REVIEW 3 major objections 4 minor 7 references
On weakly Gorenstein algebras
T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read Finitely many semi-Gorenstein projective syzygy classes force every such module to be Gorenstein projective.
desk verdict Main theorem is a genuine generalization of Ringel-Zhang and looks correct; the Auslander-Reiten conjecture section rests on false isomorphisms and needs a rewrite. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The engine is the subcategory $\varphi_n(A) = {}^\perp A \cap \Omega^n(A)$, the collection of semi-Gorenstein projective modules that appear as $n$-th syzygies. Finiteness of this subcategory is the hypothesis, and the proof's key move is that the syzygy operator $\Omega$ keeps any semi-Gorenstein projective module inside $\varphi_n(A)$ after $n$ steps while preserving indecomposability. Once two syzygies coincide, one of them is periodic, and the paper invokes the lemma that a periodic semi-Gorenstein projective module is Gorenstein projective. The dimension formula for Gorenstein projective dimension then rules out any positive finite Gorenstein projective dimension for a module with no positive extensions into $A$, closing the argument.
What would settle it
Exhibit a $\varphi_n$-finite algebra (for example, a small monomial quiver algebra) with a non-projective indecomposable module $M$ such that $\mathrm{Ext}^i_A(M,A)=0$ for all $i>0$ but $M$ is not Gorenstein projective. The theorem predicts no such module exists; a search over indecomposable modules of a known monomial algebra using the same computational tools as the paper's example could look for one.
Extended reading notes
Core claim
The central claim of the paper is Theorem 1.2: if $A$ is $\varphi_n$-finite for some $n \geq 1$, then $A$ is left weakly Gorenstein, so that every module $M$ with $\mathrm{Ext}^i_A(M,A)=0$ for all $i>0$ lies in the Gorenstein projective category. The proof considers an arbitrary non-projective indecomposable semi-Gorenstein projective module $M$. Its syzygies $\Omega^k(M)$ are indecomposable semi-Gorenstein projective modules for all $k \geq n$, so they all belong to the finite set $\varphi_n(A)$; hence $\Omega^{l+r}(M) \cong \Omega^l(M)$ for some $l \geq n$ and $r \geq 1$. Known results turn this periodic syzygy into a Gorenstein projective module, which gives $M$ finite Gorenstein projective dimension; the formula $\mathrm{Gpd}(M) = \sup\{t \geq 0 \mid \mathrm{Ext}^t_A(M,A)\neq 0\}$ then forces that dimension to be zero, so $M$ itself is Gorenstein projective. Corollary 1.3 derives weak Gorensteiness for monomial quiver algebras and for endomorphism rings of modules over representation-finite algebras, and Proposition 1.5 proves the Auslander-Reiten conjecture for left weakly Gorenstein CM-finite algebras.
Load-bearing premise
The load-bearing external premise is the lemma that a module with no positive $\mathrm{Ext}$ groups into the algebra, whose syzygy eventually repeats, must be Gorenstein projective; the paper cites this result rather than proving it, and the pigeonhole step would be useless without it.
Editorial extensions
If this is right
- Every monomial quiver algebra is weakly Gorenstein, because monomial algebras are $\Omega^2$-finite and their opposite algebras are again monomial.
- Every endomorphism algebra of a module over a representation-finite algebra is weakly Gorenstein, because such endomorphism algebras are $\Omega^2$-finite and duality preserves the relevant finiteness.
- The Auslander-Reiten conjecture holds for every $\varphi_n$-finite algebra: Theorem 1.2 makes the algebra left weakly Gorenstein, $\varphi_n$-finiteness forces CM-finiteness, and Proposition 1.5 then applies.
- The result strictly generalizes the previously known $n=1$ case, since $\varphi_k(A) \subseteq \varphi_l(A)$ for $k \geq l$ makes the hypothesis easier to satisfy for larger $n$.
- Weak Gorensteiness can occur in non-Gorenstein and representation-wild settings, as the paper's local endomorphism algebra example shows.
Reading between the lines
- (Editorial inference) A weaker hypothesis than full $\varphi_n$-finiteness would probably suffice: the pigeonhole step only needs every infinite tail of indecomposable $n$-th syzygies to contain a repeat, not the whole subcategory to be finite.
- (Editorial inference) The argument suggests a quantitative version: for a $\varphi_n$-finite algebra, the number of indecomposable modules in $\varphi_n(A)$ bounds the syzygy index at which a semi-Gorenstein projective module becomes Gorenstein projective; computations on small monomial algebras could test whether that bound is sharp.
- (Editorial inference) If other natural classes of algebras were shown to be $\varphi_n$-finite for some $n$, the same theorem would automatically make them weakly Gorenstein; special biserial algebras or higher-dimensional generalizations where syzygy categories stabilize are plausible candidates.
