{"id":"df916605-5c3f-4f6d-be36-c4d61e3f53e8","arxiv_id":"2411.16179","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Self-injective radical-cube-zero algebras satisfy the finite generation condition (Fg) exactly when their type is Dynkin, extended Dynkin other than ~An, or ~An with a finite-order Nakayama automorphism, assuming the characteristic is good.","lead":"This paper settles, for most ground fields, which radical-cube-zero self-injective algebras have a good theory of support varieties via Hochschild cohomology. The answer is a clean criterion in terms of a graph called the type, with one exceptional family depending on a root-of-unity condition.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 2.15(4) uses finite outer order of the Nakayama automorphism, but the sufficiency proof via Theorem 2.13 and Proposition 2.4 needs an actual finite-order representative to build the cyclic skew group algebra; the missing lifting lemma is the chief gap.","rationale":"The reader identified the invertibility assumptions, char ≠ 2 and the order of the Nakayama automorphism, as the weakest point. I agree those are real: Proposition 2.8 needs 2 ∈ k^×, and the phrase 'order of the Nakayama automorphism' is ambiguous because ν is only defined up to inner automorphisms. However, the paper is explicit that the classification is 'up to assumptions on the characteristic,' so the char-2 gap is a stated limitation rather than an internal mismatch. The more damaging issue is the slide from outer finite order to actual finite order in the one case that Theorem 2.13 does not itself cover, namely quiver Prop. 5.1 of [8]. The proof of Corollary 2.15(4) proves only that (Fg) implies the outer class has finite order; the converse is needed for sufficiency and is not established. This is exactly the kind of hidden representative-picking obstruction that can break a classification, and it sits at the hinge of the final ~An case. The omitted proof of Proposition 2.10 and the compression in Proposition 2.12 are secondary; they seem fillable by routine arguments, whereas the finite-order lifting problem is a genuine mathematical condition. If a lifting lemma is true, the central claim survives; if not, Corollary 2.15(4) needs to be weakened or its hypothesis changed. I therefore keep the paper conditional on supplying this lemma, or a citation to one, and on resolving the inner-versus-outer ordering ambiguity in the standing hypothesis.","tokens_in":11241,"tokens_out":12293,"duration_ms":126002,"concrete_test":"Compute, for the ~An quiver of [8, Prop. 5.1] with commutativity coefficient q, the order of the outer class of the Nakayama automorphism in Out(Λ) as a function of q; in particular test a non-root-of-unity q by solving ν^m = Inn(u) for u ∈ Λ^*. If a non-root-of-unity q has finite outer order, Corollary 2.15(4) fails as stated; if none exists, prove the lifting lemma and insert it before Theorem 2.13. A small ~A_3 example, checked with Gröbner basis computations and then verified analytically, would settle the question.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Corollary 2.15(4) asserts that a type-~An algebra whose quiver is as in [8, Prop. 5.1] is (Fg) exactly when its Nakayama automorphism has finite order as an outer automorphism. The 'if' direction is referred to Theorem 2.13, but Theorem 2.13(2) and Proposition 2.4 require the Nakayama automorphism itself to have finite order: the group G generated by ν must be finite for ΛG to be formed, and |G| must be invertible in k. Finite order of an outer class [ν] in Out(Λ) only gives ν^m = Inn(u); it does not by itself provide a finite-order automorphism representing [ν]. No lifting lemma is supplied for the ~An, Prop. 5.1 case, in contrast to Proposition 2.12, which explicitly produces finite-order representatives outside ~An. The 'only if' direction only shows that (Fg) forces the existence of a finite-order representative with q a root of unity, not the converse implication used in the 'if' direction. So either the statement should be strengthened to 'there exists a finite-order Nakayama automorphism,' or a proof is needed that finite outer order implies such a representative exists and, equivalently, that q is a root of unity. This is the load-bearing step for the final ~An case of the classification.