{"id":"5f5f3f88-06bf-452b-a0a7-0371a66ab042","arxiv_id":"1908.04738","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An algebra whose syzygy-intersection category has only finitely many indecomposables is left weakly Gorenstein; this covers monomial algebras and endomorphism rings of modules over representation-finite algebras.","lead":"The paper proves a broad finiteness condition under which finite-dimensional algebras are weakly Gorenstein, and applies it to monomial algebras and endomorphism rings over representation-finite algebras. It also derives the Auslander-Reiten conjecture for a class of weakly Gorenstein algebras.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection to Theorem 1.2 identified; main proof is sound. Separate issue: Prop. 1.1(3) is false, invalidating Prop. 1.5.","rationale":"The reader correctly identifies Proposition 1.1(4) as the load-bearing external input for Theorem 1.2. I agree that this is the main theorem's point of dependence, but I find no internal error there: the pigeonhole step is valid and the Gpd contradiction is sound. My review instead found a separate, non-central flaw: Proposition 1.1(3) and (5) are false as stated, with an explicit counterexample over K[x]/(x^2). These statements are not used in Theorem 1.2, so the central claim stands. However, Proposition 1.5 relies on them, so its proof is invalid as written. Since the reader already recommended conditional acceptance pending clarification of Proposition 1.5, my finding does not change the verdict: the paper remains conditional on correcting the secondary argument.","tokens_in":3860,"tokens_out":24025,"duration_ms":253747,"concrete_test":"Check Proposition 1.1(3) for A=K[x]/(x^2), M=K, N=A: compute Ext^1(M,A)=0 and Hom(ΩM,A)≅K; if the claimed isomorphism fails (as it does), the lemma is false and Proposition 1.5's chain of isomorphisms must be replaced by a stable-category argument. This settles whether the paper's secondary claim is proven.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing flaw in the central theorem. The proof of Theorem 1.2 is logically coherent: for non-projective indecomposable semi-Gorenstein projective M, each Ω^k(M) (k≥n) is an indecomposable object of φ_n(A); φ_n-finiteness forces Ω^{l+r}(M)≅Ω^l(M); then [Che, 2.2.17] makes Ω^l(M) Gorenstein projective, and the Gorenstein-projective-dimension argument correctly eliminates the finite positive Gpd case. The main theorem therefore rests on standard external facts (1.1(2), 1.1(4)), which I see no reason to doubt.\n\nThe most serious correctness issue is not in Theorem 1.2 but in Proposition 1.1(3)/(5), both false as stated. Over A=K[x]/(x^2), every module is semi-Gorenstein projective; for M=K and N=A, Ext^1(M,A)=0 but Hom(ΩM,A)≅K. Thus (3) fails, and (5) fails for N=A. These false identities are used only in Proposition 1.5, so the Auslander-Reiten conjecture proof is invalid as written; this does not undermine the main theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":4084,"tokens_out":10068,"duration_ms":99222,"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":[{"comment":"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.","section":"§1, Proposition 1.1(3)"},{"comment":"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.","section":"§1, Proposition 1.1(5)"},{"comment":"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.","section":"§1, Proposition 1.5"}],"minor_comments":[{"comment":"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).","section":"§1, Theorem 1.2 proof"},{"comment":"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.","section":"Introduction"},{"comment":"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.","section":"§1, Corollary 1.3(2)"},{"comment":"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.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the central theorem of the paper appears sound and is a worthwhile extension of prior work, but the false statements in Proposition 1.1 are a serious correctness issue for the secondary claim about the Auslander-Reiten conjecture. The revision should either repair these statements and the proof of Proposition 1.5, or clearly separate the AR-conjecture claim and present only the weakly-Gorenstein theorem as the main contribution. The paper is otherwise publishable in principle."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you the short version. The main theorem is real: Theorem 1.2 generalizes Ringel-Zhang's criterion from n=1 to all n, and the proof is a straightforward pigeonhole argument that works. The corollaries—monomial algebras and endomorphism rings over representation-finite algebras are weakly Gorenstein—are worth having, and example 1.4 shows the latter class can be wild, which is a nice touch.\n\nThe paper also contains a claim that does not hold up. Proposition 1.1(3) says that for semi-Gorenstein projective M, Ext^i(M,N) ≅ Hom(Ω^i M, N). That is false. Take A=K[x]/(x^2), M=K, N=A. All modules are semi-Gorenstein projective over this self-injective algebra; Ext^1(K,A)=0, but Ω(K)=K so Hom(ΩK,A)=Hom(K,A)≠0. The same example kills (5), Hom(M,N) ≅ Hom(Ω^i M, Ω^i N), since Ω(A)=0.