{"id":"e92531d8-b50d-44ba-8a05-3e4cda423bf8","arxiv_id":"1908.04898","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Small finite group actions on two-dimensional Artin-Schelter regular algebras satisfy the Auslander map isomorphism.","lead":"This paper proves a noncommutative version of Auslander's theorem: every finite 'small' group action on a two-dimensional quantum plane or Jordan plane yields an isomorphism A#G ≅ End_{A^G}(A). The proof classifies all such actions and then checks each case.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Classification of 2D AS regular algebras omits weighted algebras with generator degrees not both 1, so the proof of Theorem 1.3 does not cover all claimed cases.","rationale":"The reader correctly identified the assumption that every 2D AS regular algebra is generated in degree one, but treated it as a missing proof or citation. In fact, the statement is false: weighted polynomial and skew-polynomial algebras with generator degrees (1,2) satisfy the paper's Definition 2.1 of AS regular and admit small group actions, yet they are not among the algebras classified in Theorem 3.13. Consequently, the classification theorem itself is incorrect, and Theorem 1.3, while possibly true, is not established by the paper's arguments. This is a load-bearing concern because the proof of the main theorem depends entirely on the exhaustiveness of the classification. The reader's verdict of CONDITIONAL understates the issue: the missing family is not a small gap but a whole class of algebras with a different Hilbert series and different automorphism structure. I recommend REJECT as written, since the paper's claimed classification is false and the main theorem requires substantial additional work to be proven for the omitted cases.","tokens_in":40654,"tokens_out":51472,"duration_ms":495700,"concrete_test":"Verify that A = k⟨x,y⟩/(yx-qxy) with deg x = 1, deg y = 2, q ≠ 0,1, satisfies Definition 2.1: compute its Hilbert series 1/((1-t)(1-t^2)), exhibit the resolution 0 → A(-3) → A(-1)⊕A(-2) → A → k → 0, and confirm Ext^2(k,A) ≅ k[-3] while gl.dim A = 2. Then take the small subgroup G = ⟨τ⟩ with τ(x) = -x, τ(y) = -y, and check Tr_A(τ) = 1/((1+t)(1+t^2)); since this has no simple pole at t = 1, G is small. This concrete example is AS regular of dimension 2 and is absent from Theorem 3.13, demonstrating that the proof of Theorem 1.3 does not cover all pairs it claims to cover.","verdict_should_be":"REJECT","load_bearing_attack":"The proof of Theorem 1.3 reduces at the start of Section 3 to 'A is either a quantum plane or the Jordan plane', based on the claim that every two-dimensional AS regular algebra is generated in degree one. This is false under the paper's own Definition 2.1, which does not require generation in degree one. A counterexample is the weighted polynomial ring A = k[x,y] with deg x = 1 and deg y = 2. It is connected graded, finitely generated, has Hilbert series 1/((1-t)(1-t^2)), global dimension 2, and the standard Koszul-type resolution 0 → A(-3) → A(-1)⊕A(-2) → A → k → 0 gives Ext^2(k,A) ≅ k[-3] with Ext^1 = 0, so it is AS regular of dimension 2 by Definition 2.1. Similarly, the weighted quantum plane k⟨x,y⟩/(yx-qxy) with deg x=1, deg y=2 is a noncommutative 2D AS regular algebra not isomorphic to k_q[u,v] or kJ[u,v]. These algebras admit small group actions, e.g. A = k[x,y] (deg x=1, deg y=2) with G = ⟨-1,-1⟩; the trace of the nontrivial element is 1/((1+t)(1+t^2)), which has no pole of order 1 at t=1, so G is small. This pair is not in the list of Theorem 3.13. Hence the classification that supports Theorem 1.3 is incomplete, and the central theorem is unproven for an infinite family of missing cases. This is not merely a missing citation: the claimed completeness is false.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a noncommutative analogue of Auslander's theorem for two-dimensional Artin–Schelter regular algebras: for a small finite subgroup G of graded automorphisms of such an algebra A, the natural map φ: A#G → End_{A^G}(A) is an isomorphism. The proof proceeds by classifying, up to conjugation, all small subgroups of Aut_gr(A) when A is a quantum plane or the Jordan plane (Theorem 3.13), and then verifying the Chan–Kirkman–Walton–Zhang criterion that (A#G)/⟨g⟩ is finite dimensional (Theorems 4.3 and 4.8). The paper also gives presentations for many of the resulting invariant rings and identifies commutative ones with quotient surface singularities.","tokens_in":41013,"tokens_out":18159,"duration_ms":175312,"significance":"If the main theorem is established in the