input by construction, and no fitted parameter is renamed as a prediction. The score of 1 merely records the manuscript's heavy reliance on prior literature, including coauthor work, for peripheral classification results; this is normal scholarly dependence, not circularity.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

No free parameters: the constructions are explicit with no fitted constants. The axioms listed are standard background results and standing hypotheses, including the existence of a dualizing complex, condition (#), and several deep theorems from the published literature. The paper defines new terminology but no new mathematical entities.

assumptions (9)
  • domain assumption R admits a dualizing complex D (standing hypothesis in Sections 2 and 3)
    Used to define quasi-dominance (Definition 2.2) and to state condition (#).
  • domain assumption Condition (#): the Verdier quotient D^b(R)/thick{R,D} has finite dimension
    Definition 3.6(4); assumed in Theorem 3.8 and Corollary 3.11. Holds for excellent rings by Aoki [1] per Remark 3.7(2).
  • standard math Aoki [1]: quasi-excellent rings admit strong generators, so (#) holds
    Invoked in Remark 3.7(2) to ensure (#) for excellent rings.
  • standard math Iyengar-Krause [35, Theorem 5.3]: K_tac(ProjR)=0 iff thick{R,D}=D^b(R)
    Proof of Theorem 3.8, equivalence (2) iff (3).
  • standard math Dey-Mifune [28, Corollary 3.3]: annihilator description of the non-strongly G-regular locus
    Used in proving equality (3.8.1) in Theorem 3.8.
  • standard math Atkins-Vraciu [2, Example 2.5(a)]: G-regularity and (tr) of R=A⊗B with square-zero maximal ideals
    Final step of Theorem 5.7; the counterexample's G-regularity depends on it.
  • standard math Krause-Stevenson [40, Theorem 1] and Takahashi [50] bijections between thick subcategories
    Used in Proposition 2.10 and Theorem 2.11.
  • standard math Takahashi [49, Theorem 1.4] on contravariantly finite resolving subcategories
    Used in the proof of Theorem 4.5.
  • standard math Takahashi [50, Theorem 5.6] preservation of dominance under quotient by a regular element
    Used in Propositions 2.7, 5.1, and 5.3.

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Pith. "Pith review of On strongly G-regular rings." pith.science (2026). https://pith.science/paper/F5YXI3LV

@misc{pith2026260808228,
  author       = {Pith},
  title        = {Pith review of: On strongly G-regular rings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F5YXI3LV}},
  note         = {Machine review of arXiv:2608.08228}
}
read the original abstract

A noetherian ring is called G-regular when all finitely generated Gorenstein projective modules are projective. In this paper, we study rings satisfying the stronger condition that all Gorenstein projective modules are projective, which we call strongly G-regular. We show that the notion of strongly G-regular rings is closely related to that of quasi-dominant rings introduced by Takahashi and to the covariant/contravariant finiteness of a certain thick subcategory. We also answer a series of questions due to Chen in the negative, showing that the Gorenstein projective analogue of the Auslander-Ringel-Tachikawa theorem fails even for commutative local artin algebras which are weakly Gorenstein in the sense of Ringel and Zhang.

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