{"id":"1c441aa4-1713-48ce-8ec8-6a7ef7837db6","arxiv_id":"2412.04511","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For geodesic ghor algebras on genus g at least 1 surfaces, every cyclic localization has global dimension at most 2g+1, with equality precisely over the noetherian locus of the center.","lead":"This paper proves a bound on the global dimension of localizations of a class of surface algebras called geodesic ghor algebras: on a genus g surface the bound is 2g+1, with equality exactly at points where the algebra's center is noetherian. The result links the homological size of a noncommutative algebra to the topology of the surface it is drawn on.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 6.3's global-dimension bound does not follow from finite-dimensional simples; no argument controls infinite-dimensional modules.","rationale":"The reader's CONDITIONAL verdict is well-founded. Among the three gaps they list, I consider the reduction from finite-dimensional simples to global dimension the most load-bearing for the stated theorem, because it is a logical prerequisite for the upper bound and is asserted in a single sentence in the proof of Theorem 6.3. Proposition 3.2 is explicitly about finite-dimensional indecomposable modules; Lemma 3.3 only shows simples are finite-dimensional, not that all modules are filtered by simples. Since A_n is not shown to be left artinian—and is non-noetherian off the noetherian locus—the supremum defining gldim can be attained on infinite-dimensional modules. No direct-limit or filtration argument is supplied, and direct limits do not preserve projective dimension. Thus even a correct resolution of every simple module (Theorem 6.2) would not yield the theorem. This concern does not contradict the reader's weakest assumption about the exactness of (6.4); both are genuine. If the exactness gap were fixed, the global-dimension gap would remain. Therefore the verdict stays CONDITIONAL: the result is plausible, but the proof as written is incomplete in a way that affects the main claim.","tokens_in":14113,"tokens_out":17882,"duration_ms":171219,"concrete_test":"On the geodesic genus-2 example (Baur–Beil, Fig. 2) with n in the noetherian locus, compute the projective dimension of a cyclic module A_n / I where I is a left ideal not contained in the annihilator of any simple module—for instance, I = A_n p for a path p whose scalar is nonzero but which is not a cycle. If pd(A_n/I) > N+1 while all simples have pd ≤ N+1, the reduction used in Theorem 6.3 is invalid. If no such module can be found and a general lemma closing the gap is supplied, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 6.3 is 'Follows from Proposition 3.2 and Theorem 6.2.' This does not establish the upper bound on gldim. Proposition 3.2 bounds pd only for finite-dimensional indecomposable modules, using a composition series; it says nothing about modules of infinite length. Global dimension is the supremum of projective dimensions over all left modules. A_n is not shown to be left artinian (and is nonnoetherian outside the noetherian locus), so not every module has a finite filtration by the simple modules of Lemma 3.3. Direct limits do not preserve projective dimension, so one cannot pass from finite-dimensional modules to all modules. Therefore, even if the complex (6.4) were a correct projective resolution of every simple A_n-module, the claimed inequality gldim A_n ≤ dim R = N+1 would not follow. The equality statement is likewise unsupported for modules that are not finite-dimensional. To close the gap the authors would need an additional argument (or a citation) showing that for these cyclic localizations, sup_M pd(M) equals sup over simple modules.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies geodesic ghor algebras A on compact surfaces obtained by identifying opposite sides and vertices of a convex 2N-gon, with center R and cycle algebra S. Its main result, Theorem 6.3, asserts that for every maximal ideal n in Max S, the cyclic localization A_n has global dimension at most dim R = dim S = N+1, with equality precisely when n lies in the noetherian locus U_{S/R} (for N >= 3). The proof strategy is to construct, in Section 6, a finite projective resolution for each simple A_n-module, of length N+1 for modules of maximal dimension vector and of length at most max{3,N} otherwise, and to combine this with Proposition 3.2, which bounds projective dimensions of finite-dimensional indecomposable modules in terms of simple modules.","tokens_in":14438,"tokens_out":4681,"duration_ms":44551,"significance":"If correct, the theorem establishes a striking and genuinely new relation between homological dimension of noncommutative localizations and the topology of the underlying surface: gldim A_n <= rank H_1(Sigma)+1, with equality exactly over the noetherian locus. The paper also offers a concrete conjectural refinement via geometric height, and the adaptation of the Berenstein-Douglas complex to higher genus surfaces is a valuable idea. The authors are careful to situate the result within their earlier work on geodesic structure and depictions of the center. However, the proof as written contains several load-bearing gaps, so the main theorem is not yet established to the standard required for publication.","major_comments":[{"comment":"The proof says 'Follows from Proposition 3.2 and Theorem 6.2', but Proposition 3.2 bounds projective dimension only for finite-dimensional indecomposable A-modules. Global dimension is the supremum of projective dimensions over all left modules, and A_n is not shown to be left artinian (indeed it is nonnoetherian outside U_{S/R}). No