stated generality, this is a valuable extension of both Auslander's classical theorem and the Hopf-action results of Chan–Kirkman–Walton–Zhang in dimension 2. The paper is carefully written and contains many explicit, detailed computations, including a classification of small group actions on the quantum and Jordan planes, and substantial information about the invariant rings, including realizations as factors of AS regular algebras. The main obstruction to accepting the paper in its current form is that the classification step is not complete under the paper's own Definition 2.1, so the central theorem is not proved for all claimed cases.","major_comments":[{"comment":"The reduction to A being either the quantum plane k_q[u,v] or the Jordan plane k_J[u,v] is made without justification and is false under Definition 2.1, which does not require generation in degree one. For example, A = k[x,y] with deg x = 1 and deg y = 2 is connected graded and has Hilbert series 1/((1-t)(1-t^2)); the resolution 0 → A(-3) → A(-1)⊕A(-2) → A → k → 0 shows that it has global dimension 2 and is AS regular of dimension 2, and it is not graded-isomorphic to k[u,v] with the standard grading. The same holds for the weighted quantum plane k⟨x,y⟩/(yx - qxy) with deg x = 1, deg y = 2. On such an A, the action of G = ⟨diag(-1,-1)⟩ has graded trace Tr_A(g) = 1/((1+t)(1+t^2)), which is regular at t = 1; by Definition 2.2 this element is not a quasi-reflection, so G is small. This pair is not in the list of Theorem 3.13. Since Theorem 1.4 and the proof of Theorem 1.3 are predicated on the exhaustiveness of that list, the main theorem is unproven for an infinite family of 2D AS regular algebras. The authors should either explicitly restrict the main theorem to algebras generated in degree one or extend the classification to all 2D AS regular algebras; the present statement is not supported.","section":"Section 2.3 and Section 3, opening; Theorems 1.3, 1.4, 3.13"}],"minor_comments":[{"comment":"In the paragraph after Lemma 4.5, the text says that multiplying (u^{2(ℓ+r)} - v^{2(ℓ+r)})G_ℓ on the left by (u^{2(ℓ+r)} + v^{2(ℓ+r)}) shows that (u^{4(ℓ+r)} - v^{4(ℓ+r)}) ∈ ⟨g⟩. As written, the factor G_ℓ is dropped. The correct conclusion is (u^{4(ℓ+r)} - v^{4(ℓ+r)})G_ℓ ∈ ⟨g⟩, which is sufficient because summing over ℓ and applying Lemma 4.4(3) then removes G_ℓ. This should be corrected.","section":"Theorem 4.8 proof"},{"comment":"The phrase \"i.e. A is either a quantum plane or the Jordan plane\" in the first paragraph of Section 3 overstates what was shown in Section 2.3, where the classification is explicitly only for algebras generated in degree one. The paper should state clearly wherever this generation hypothesis is being used, and adjust the statements of Theorems 1.3 and 1.4 accordingly.","section":"Section 2.3; Section 3 opening"},{"comment":"There are minor typographical issues, such as \"A uslander\" in the abstract and \"of of\" in Section 2.2.1; these are easily fixed.","section":"Abstract and Section 1"}],"recommendation":"major_revision","confidential_remarks":"The central issue is the completeness of the classification of 2D AS regular algebras. The weighted examples are genuine under the paper's Definition 2.1, so the main theorem as stated is not proved. If the authors cannot extend the classification to the weighted cases, they should narrow the main theorem to AS regular algebras generated in degree one; the rest of the paper's contributions would remain substantial. I would not recommend acceptance before this is resolved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Crawford's paper is a solid piece of invariant theory for the standard degree-one-generated 2D AS regular algebras, but the main theorem is overclaimed: the classification silently assumes every 2D AS regular algebra is generated in degree one, which is false under his own Definition 2.1. That means Theorem 1.3, as stated, is unproven for an infinite family of missing cases.\n\nWhat is actually new and good: the classification of small subgroups for the quantum plane (q ≠ ±1), the (-1)-quantum plane, and the Jordan plane is careful and useful. The proof that the Auslander map is an isomorphism in those cases follows the CKWZ criterion and, apart from one repairable typo, goes through. The invariant ring results are also substantial: the Jordan plane Veronese presentations, the identification of commutative invariant rings of the (-1)-quantum plane with type A and D quotient singularities, and the generator formulas for the noncommutative cases. Anyone working in noncommutative invariant theory will find these computations valuable.