- (Editorial inference) Because the proof of the Auslander-Reiten conjecture only needs the Gorenstein projective category to be finite, any algebra whose $\varphi_n(A)$ is finite satisfies the conjecture; the paper's examples are consequences, not the limit.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies finite-dimensional algebras over a field and introduces the subcategory φ_n(A) = ⊥A ∩ Ω^n(A). It proves that if φ_n(A) is representation-finite for some n ≥ 1, then A is left weakly Gorenstein, i.e. every semi-Gorenstein projective module is Gorenstein projective. The proof combines the indecomposability of syzygies of non-projective semi-Gorenstein projective modules, finiteness to obtain periodicity of a high syzygy, and a known result that periodic semi-Gorenstein projective modules are Gorenstein projective. Corollary 1.3 applies this to show that all monomial algebras and all endomorphism algebras of modules over representation-finite algebras are weakly Gorenstein; an explicit QPA example illustrates the latter class. The paper also claims, in Proposition 1.5, a proof of the Auslander-Reiten conjecture for left weakly Gorenstein, CM-finite algebras.
Significance. If Theorem 1.2 stands, it is a genuine extension of the n = 1 case proved by Ringel and Zhang, with a short and largely transparent argument. The applications to monomial algebras and to endomorphism algebras over representation-finite algebras are concrete, and the example in Section 1.4 is useful. A notable strength is that the main proof does not depend on any fitted parameters or on the paper's own earlier claims; it relies on standard external results that are cited. However, the paper contains two false identities in Proposition 1.1, and the proof of Proposition 1.5 depends on them. Since Proposition 1.5 is advertised in the abstract as a proof of the Auslander-Reiten conjecture, this is a load-bearing flaw in the paper as submitted, even though it does not affect the central weakly-Gorenstein theorem. The manuscript is valuable and repairable, but it needs substantive correction.
major comments (3)
- [§1, Proposition 1.1(3)] Proposition 1.1(3) is false as stated. Let A = K[x]/(x^2), M = K, and N = A. Then A is self-injective, so Ext^1_A(K, A) = 0, while ΩK ≅ K and Hom_A(ΩK, A) ≅ Hom_A(K, A) ≅ K. Thus Ext^i_A(M, N) ≅ Hom_A(Ω^i M, N) does not hold in general. This identity is used in the proof of Proposition 1.5, so the failure is not merely cosmetic.
- [§1, Proposition 1.1(5)] Proposition 1.1(5) is also false as stated. With the same example, A = K[x]/(x^2), M = K, N = A, one has Hom_A(M, N) ≅ Hom_A(K, A) ≅ K, whereas Hom_A(ΩM, ΩN) ≅ Hom_A(K, 0) = 0. The reference to [Iya, section 2.1] cannot justify the displayed identity in this generality. This statement is used in the final chain of equalities in Proposition 1.5.
- [§1, Proposition 1.5] Because the proof of Proposition 1.5 uses the false identities 1.1(3) and 1.1(5), the claimed proof of the Auslander-Reiten conjecture is invalid as written. In addition, the pigeonhole step 'there must exist integers l, t with Ω^{l+t}(M) ≅ Ω^l(M)' is not justified by CM-finiteness alone: CM-finiteness gives only finitely many indecomposable Gorenstein projective modules, while Ω^k(M) may be decomposable or have varying numbers of indecomposable summands. One would need an additional argument, for example splitting off projective summands and using Proposition 1.1(2) on the remaining non-projective indecomposable Gorenstein projective summands, before a repetition of whole syzygies can be inferred. The statement may be true, but the present proof does not establish it.
minor comments (4)
- [§1, Theorem 1.2 proof] In the proof of Theorem 1.2, the sentence beginning 'By 1.1 (2), Gpd(M) = ...' should refer to Proposition 1.1(1), not Proposition 1.1(2).
- [Introduction] The sentence after the main theorem says monomial algebras and endomorphism rings are 'left nearly Gorenstein'; this should presumably read 'left weakly Gorenstein', since 'left nearly Gorenstein' is not defined in the paper.
- [§1, Corollary 1.3(2)] In the proof of Corollary 1.3(2), after setting A = End_B(M), the text 'the opposite algebra of End_A(M)' and 'A^op' uses the letter A inconsistently; it should refer to B, e.g. End_B(M)^op and B^op.
- [General] There are minor typographical errors, including 'Gorenstien' in the definition of weakly Gorenstein algebras and the repeated phrase 'left nearly Gorenstein'; these should be corrected in a revision.