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a reduction strategy for deciding whether a self-injective radical-cube-zero algebra satisfies the finite-generation condition (Fg). The main idea is to pass from a Frobenius algebra Λ to its 2-quasi-Veronese Λ[2], identify Λ[2] as a Z2-smash product, and then use skew group algebras and separable equivalence to reduce the problem to the weakly symmetric case, where the classification is known. The paper states a reduction theorem (Theorem 2.13) and a full classification (Corollary 2.15) in terms of the Dynkin type of the separated quiver, the quiver shape in the ~A_n case, and the Nakayama automorphism. The exposition is concise and depends on several external results, with some proofs sketched or omitted.","tokens_in":11434,"tokens_out":19375,"duration_ms":180423,"significance":"If the stated classification is correct, the paper gives a short, conceptual proof of a result that previously required long case-by-case computations in Said's thesis, and it covers the exceptional types that were not accessible there. The use of separable equivalence and the 2-quasi-Veronese to transfer (Fg) is a genuine methodological contribution. The paper is also careful about characteristic assumptions and is explicit that the complete answer is conditional on them. However, the final ~A_n case contains a load-bearing gap in the proof of Corollary 2.15(4), so the classification is not fully established as written.","major_comments":[{"comment":"The 'if' direction of Corollary 2.15(4) is not proved. Theorem 2.13(2) requires a Nakayama automorphism of finite order as an automorphism, so that the cyclic group G generated by it can be formed and Proposition 2.4 can be applied. Finite order of the outer class [ν] in Out(Λ) only yields ν^m = Inn(u) for some unit u; it does not by itself produce a finite-order representative of that outer class. The proof of the 'only if' direction shows that (Fg) implies finite outer order, but the converse is exactly what the 'if' direction needs. The text also does not connect the finite outer order condition to the root-of-unity coefficient q of [21], which is the criterion that actually gives (Fg) in the Prop. 5.1 case. Please either prove that finite outer order implies the existence of a finite-order representative in this setting (equivalently, that q is a root of unity), or reformulate condition (4) in terms of q or of a finite-order representative.","section":"Corollary 2.15(4), Theorem 2.13(2), Proposition 2.4"},{"comment":"The reduction for types other than ~A_n depends on Proposition 2.12, whose proof is only sketched. The claim that any twisted trivial extension of a tame hereditary algebra not of type ~A_n can be endowed with a finite-order Nakayama automorphism rests on Proposition 2.10, whose proof is explicitly omitted, and on an unproved extension of [19, Proposition 1.7] to disconnected algebras. Since this is a load-bearing step in the proof of Theorem 2.13(1), the omitted proof and the disconnected-case justification should be supplied, or the relevant statements should be quoted with precise references verifying all hypotheses.","section":"Proposition 2.12, Proposition 2.10"},{"comment":"The proof of Proposition 2.1 is a sketch: it cites [16, Theorem 4.1] and asserts that the symmetric assumption can be circumvented by substituting N for D(M), but it does not carry out the functorial correspondence between the Ext modules or verify the Noetherianity transfer in the non-symmetric setting. Since all subsequent transfer steps (Proposition 2.8 and Theorem 2.13) rely on this proposition, the argument should be written out in full or the proposition should be stated as a known theorem with all hypotheses explicitly checked.","section":"Proposition 2.1"}],"minor_comments":[{"comment":"The phrase 'order of the Nakayama automorphism' is ambiguous: it could mean the order of the automorphism itself or the order of its outer class. Proposition 2.4 requires the former, while Corollary 2.15(4) is stated in terms of the latter. This ambiguity should be resolved explicitly.","section":"Theorem 2.13, Corollary 2.15"},{"comment":"The proof of Proposition 2.10 is omitted. If the statement is as elementary as it appears, a short proof should be included; if it is meant to be quoted, a reference should be given.","section":"Proposition 2.10"},{"comment":"The assertion that the split exact sequence of [19, Proposition 1.7] remains valid without the connectedness assumption whenever Out0(A) is trivial is made without proof. Please add a justification or a precise reference.","section":"Section 2, paragraph on [19, Proposition 1.7]"},{"comment":"The sentence declaring the 'if' direction for items (2)-(4) to follow from 'the above theorem in combination with the main result of [21]' is imprecise: item (3) is a direct consequence of [21], while item (4) is not a consequence of Theorem 2.13 as written.","section":"Proof of Corollary 2.15"},{"comment":"There are several minor typos: 'for all all λ′' in the nondegeneracy check, 'autmorphism' in the proof of Corollary 2.15, 'any twisted trivial extensions' in Proposition 2.12, and a duplicated 'Acknowledgments' heading. These should be corrected.