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When do pseudo-Gorenstein rings become Gorenstein?

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Pith's one-line read Pseudo-Gorenstein rings become Gorenstein once the trace ideal of the canonical module contains a two-element regular sequence in the smallest positive degree; for standard graded nearly Gorenstein domains of dimension at least two, this…

desk verdict Solid paper with a true main theorem: pseudo-Gorenstein plus nearly Gorenstein forces Gorenstein under a mild regular-sequence condition; the abstract overstates it and Theorem 3.4 has two mechanical proof slips, but the paper deserves refereeing. read the letter →

arxiv 2502.01133 v6 pith:HCYQC2ND submitted 2025-02-03 math.AC

Sora Miyashita This is my paper · ORCID
classification math.AC MSC 13H1013A0205E40
keywords Pseudo-GorensteinringNearlyGorensteinAlmostLevelTraceidealCanonicalmoduleQuasi-GorensteinVeronesesubalgebra
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 investigates when the weak Gorenstein-like condition called pseudo-Gorensteinness actually forces a graded ring to be Gorenstein. Its main theorem says that if the ring is generically Gorenstein, its canonical module contains a torsion-free element of lowest degree, and the trace ideal of the canonical module contains a two-element regular sequence lying in the smallest positive degree of the ring, then pseudo-Gorensteinness implies Gorensteinness. For standard graded nearly Gorenstein domains of dimension at least two these hypotheses are automatically satisfied, so in that class every pseudo-Gorenstein ring is Gorenstein. The paper applies this to sort out how nearly Gorenstein, almost Gorenstein, and level rings relate in higher-dimensional graded domains, and it constructs quasi-Gorenstein rings as Veronese subalgebras of certain non-Cohen-Macaulay rings.

What carries the argument

The machinery is the trace ideal $\operatorname{tr}_R(\omega_R)$, the ideal built from all graded module maps $\omega_R\to R$, together with a two-element regular sequence — a pair of elements each of which is a non-zero-divisor modulo the previous one — living in the smallest positive degree of $R$. The key step is Lemma 3.1: with two homogeneous non-zero-divisors $f_1,f_2$ in an ideal $I$ with $f_2\notin Rf_1$, a regular sequence $\theta_1,\theta_2$ in $R_{\operatorname{indeg}_R(\mathfrak m_R)}$ forces one of the fractions $\theta_i f_2/f_1$ to lie outside $R$, meaning the trace ideal contains an element that cannot come from inside $R$. Feeding this into the identity $\operatorname{tr}_R(I)=I\cdot I^{-1}$ for ideals containing a non-zero divisor, Theorem 3.4 shows that the canonical module's ideal summand has an $R$-free summand; then the trace ideal is the whole ring, so the ring is quasi-Gorenstein. In the Cohen-Macaulay setting, pseudo-Gorensteinness is read off the h-vector as $h_{s(R)}=1$, which is what connects the trace computation to the Gorenstein conclusion.

What would settle it

Find a standard graded nearly Gorenstein domain of dimension at least two whose last h-vector entry is 1 (so the ring is pseudo-Gorenstein), in which two degree-one elements of the trace ideal form a regular sequence, but which is not Gorenstein; the theorem says no such ring exists. A concrete place to look is a normal affine semigroup ring such as $R=\mathbb{Q}[x,xy,x^2y^3,x^3y^5]$ from Remark 4.5, where computing the trace ideal and checking the degree-one regular-sequence condition would test the boundary of the hypothesis.