Circularity Check
No circularity found: Theorem 1.2 follows from independent external lemmas; the only notable flaw is a non-circular correctness problem in Proposition 1.5.
full rationale
The derivation chain is self-contained in the relevant sense. Theorem 1.2 does not define its conclusion into its hypothesis, and the proof does not rely on the author's own prior work. The load-bearing inputs are external: Proposition 1.1(2) (indecomposability of syzygies of non-projective indecomposable semi-Gorenstein projective modules) is cited to Ringel-Zhang corollary 3.3; Proposition 1.1(4) (semi-Gorenstein projective plus periodic implies Gorenstein projective) is cited to Chen proposition 2.2.17; and the Gorenstein-projective-dimension formula is cited to Chen proposition 3.2.2. None of these citations is a self-citation, none is imported from the present author, and none assumes the target result that the algebra is weakly Gorenstein. The phi_n-finiteness hypothesis is a genuine finiteness condition: it is used only to force repetition among indecomposable syzygies, after which a periodic syzygy is handled by the cited external lemma. Corollary 1.3 likewise adds no circular step: monomial Omega^2-finiteness is cited to Zimmermann-Huisgen, and the endomorphism-ring argument is proved locally from representation-finiteness of the base algebra. A separate correctness concern, not a circularity, is that Proposition 1.1(3) and (5) are false as stated; for example, over K[x]/(x^2), M=K and N=A give Ext^1(M,A)=0 but Hom(Omega M,A) is nonzero, so the use of 1.1(5) in Proposition 1.5 is unsound. This affects Proposition 1.5 but does not make the main theorem circular, and under the review rules correctness objections are kept distinct from the circularity score.
Assumptions & free parameters
assumptions (6)
- standard math Gorenstein projective dimension formula: if Gpd(M) is finite then Gpd(M) = sup {t≥0 | Ext^t_A(M,A) ≠ 0}.
- standard math Non-projective indecomposable semi-Gorenstein projective modules have indecomposable syzygies Ω^k(M) for all k > 0.
- standard math A semi-Gorenstein projective periodic module is Gorenstein projective.
- standard math For semi-Gorenstein projective M, Ext^i_A(M,N) ≅ Hom_A(Ω^i(M),N) and Hom_A(M,N) ≅ Hom_A(Ω^i(M),Ω^i(N)).
- standard math Monomial algebras are Ω^2-finite.
- standard math For A = End_B(M), the functor Hom_B(M,-) gives an equivalence add(M) ≅ proj-A.
Cite this review
Pith. "Pith review of On weakly Gorenstein algebras." pith.science (2026). https://pith.science/paper/FM6SHC6S
@misc{pith2026190804738,
author = {Pith},
title = {Pith review of: On weakly Gorenstein algebras},
year = {2026},
howpublished = {\url{https://pith.science/paper/FM6SHC6S}},
note = {Machine review of arXiv:1908.04738}
}
abstract
We prove that algebras are left weakly Gorenstein in case the subcategory $^{\perp}A \cap \Omega^n(A)$ is representation-finite. This applies in particular to all monomial algebras and endomorphism algebras of modules over representation-finite algebras. We also give a proof of the Auslander-Reiten conjecture for such algebras.
Reference graph
Works this paper leans on
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[1]
Cambridge University Press, Cambridge, 1997
Auslander, M.; Reiten, I.; Smalo, S.: Representation Theory of Artin Algebras Cambridge Studies in Advanced Mathematics, 36. Cambridge University Press, Cambridge, 1997. xiv+425 pp
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[2]
https://arxiv.org/abs/1712.04587
Chen, X.: Gorenstein Homological Algebra of Artin Algebras. https://arxiv.org/abs/1712.04587
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[3]
Advances in Mathematics Volume 210, Issue 1, 20 March 2007, Pages 51-82
Iyama, O.: Auslander correspondence. Advances in Mathematics Volume 210, Issue 1, 20 March 2007, Pages 51-82
work page 2007
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[4]
The QPA-team, QPA - Quivers, path algebras and representations - a GAP package, Version 1.25; 2016 (https://folk.ntnu.no/oyvinso/QPA/)
2016
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[5]
Gorenstein-projective and semi-Gorenstein-projective modules
Ringel, C. M.; Zhang, P.: Gorenstein-projective and semi-Gorenstein-projective modules. https://arxiv.org/abs/1808.01809
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[6]
Contemporary Mathematics Volume 406, 2006
Skowronski, A.: Selfinjective algebras: Finite and tame type. Contemporary Mathematics Volume 406, 2006
work page 2006
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[7]
manuscripta mathematica, December 1991, Volume 70, Issue 1, pp 157-182
Zimmermann-Huisgen, B.: Predicting syzygies over monomial relations algebras. manuscripta mathematica, December 1991, Volume 70, Issue 1, pp 157-182
work page 1991
Reviewed August 14, 2026 · model on record in the stance chip above.
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