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The central reduction strategy is plausible and the paper is likely to be a useful contribution once the gap in Corollary 2.15(4) is addressed. The omitted proof of Proposition 2.10 and the compressed proof of Proposition 2.1 should also be completed, since they support the main reduction. There are no concerns about attribution or scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the punchline: this note does something genuinely new. The reduction from general radical-cube-zero self-injective algebras to the weakly symmetric case via skew group algebras and separable equivalence is clean and likely correct. It handles the exceptional extended Dynkin types that Said's thesis did not, and the 2-quasi-Veronese trick is elegant. Proposition 2.1 and the transfer arguments look fine. If the main reduction holds, it is a real contribution.\n\nThe soft spots are where you would expect them. The final case of the classification, Corollary 2.15(4), has a genuine gap. The statement requires only that the Nakayama automorphism have finite order as an outer automorphism, but the proof of the \"if\" direction invokes Theorem 2.13, which needs a finite-order representative to build the cyclic skew group algebra. Finite outer order gives ν^m = Inn(u), not a finite-order automorphism, and no lifting lemma is supplied for the ~An, Prop. 5.1 case. Proposition 2.12 does the lifting outside ~An, but that is precisely where the problem is hardest. A referee will need to see a proof that finite outer order implies a finite-order representative (equivalently, that the relevant coefficient q is a root of unity) or a direct reference to Said's criterion.\n\nMinor issues: the standing assumption \"both 2 and the order of the Nakayama automorphism is invertible\" is ambiguous if the order is infinite; it should say \"if finite, its order is invertible.\" Proposition 2.10's proof is omitted, and the extension of Miyachi–Yekutieli to disconnected cases is compressed. These are minor and plausible. The characteristic 2 limitation is explicit and not a defect, but it does mean the title's \"complete answer\" is only for good characteristic.\n\nThis deserves a serious referee. The reduction and the exceptional-type results are worth publishing, and the gap in the ~An case is fixable rather than fatal. I would send it out, and ask the referee to focus on Corollary 2.15(4) and the lifting question.","headline":"A useful reduction and a real step toward completing the (Fg) classification, but the final ~An case in Corollary 2.15 has a proof gap that needs a lifting lemma.","tokens_in":12053,"tokens_out":4135,"would_cite":true,"duration_ms":48107,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16D50","16E40","16S35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that self-injective algebras with radical cubed zero satisfy the (Fg) finite-generation condition exactly when their type is Dynkin or extended Dynkin, with a single outer-automorphism condition in the remaining tilde-A…","keywords":["self-injective algebra","radical cube zero","(Fg) property","Hochschild cohomology","support varieties","skew group algebra","separable equivalence","Nakayama automorphism"],"falsifier":"Take a type $\\tilde A_n$ radical-cube-zero self-injective algebra whose quiver matches [8, Proposition 5.4], choose a commutativity relation with a coefficient that is not a root of unity, and compute directly the Noetherianity of its Hochschild cohomology; the paper predicts (Fg) holds, so a failure would refute Corollary 2.15(3).","tokens_in":10931,"feed_emoji":"🧮","tokens_out":17680,"duration_ms":134087,"temperature":0.7,"pith_summary":"The paper completes the classification of self-injective algebras whose radical cubed is zero and which satisfy (Fg), a finite-generation condition on Hochschild cohomology needed to define support varieties. For fields of characteristic not 2 in which the order of the Nakayama automorphism is invertible, the answer is: an algebra of this kind is (Fg) if and only if its type is Dynkin, or extended Dynkin other than $\\tilde A_n$, or a specific $\\tilde A_n$ case governed by the quiver shape and the outer order of the Nakayama automorphism. The main step is a reduction: each such algebra is (Fg) if and only if a certain symmetric radical-cube-zero algebra built from it by a skew group algebra construction is (Fg), and the symmetric case was already classified. This reduction, combined with prior work on the $\\tilde A_n$ family, gives the complete answer promised in the title.","feed_headline":"Complete (Fg) classification of rad-cube-zero algebras","feed_subtitle":"A reduction to weakly symmetric algebras finishes the if-and-only-if answer, except in characteristic 2.","key_machinery":"The argument relies on three constructions. First, separable equivalence, defined in [16], transfers the (Fg) property between an algebra and a skew group algebra $\\Lambda G$ when the order of $G$ is invertible in $k$ (Proposition 2.1). Second, the 2-quasi-Veronese $\\Lambda^{[2]}$ is a smash product $\\Lambda\\#\\mathbb Z_2^*$, so for $\\operatorname{char} k \\neq 2$, $\\Lambda$ is (Fg) if and only if $\\Lambda^{[2]}$ is (Proposition 2.8). Third, $\\Lambda^{[2]}$ is a twisted trivial extension $\\Delta_\\sigma A$ of a bipartite hereditary algebra $A$, whose Nakayama automorphism is $\\sigma^{-1}$. For $A$ tame hereditary of type other than $\\tilde A_n$, $\\sigma$ can be chosen of finite order because the vertex-fixing part of the outer automorphism group is trivial; the cyclic group generated by the Nakayama automorphism then makes the skew group algebra symmetric (Proposition 2.4), putting one in the already-classified weakly symmetric case [9]. This chain of equivalences produces the theorem.","core_discovery":"The central claim is that (Fg) is invariant under separable equivalence and under passage to the 2-quasi-Veronese, a graded normal form, when the characteristic differs from 2. For a connected Frobenius algebra $\\Lambda$ with $\\operatorname{rad}^3\\Lambda = 0 \\neq \\operatorname{rad}^2\\Lambda$, the 2-quasi-Veronese $\\Lambda^{[2]}$ is a twisted trivial extension of a bipartite hereditary algebra $A$, and its Nakayama automorphism is essentially the inverse of the twisting automorphism. Since the vertex-fixing part of the outer automorphism group of a tame hereditary algebra of type other than $\\tilde A_n$ is trivial, the twisting automorphism can be chosen of finite order; adjoining the cyclic group it generates makes the skew group algebra symmetric radical-cube-zero, hence (Fg) by the weakly symmetric classification in [9]. For type $\\tilde A_n$, the earlier thesis [21] provides the precise boundary. Corollary 2.15 states the resulting classification: (Fg) holds exactly for Dynkin type, for extended Dynkin type other than $\\tilde A_n$, and for $\\tilde A_n$ either when the quiver matches one of the two shapes in [8, Proposition 5.4 or 6.4], or when the quiver is the remaining shape and the Nakayama automorphism has finite order as an outer automorphism.","pith_inferences":["Beyond the paper: if one can show that the smash-product equivalence in Proposition 2.7 preserves (Fg) even in characteristic 2, the whole classification would extend verbatim; a counterexample would show the restriction is intrinsic.","Beyond the paper: the reduction suggests a general recipe — for any self-injective algebra whose Nakayama automorphism generates a finite group up to inner automorphisms, (Fg) might be equivalent to (Fg) of a symmetric skew group algebra, potentially beyond radical-cube-zero algebras.","Beyond the paper: the outer-order condition in the $\\tilde A_n$ case may be a repackaging of the root-of-unity condition on the single non-$\\pm1$ commutativity coefficient known from [21]; a direct comparison of the two invariants would make the classification more transparent."],"forward_implications":["Every connected Frobenius algebra with $\\operatorname{rad}^3\\Lambda = 0 \\neq \\operatorname{rad}^2\\Lambda$ of extended Dynkin type other than $\\tilde A_n$ satisfies (Fg), provided the field is not characteristic 2 and the Nakayama order is invertible.","For type $\\tilde A_n$, (Fg) is equivalent to having one of the two quiver shapes from [8, Proposition 5.4 or 6.4], or the remaining shape with the Nakayama automorphism of finite order as an outer automorphism.","Because (Fg) implies finite complexity, the listed algebras are exactly the ones in this family with a well-behaved