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Extended reading notes

Core claim

The central discovery is a sufficient condition for pseudo-Gorenstein rings to be Gorenstein, stated as Theorem 1.2. Let $R$ be a Cohen-Macaulay generically Gorenstein graded ring whose canonical module $\omega_R$ contains a torsion-free homogeneous element of degree $-a_R$, and suppose $\operatorname{tr}_R(\omega_R)$ contains an $R$-regular sequence $\theta_1,\theta_2$ in $R_{\operatorname{indeg}_R(\mathfrak m_R)}$. If $R$ is pseudo-Gorenstein, meaning $\dim_{R_0}([\omega_R]_{-a_R})=1$, then $R$ is Gorenstein. The proof actually establishes a stronger non-Cohen-Macaulay statement, Theorem 3.7: under the same hypotheses, $\dim_{R_0}([\omega_R]_{-a_R})=1$ forces $R$ to be quasi-Gorenstein, i.e. the canonical module is a shifted copy of $R$. In the Cohen-Macaulay case quasi-Gorenstein is Gorenstein. The paper's applications show that in standard graded nearly Gorenstein domains of dimension at least two, the hypotheses are automatic, and consequently pseudo-Gorensteinness alone becomes a Gorenstein certificate.

Load-bearing premise

The proof depends on finding two elements inside the trace ideal that both lie in the smallest positive degree of the ring and that remain non-zero-divisors after dividing out by one another; if only a single such element is available, the conclusion can fail in dimension one.

Editorial extensions

If this is right

  • In any standard graded nearly Gorenstein domain of dimension at least two, pseudo-Gorensteinness is equivalent to Gorensteinness; in particular such a domain with last h-vector entry 1 must be Gorenstein.
  • In a nearly Gorenstein graded domain of dimension at least two whose Cohen-Macaulay type is 2, the canonical module is generated in a single degree, so the ring is level (Corollary 4.6(2)).
  • An almost Gorenstein standard graded domain over an algebraically closed field of characteristic zero is either Gorenstein or has minimal multiplicity, and in either case it is level (Corollary 4.7(3)).
  • For a Noetherian standard graded domain whose trace ideal contains the maximal ideal and whose lowest nonzero canonical-module piece is one-dimensional, a Veronese subalgebra of index $k$ with depth at least two is quasi-Gorenstein; if that subalgebra is Cohen-Macaulay, it is Gorenstein (Corollary 4.9).
  • Under the weaker radical condition $\sqrt{[\operatorname{tr}_R(\omega_R)]_{\operatorname{indeg}_R(\mathfrak m_R)}R}\supseteq\mathfrak m_R$, a Cohen-Macaulay ring of dimension at least two that is pseudo-Gorenstein is Gorenstein, and type 2 forces levelness (Corollary 4.4).

Reading between the lines

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

  • An implicit combinatorial consequence of Corollary 4.7 is that for Ehrhart rings of lattice polytopes, which are standard graded domains, a nearly Gorenstein Ehrhart ring whose h-vector ends in 1 must be Gorenstein; this can be tested on families of polytopes with known non-symmetric h-vectors.
  • The proof suggests that the real engine is low-degree abundance of the trace ideal: two regular-sequence elements in the minimal degree already make the trace ideal the whole ring. A natural test is whether the two-element regular sequence can be relaxed to a weaker condition in dimension at least two, since Remark 3.10 shows dimension one is genuinely exceptional.
  • The Veronese construction offers a recipe for producing quasi-Gorenstein rings: any standard graded domain whose canonical module has a one-dimensional bottom piece and whose trace ideal contains the maximal ideal yields quasi-Gorenstein Veronese subalgebras without requiring the original ring to be Cohen-Macaulay; applying this to concrete semigroup rings could produce new quasi-Gorenstein but no
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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

4 major / 2 minor

Summary. The paper studies when pseudo-Gorenstein graded rings are Gorenstein. The main result, Theorem 3.7 and Corollary 3.9, says that under a trace-ideal hypothesis—an R-regular sequence of two elements of degree indeg_R(m_R) inside tr_R(omega_R)—a generically Gorenstein ring that is pseudo-Gorenstein is quasi-Gorenstein, and if Cohen–Macaulay it is Gorenstein. The proof proceeds by showing that the canonical module, viewed as a graded ideal, has a free summand (Theorem 3.4). The paper then applies this to compare nearly Gorenstein, almost Gorenstein, and level rings for standard/semi-standard graded domains, and gives a Veronese-subalgebra criterion for quasi-Gorensteinness (Theorem 4.8, Corollary 4.9). Remark 3.10 explicitly notes that the two-element regular sequence condition cannot be weakened to one non-zero divisor, as one-dimensional pseudo-Gorenstein nearly Gorenstein domains show.