\n\nThe soft spots are real. Section 3 opens with 'A is either a quantum plane or the Jordan plane' based on Section 2.3, which only classifies algebras generated in degree one. Under Definition 2.1, the weighted polynomial ring k[x,y] with deg x = 1, deg y = 2 is connected graded, finitely generated, has Hilbert series 1/((1-t)(1-t^2)), global dimension 2, and the Koszul resolution shows AS regularity with AS index 3. It is not isomorphic to k_q[u,v] or k_J[u,v]. Same for the weighted quantum plane. These admit small group actions (e.g., G = ⟨-1,-1⟩ on the deg 1,2 polynomial ring) that are not in Theorem 3.13. So Theorem 1.3 is unproven for an infinite family; this is a claim of completeness that is false, not a missing citation. The fix is either to extend the classification or to restrict the paper's scope to algebras generated in degree one.\n\nThe Theorem 4.8 concern is smaller. The sentence claiming to get rid of G_ℓ by multiplying with (u^{2a}+v^{2a}) is wrong; the G_ℓ survives. But the final summing argument only needs (u^s - v^s)G_ℓ ∈ ⟨g⟩, so keeping the factor throughout repairs the proof. The ℓ=0 case has enough ingredients via Lemma 4.5(1). So that's a repairable gap.\n\nWho is this for: people working on noncommutative Auslander theorems, Hopf actions on AS regular algebras, and invariant ring presentations. It deserves a serious referee, but I'd want the authors to fix the scope overclaim before publication. As stated, I wouldn't cite it for the general Theorem 1.3.","headline":"Solid invariant theory for the degree-one-generated case, but the main theorem overclaims: the classification omits weighted 2D AS regular algebras, so Theorem 1.3 is unproven for an infinite family.","tokens_in":41528,"tokens_out":6105,"would_cite":false,"duration_ms":56864,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J17","16S35","16W22"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every small group of graded symmetries of a two-dimensional quantum or Jordan plane, the Auslander map is an isomorphism.","keywords":["Artin-Schelter regular algebras","Auslander map","invariant rings","small groups","quasi-reflections","quantum plane","Jordan plane","continued fractions"],"falsifier":"For a candidate pair $(A,G)$, compute the growth dimension of the quotient $(A\\#G)/\\langle \\sum_{g\\in G} g\\rangle$; the criterion used in the paper makes the Auslander map an isomorphism exactly when this dimension is at most $0$, so any small action for which this quotient has growth dimension $1$ would disprove the theorem. The most likely place to look is a two-dimensional AS regular algebra not generated in degree one, since the classification of pairs assumes but does not prove that the degree-one list is complete.","tokens_in":40440,"feed_emoji":"🔄","tokens_out":13422,"duration_ms":135238,"temperature":0.7,"pith_summary":"The paper proves that Auslander's classical isomorphism between a skew group algebra and the endomorphism ring of an invariant ring survives in dimension two when the ring is noncommutative. Precisely: if $A$ is a two-dimensional Artin-Schelter regular algebra, a noncommutative stand-in for $\\Bbbk[u,v]$, and $G$ is a finite group of graded automorphisms containing no quasi-reflections, then the natural map $A \\# G \\to \\mathrm{End}_{A^G}(A)$ is an isomorphism. This matters because in the commutative setting that isomorphism underlies the McKay correspondence and the study of surface quotient singularities, and there was no general version for quantum or Jordan planes. To prove it, the paper classifies all possible small group actions, up to conjugation, and then checks the resulting pairs against a known growth-dimension criterion. For most cases it also writes down explicit presentations of the invariant rings and identifies them as factors of AS regular algebras.","feed_headline":"Small 2-D actions keep the Auslander map an isomorphism","feed_subtitle":"For every quantum or Jordan plane, small symmetry groups inherit the classical invariant-theory dictionary.","key_machinery":"The argument is carried by three objects. First, the Auslander map $\\varphi(ag)(b)=a(g\\cdot b)$ from the skew group algebra $A\\#G$ to the endomorphism ring $\\mathrm{End}_{A^G}(A)$, whose isomorphism is the goal. Second, the trace $\\mathrm{Tr}_A(g)$ of a graded automorphism, used to define quasi-reflections, in dimension $2$ meaning $\\mathrm{Tr}_A(g)=\\frac{1}{(1-t)(1-\\lambda t)}$ with $\\lambda\\neq 1$, and to compute Hilbert series of invariant rings by averaging traces. Third, the