\n\nThose false identities are used only in the proof of Proposition 1.5, the Auslander-Reiten conjecture claim. So that proof is invalid as written. There is also a smaller gap in the same proof: CM-finiteness alone doesn't immediately make the syzygy sequence repeat unless you note that Ω preserves the number of indecomposable summands; that part is repairable. But the false isomorphisms are not a minor issue. The proposition may still be true, but it needs a correct proof or a correction to the statements (e.g., using stable Hom).\n\nNone of this touches the main theorem. Theorem 1.2 only needs 1.1(2) and (4), which look fine. The referee should ask the author to fix Proposition 1.5 or delete it, and re-check all the cited lemmas for correct hypotheses. I would send this to peer review: the main result is solid and the errors, while embarrassing, are confined to a secondary claim that is not load-bearing.","headline":"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.","tokens_in":4616,"tokens_out":10203,"would_cite":true,"duration_ms":93007,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G10","16E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Finitely many semi-Gorenstein projective syzygy classes force every such module to be Gorenstein projective.","keywords":["weakly Gorenstein algebra","semi-Gorenstein projective module","Gorenstein projective module","syzygy finiteness","monomial quiver algebra","endomorphism algebra","Auslander-Reiten conjecture","CM-finite algebra"],"falsifier":"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.","tokens_in":3636,"feed_emoji":"🔁","tokens_out":13137,"duration_ms":110597,"temperature":0.7,"pith_summary":"The paper tries to establish that a finite-dimensional algebra $A$ is left weakly Gorenstein whenever the subcategory $\\varphi_n(A) = {}^\\perp A \\cap \\Omega^n(A)$ is representation-finite for some $n \\geq 1$, meaning only finitely many indecomposable semi-Gorenstein projective modules occur as $n$-th syzygies. The proof runs a pigeonhole argument on the iterated syzygies of any semi-Gorenstein projective module, so the same finite list must repeat; a repeated periodic syzygy is then, by a known lemma, Gorenstein projective. This upgrades every semi-Gorenstein projective module to a Gorenstein projective module. The finiteness condition is shown to hold for monomial quiver algebras and for endomorphism algebras of modules over representation-finite algebras, so those algebras are weakly Gorenstein, and the Auslander-Reiten conjecture follows for them.","feed_headline":"A syzygy-counting condition forces weakly Gorenstein algebras","feed_subtitle":"Covers monomial algebras and endomorphism rings, and proves the Auslander-Reiten conjecture for them.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["(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."],"forward_implications":["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."],"supporting_citations":[{"why":"Provides the two external lemmas: a periodic semi-Gorenstein projective module is Gorenstein projective, and Gorenstein projective dimension equals the largest positive Ext degree into the algebra.","marker":"[Che]"},{"why":"Introduced weakly Gorenstein algebras and proved the n=1 case this paper generalizes; also supplies the indecomposability of syzygies of semi-Gorenstein projective modules.","marker":"[RZ]"},{"why":"Shows monomial quiver algebras are Omega^2-finite, the finiteness input used to apply Theorem 1.2 to monomial algebras.","marker":"[Z]"},{"why":"Provides the add(M)-to-projectives equivalence over the endomorphism ring, used to show endomorphism algebras are Omega^2-finite; also states the Auslander-Reiten conjecture.","marker":"[ARS]"},{"why":"Supplies the Hom isomorphism that identifies syzygy-extensions with original extensions in the Auslander-Reiten proof.","marker":"[Iya]"},{"why":"Used to compute the paper's example of a weakly Gorenstein local non-Gorenstein endomorphism algebra, showing the result covers wild-type cases.","marker":"[QPA]"}],"fun_headline_variants":["Finite syzygies force weakly Gorenstein algebras","Syzygy finiteness yields weakly Gorenstein structure","Weak Gorenstein from finite syzygies, with AR conjecture","Finite syzygy category forces weak Gorenstein, AR conjecture","Syzygy finiteness proves Auslander-Reiten for weakly Gorenstein"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Finite syzygies force weakly Gorenstein algebras","Syzygy finiteness yields weakly Gorenstein structure","Weak Gorenstein from finite syzygies, with AR conjecture","Finite syzygy category forces weak Gorenstein, AR conjecture","Syzygy finiteness proves Auslander-Reiten for weakly Gorenstein"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000662,"raw_usage":{"total_tokens":2992,"prompt_tokens":882,"completion_tokens":2110,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":498,"completion_tokens_details":{"reasoning_tokens":2020}},"tokens_in":498,"tokens_out":2110,"duration_ms":16639,"temperature":1.0,"reasoning_tokens":2020,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:35:26.001423+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}