support variety theory via Hochschild cohomology.","The classification is independent of the choice of Nakayama automorphism within its inner class; only the outer order matters."],"supporting_citations":[{"why":"Introduces separable equivalence and the result that it preserves finiteness of Hochschild cohomology generation; Proposition 2.1 adapts this to non-symmetric algebras.","marker":"[16]"},{"why":"Provides the bimodule direct-summand structure that makes an algebra and its skew group algebra separably equivalent, and shows the skew group algebra of a Frobenius algebra is Frobenius.","marker":"[20]"},{"why":"Classifies (Fg) for weakly symmetric radical-cube-zero algebras; this is the base case to which the paper reduces.","marker":"[9]"},{"why":"Gives the quiver and type description of radical-cube-zero self-injective algebras and the Dynkin-or-extended-Dynkin restriction imposed by finite complexity.","marker":"[8]"},{"why":"Provides the partial classification for types $\\tilde A_n$ and $\\tilde D_n$, including the root-of-unity criterion used in Corollary 2.15.","marker":"[21]"},{"why":"Shows (Fg) is invariant under Morita equivalence, used to pass between an algebra and its basic version.","marker":"[14]"},{"why":"Identifies the 2-quasi-Veronese with a smash product and gives the Morita equivalence connecting a smash product to the original skew group algebra.","marker":"[5]"}],"fun_headline_variants":["Full (Fg) classification for rad-cube-zero algebras","Rad-cube-zero algebras: complete (Fg) answer","Exact (Fg) boundary for self-injective rad-cube-zero","Solving (Fg) for radical-cube-zero algebras","Rad-cube-zero: (Fg) characterized completely"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that 2 and the order of the Nakayama automorphism are invertible in the field, since the proof passes (Fg) through a $\\mathbb Z_2$ smash product and a cyclic skew group algebra whose group order must be invertible.","fun_headline_variants_meta":{"raw":{"variants":["Full (Fg) classification for rad-cube-zero algebras","Rad-cube-zero algebras: complete (Fg) answer","Exact (Fg) boundary for self-injective rad-cube-zero","Solving (Fg) for radical-cube-zero algebras","Rad-cube-zero: (Fg) characterized completely"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000766,"raw_usage":{"total_tokens":3397,"prompt_tokens":945,"completion_tokens":2452,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":561,"completion_tokens_details":{"reasoning_tokens":2365}},"tokens_in":561,"tokens_out":2452,"duration_ms":17467,"temperature":1.0,"reasoning_tokens":2365,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:24:59.431720+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a type $\\tilde A_n$ radical-cube-zero self-injective algebra whose quiver matches [8, Proposition 5.4], choose a commutativity relation with a coefficient that is not a root of unity, and compute directly the Noetherianity of its Hochschild cohomology; the paper predicts (Fg) holds, so a failure would refute Corollary 2.15(3).","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces separable equivalence and the result that it preserves finiteness of Hochschild cohomology generation; Proposition 2.1 adapts this to non-symmetric algebras."},{"cited_title":"Algebra 92 (1985), no","cited_arxiv_id":null,"evidence_quote":"Provides the bimodule direct-summand structure that makes an algebra and its skew group algebra separably equivalent, and shows the skew group algebra of a Frobenius algebra is Frobenius."},{"cited_title":"Pure Appl","cited_arxiv_id":null,"evidence_quote":"Classifies (Fg) for weakly symmetric radical-cube-zero algebras; this is the base case to which the paper reduces."},{"cited_title":"Pure Appl","cited_arxiv_id":null,"evidence_quote":"Gives the quiver and type description of radical-cube-zero self-injective algebras and the Dynkin-or-extended-Dynkin restriction imposed by finite complexity."},{"cited_title":"Thesis, 2015","cited_arxiv_id":null,"evidence_quote":"Provides the partial classification for types $\\tilde A_n$ and $\\tilde D_n$, including the root-of-unity criterion used in Corollary 2.15."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows (Fg) is invariant under Morita equivalence, used to pass between an algebra and its basic version."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Identifies the 2-quasi-Veronese with a smash product and gives the Morita equivalence connecting a smash product to the original skew group algebra."}],"review_version":1}