criterion from the literature that the Auslander map is an isomorphism for AS regular, GK-Cohen-Macaulay algebras with $\\mathrm{GKdim}\\,A \\geq 2$ exactly when $\\mathrm{GKdim}\\big((A\\#G)/\\langle \\sum_{g\\in G} g\\rangle\\big) \\leq \\mathrm{GKdim}\\,A - 2$; since $\\mathrm{GKdim}\\,A=2$, the proof reduces to showing the quotient is finite dimensional.","core_discovery":"The central claim is Theorem 1.3: for any two-dimensional AS regular algebra $A$ and any small subgroup $G$ of $\\mathrm{Aut}_{\\mathrm{gr}}(A)$, the graded map $\\varphi: A \\# G \\to \\mathrm{End}_{A^G}(A)$ defined by $\\varphi(ag)(b) = a(g\\cdot b)$ is an isomorphism. The proof rests on a classification, Theorem 3.13: up to conjugation, every noncommutative pair $(A,G)$ is either a diagonal cyclic group $\\frac{1}{n}(1,a)$ acting on the quantum plane $\\Bbbk_q[u,v]$ with $\\gcd(a,n)=1$, a group $G_{n,k}$ acting on the $(-1)$-quantum plane $\\Bbbk_{-1}[u,v]$ with $\\gcd(n,k)=1$ and $k \\not\\equiv 2 \\bmod 4$, or the scalar cyclic group $\\frac{1}{n}(1,1)$ acting on the Jordan plane $\\Bbbk_J[u,v]$. For each family the paper verifies the criterion that $(A \\# G)/\\langle \\sum_{g\\in G} g \\rangle$ has growth dimension at most $0$, which forces the Auslander map to be an isomorphism.","pith_inferences":["If the suspected converse also holds, dimension two would be an exact noncommutative analogue of the commutative theorem: the Auslander map is an isomorphism if and only if the acting group is small, and the classification in Theorem 3.13 would double as a classification of all pairs with an isomorphism.","The continued-fraction description of generators for noncommutative $\\Bbbk_{-1}[u,v]^{G_{n,k}}$ suggests that, for all coprime odd $n$ and $k$, these invariant rings are factors of AS regular algebras obtained by adjoining a central element; the fully worked $(n,k)=(7,3)$ example provides a template for such a proof.","The module bijections of Theorem 4.9 could seed a noncommutative Auslander-Reiten theory for these invariant rings, with the continued-fraction data playing a role similar to the dual graphs that encode commutative surface quotient singularities.","Because the commutative invariant rings in case (ii) realize every type D quotient singularity, properties visible in the explicit noncommutative presentations might transfer information back to the classical singularity theory, and vice versa."],"forward_implications":["Every pair in Theorem 3.13 has an isomorphic Auslander map, so the corresponding invariant rings have the graded-isolated-singularity property and satisfy the module bijections of Theorem 4.9 linking simple $kG$-modules, indecomposable summands of $A$ over $A^G$, and maximal Cohen-Macaulay $A^G$-modules.","For diagonal actions on $\\Bbbk_q[u,v]$, the invariant ring has an explicit presentation as a $q$-deformed version of the commutative quotient-singularity presentation, and it is a factor of a quantum polynomial ring.","For scalar actions on the Jordan plane, the invariant ring is generated by $n+1$ elements and is a factor of an AS regular algebra of dimension $n+1$, with relations obtained from binomial-coefficient identities.","For the nondiagonal actions on $\\Bbbk_{-1}[u,v]$, the invariant ring is commutative exactly when $n$ or $k$ is even, and in that case it is a cyclic quotient singularity or a type D surface quotient singularity; when $n$ and $k$ are both odd, explicit generators are written down using a continued fraction expansion of $n/\\frac{1}{2}(n+k)$.","Smallness is sufficient in dimension two even when the homological determinant, the noncommutative analogue of the usual determinant, is nontrivial, extending the previously known trivial-determinant case."],"supporting_citations":[{"why":"It supplies the GK-dimension criterion that reduces the Auslander isomorphism to finite dimensionality of the quotient $(A\\#G)/\\langle \\sum_{g\\in G} g\\rangle$.","marker":"[2]"},{"why":"It established the Auslander map for semisimple Hopf actions on two-dimensional AS regular algebras with trivial homological determinant, the special case this paper extends, and it supplies the generator comparison for one dihedral example.","marker":"[6]"},{"why":"It gives the generator and relation formulas for invariant rings of finite subgroups of $\\mathrm{GL}(2,\\Bbbk)$ via continued fractions, which the paper adapts to the quantum and Jordan planes.","marker":"[18]"},{"why":"It supplies the trace formalism and the Koszul trace computation used to identify quasi-reflections, as well as the averaging formula for Hilbert