Significance. If the main theorem is correct, the paper gives a clean and usable condition under which pseudo-Gorenstein implies Gorenstein, a question that has been studied in several classes of graded rings. The application to nearly Gorenstein standard graded domains of dimension at least two is valuable, as it clarifies the hierarchy among generalizations of Gorensteinness in higher dimension. The author also honestly records the sharpness boundary of the hypothesis in Remark 3.10, which is a useful service to the community. The central argument is plausible and the overall strategy—passing through a free summand of the canonical module—is coherent. However, as written, several load-bearing proofs contain concrete errors that must be repaired before the claims can be accepted.

major comments (4)
  1. [Theorem 3.4, proof (first paragraph)] The proof asserts 'indeg_R(I)=1' from the graded isomorphism M ≅ I⊕N and indeg_R(I)<indeg_R(N). This is not a hypothesis: Theorem 3.4 only assumes indeg_R(M)<∞, and indeg_R(I) need not be 1. The subsequent argument uses f as an R0-basis of I_1 and compares degrees against 1, so the proof does not cover the stated generality as written. The theorem appears repairable by setting d=indeg_R(I), using dim_{R0} I_d=1 (which follows from dim_{R0} M_indeg(M)=1 together with the shift), and replacing the occurrences of 1 by d; nevertheless the current text is invalid.
  2. [Theorem 3.4, proof (final paragraph)] The splitting map ψ is misdefined. Setting ψ(1_R)=fg_i0 gives, for u=fg_i0, ψφ(f)=ψ(u)=u·fg_i0=u^2, not f, so ψφ≠id_I in general. The correct definition is ψ(1_R)=f/(fg_i0), which lies in I because fg_i0 is a unit in R0. This is not cosmetic: the existence of the R(−indeg_R(I))-free summand is exactly the conclusion of the theorem.
  3. [Lemma 2.20, proof] The proof reuses the letter I for the ideal isomorphic to omega_R, after the statement has already used I for the arbitrary ideal generated by a subset of R1. As a result, the concluding inclusion (n_R^{k−1}I)^{(k)} is asserted for the canonical-model ideal, not for the I of the lemma. To prove the lemma, one must name the canonical ideal, say L, and explicitly use the hypothesis tr_R(L)=tr_R(omega_R)⊇I when applying [30, Theorem 6.1]. Since Lemma 2.20 is the key input for Theorem 4.8, this needs repair.
  4. [Theorem 4.8, proof] The proof states that from omega_R(k)=(omega_R)^{(k)} and a_R∈kZ it follows that a_R(k)=a_R. This is false for the Veronese grading [R^{(k)}]_i=R_{ik}; the correct relation is a_R(k)=a_R/k. The dimension equality used later is still correct because [omega_R(k)]_{-a_R(k)} = [omega_R]_{-a_R}, so the argument survives after this correction, but the displayed equality as written is wrong.
minor comments (2)
  1. [Definition 3.2] In the parenthetical description of [∗Hom_R(M,R)]_i, the symbol N appears in the set-builder condition and in the codomain of the maps; N is not defined in this context. The codomain should be R.
  2. [Theorem 4.2, proof] The reduction 'Without loss of generality, we may assume that R0 is infinite (see Remark 2.18)' is asserted without details. Please indicate why the hypotheses—torsion-freeness of the lowest-degree canonical module element, dimension one of that component, and the radical condition on [tr_R(omega_R)]_{indeg_R(m_R)}—are preserved under the base change R→R⊗_{R0}k.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; the main theorem is a forward derivation from trace-ideal facts and is not forced by its hypotheses.

full rationale

The paper's central claim (Corollary 3.9) is proved by a chain of reductions: Theorem 3.4 (module-theoretic free-summand criterion), Theorem 3.6 (quasi-Gorensteinness from a free summand of the canonical ideal), Theorem 3.7 (passing from ideals to omega_R via generic Gorensteinness), and finally Corollary 3.9 (Cohen-Macaulay case). The pseudo-Gorenstein hypothesis dim_{R0}([omega_R]_{-a_R})=1 is used only to make the lowest-degree piece of the canonical ideal one-dimensional, not to assume the conclusion. The regular-sequence hypothesis theta_1,theta_2 in R_{indeg(m_R)} inside tr_R(omega_R) is genuinely load-bearing: Lemma 3.1 forces an element theta_i f'/f outside R, and Theorem 3.4 uses it to control degrees and produce a free summand. Remark 3.10 explicitly shows that weakening this hypothesis to a single non-zero-divisor destroys the theorem, so the hypothesis is not a restatement of the conclusion. The proof uses self-citations [26] and [30] for auxiliary facts (canonical module as a graded ideal in the generically Gorenstein case, trace criterion for quasi-Gorensteinness, and a Veronese trace inclusion), but these are parameter-free statements whose assumptions do not contain the target result; they are not equivalent by construction to the paper's conclusions. The two slips in Theorem 3.4's written proof (the blanket assertion indeg_R(I)=1 and the definition of the splitting map psi) are correctness defects to repair, not instances of circular reasoning. Since no equation in the paper reduces a claimed prediction to a fitted parameter or to the conclusion itself, the derivation is self-contained with respect to circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central theorem rests on standard trace-ideal and canonical-module facts quoted from the literature, several from the author's own prior papers ([26], [29], [30]). No numerical parameters are fitted, and no new objects are postulated. The main assumptions, generically Gorenstein, torsion-free element, and regular sequence in tr(omega), are hypotheses, not free parameters.

assumptions (5)
  • domain assumption For a generically Gorenstein Noetherian graded ring, the canonical module is isomorphic to a graded ideal of the ring.
    Invoked in the proof of Theorem 3.7 via [26, Remark 2.13 (1)]; it is a cited theorem, not proved in this paper.
  • standard math tr_R(omega_R)=R if and only if R is quasi-Gorenstein.
    Used to conclude quasi-Gorenstein in Theorems 3.6 and 4.2, cited to [26, Remark 2.9 (4)].
  • standard math For a semi-standard graded ring, the ideal generated by R_1 is m-primary.
    Used in Corollary 4.7 and Theorem 4.8 to pass from near-Gorenstein to the radical condition; cited to [38, Chapter I, 5.2].
  • standard math omega_{R^(k)} is isomorphic to (omega_R)^(k), and the a-invariant scales by the Veronese index.
    Used in Theorem 4.8; cited to [15, Corollary 3.1.3]. The proof misstates the scaling as a_R instead of a_R/k.
  • domain assumption An almost Gorenstein standard graded domain is either pseudo-Gorenstein or has minimal multiplicity.
    Used in Corollary 4.7(3); cited to [21, Theorem 4.7].

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Pith. "Pith review of When do pseudo-Gorenstein rings become Gorenstein?." pith.science (2026). https://pith.science/paper/HCYQC2ND

@misc{pith2026250201133,
  author       = {Pith},
  title        = {Pith review of: When do pseudo-Gorenstein rings become Gorenstein?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HCYQC2ND}},
  note         = {Machine review of arXiv:2502.01133}
}
read the original abstract

We discuss the relationship between the trace ideal of the canonical module and pseudo-Gorensteinness. In particular, under certain mild assumptions, we show that every pseudo-Gorenstein nearly Gorenstein graded domain is Gorenstein. As an application, we clarify the relationships among nearly Gorensteinness, almost Gorensteinness, and levelness -- notions that generalize Gorensteinness -- in the context of standard graded domains. Moreover, we give a method for constructing quasi-Gorenstein rings by taking a Veronese subalgebra of certain Noetherian graded rings.

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