series of invariant rings.","marker":"[11]"},{"why":"It provides the criterion that $A^G$ is AS Gorenstein exactly when the action has trivial homological determinant, and the Reynolds operator used to compute bases of invariant rings.","marker":"[13]"},{"why":"It constructs the family of AS regular algebras that appear as the ambient factors for the Jordan-plane invariant rings.","marker":"[16]"},{"why":"It supplies the module-category bijections recorded in Theorem 4.9, linking simple $kG$-modules, indecomposable summands of $A$, and maximal Cohen-Macaulay $A^G$-modules.","marker":"[7]"}],"fun_headline_variants":["Auslander theorem holds for small groups on 2D AS algebras","Small 2D actions keep Auslander map isomorphic","For 2D AS regular algebras, small groups preserve the Auslander isomorphism","Auslander's map is isomorphic for small actions on 2D AS algebras","Small group actions on 2D AS algebras: Auslander map is an isomorphism"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The classification of all possible small actions assumes that every two-dimensional AS regular algebra is generated in degree one and therefore is one of the two known algebras, the quantum plane or the Jordan plane, but the paper states this only for algebras generated in degree one and does not prove or cite completeness of the list for all such algebras.","fun_headline_variants_meta":{"raw":{"variants":["Auslander theorem holds for small groups on 2D AS algebras","Small 2D actions keep Auslander map isomorphic","For 2D AS regular algebras, small groups preserve the Auslander isomorphism","Auslander's map is isomorphic for small actions on 2D AS algebras","Small group actions on 2D AS algebras: Auslander map is an isomorphism"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001305,"raw_usage":{"total_tokens":5371,"prompt_tokens":1045,"completion_tokens":4326,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":661,"completion_tokens_details":{"reasoning_tokens":4231}},"tokens_in":661,"tokens_out":4326,"duration_ms":30167,"temperature":1.0,"reasoning_tokens":4231,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:30:47.127910+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a candidate pair $(A,G)$, compute the growth dimension of the quotient $(A\\#G)/\\langle \\sum_{g\\in G} g\\rangle$; the criterion used in the paper makes the Auslander map an isomorphism exactly when this dimension is at most $0$, so any small action for which this quotient has growth dimension $1$ would disprove the theorem. The most likely place to look is a two-dimensional AS regular algebra not generated in degree one, since the classification of pairs assumes but does not prove that the degree-one list is complete.","supporting_citations":[{"cited_title":"2, 667–710","cited_arxiv_id":null,"evidence_quote":"It supplies the GK-dimension criterion that reduces the Auslander isomorphism to finite dimensionality of the quotient $(A\\#G)/\\langle \\sum_{g\\in G} g\\rangle$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It established the Auslander map for semisimple Hopf actions on two-dimensional AS regular algebras with trivial homological determinant, the special case this paper extends, and it supplies the generator comparison for one dihedral example."},{"cited_title":"Riemenschneider, Die Invarianten der endlichen Untergruppen von GL(2,C) , Mathematische Zeitschrift 153 (1977), no","cited_arxiv_id":null,"evidence_quote":"It gives the generator and relation formulas for invariant rings of finite subgroups of $\\mathrm{GL}(2,\\Bbbk)$ via continued fractions, which the paper adapts to the quantum and Jordan planes."},{"cited_title":"Jing and J","cited_arxiv_id":null,"evidence_quote":"It supplies the trace formalism and the Koszul trace computation used to identify quasi-reflections, as well as the averaging formula for Hilbert series of invariant rings."},{"cited_title":"Kirkman, J","cited_arxiv_id":null,"evidence_quote":"It provides the criterion that $A^G$ is AS Gorenstein exactly when the action has trivial homological determinant, and the Reynolds operator used to compute bases of invariant rings."},{"cited_title":"Lecoutre and S","cited_arxiv_id":null,"evidence_quote":"It constructs the family of AS regular algebras that appear as the ambient factors for the Jordan-plane invariant rings."},{"cited_title":"1, 87–114","cited_arxiv_id":null,"evidence_quote":"It supplies the module-category bijections recorded in Theorem 4.9, linking simple $kG$-modules, indecomposable summands of $A$, and maximal Cohen-Macaulay $A^G$-modules."}],"review_version":1}