argument is given that sup_M pd_{A_n}(M) equals the supremum over finite-dimensional simple modules; direct limits do not preserve projective dimension, so the desired inequality gldim A_n <= N+1 does not follow from the stated results. This is a load-bearing gap that requires an additional reduction or a different argument.","section":"Section 6, Theorem 6.3 (proof)"},{"comment":"Theorem 6.2 asserts that (6.4) is a projective resolution for every simple A_n-module, but exactness is not established. The maps (6.2) presuppose that for every alpha in T (or T^sigma) there is a geodesic cycle s_j in e_i A_n e_i with [s_j] = alpha, and for every (alpha_1, alpha_2) a cycle s homotopic to t_1 t_2; the paper does not prove that such representatives exist compatibly for the stated homology classes. Moreover, exactness of the Koszul-like part requires that the elements s_j - ~s_j e_i generate the maximal ideal n and form a regular sequence on S_n (or A_n); no regularity argument is supplied. The sentence 'Therefore ... (6.4) is a projective resolution by Theorem 5.7' invokes only the first-syzygy description and does not justify exactness at the higher syzygy modules.","section":"Section 6, Theorem 6.2 and complex (6.4)"},{"comment":"The 'if and only if' for equality is not supported by the cited results. Theorem 6.2 gives projective dimensions for simple modules in terms of their dimension vector, and Proposition 5.2 shows that a simple module with dimension vector 1_Q0 has annihilator in the noetherian locus. But the equality statement additionally requires that every n in U_{S/R} admits a simple A_n-module of dimension vector 1_Q0 with pd = N+1, and that n outside U_{S/R} forces pd < N+1 for all simple modules. Neither direction is proved or cited, so the equality claim in Theorem 6.3 is currently unsupported.","section":"Section 6, Theorem 6.3 equality statement"}],"minor_comments":[{"comment":"'filteration' should be 'filtration'.","section":"Section 3, first paragraph"},{"comment":"'minimimal' should be 'minimal'.","section":"Section 6, proof of Theorem 6.2"},{"comment":"The phrase 'a representative of p + ker eta' is ambiguous; a representative should be a path in Q, not an element of the quotient algebra.","section":"Definition 2.3"},{"comment":"The paths u_j, v_j, r_j in (6.3) are not defined with enough precision; the reference to 'Figure 5.ii' should be made consistent with the figure labels, and the minimality conditions on these paths should be stated explicitly.","section":"Section 6, equations (6.3)"}],"recommendation":"major_revision","confidential_remarks":"The reader's report and the stress-test note identify the same central gap: the passage from projective dimensions of simple modules to global dimension is missing, and the exactness of the Koszul-like complex is asserted rather than proved. I agree with that assessment. The paper would be substantially strengthened by a lemma showing that gldim A_n can be computed from simple modules (or by a direct argument for arbitrary modules), and by a detailed verification of the regular-sequence/exactness properties of the complex (6.4)."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague —\n\nThe headline is this: Baur and Beil have a genuinely new idea here, and the main theorem is likely true. They construct a projective resolution of simple modules over cyclic localizations of geodesic ghor algebras, indexed by homology classes of the surface, and use it to bound gldim by N+1 = 2g+1 with equality on the noetherian locus. The link between global dimension and the topology of the surface is nice, and the resolution is not a routine tweak of the torus case. Credit where due.\n\nBut the proof as written has a load-bearing gap. Theorem 6.3 says 'Follows from Proposition 3.2 and Theorem 6.2.' It does not follow. Proposition 3.2 bounds p.d. only for finite-dimensional indecomposable modules, via a composition series. The theorem concludes gldim A_n ≤ N+1, which is the supremum over all left modules. A_n is not shown to be left noetherian or left artinian (and is nonnoetherian outside the noetherian locus), so infinite-length modules are not covered. Direct limits don't preserve projective dimension, so you can't sneak up on them. The equality statement has the same problem. There is no argument that sup over simples equals sup over all modules. This is exactly the stress-test note, and it lands.\n\nThe second gap is the resolution itself. The exactness of (6.4) is asserted: the proof of Theorem 6.2 says the Koszul maps 'cover the relations' and 'therefore' it is a projective resolution. No check of the differentials, no regular-sequence argument for the elements s_j - \\tilde{s}_j, no proof that the chosen geodesic cycles exist for every homology class in T. That is not a small missing detail; it is the heart of the construction.\n\nMinor: the theorem silently inherits the uncountable-field hypothesis from the depiction theorem in [2]. It should be stated.\n\nOn circularity: the paper leans on the authors' earlier work, but the target theorem isn't assumed there. That is fine.\n\nVerdict: this deserves a serious referee, not desk rejection. The construction is substantial and the result is plausible. But the current proof is incomplete in a way that matters. A revision needs either a direct proof that global dimension reduces to simples in this setting, or a different argument, plus a real proof of exactness.\n\nRecommendation: send to peer review; if I were refereeing, I would ask for major revision.","headline":"Plausible, genuinely new main theorem, but the proof as written does not get from projective dimensions of simples to global dimension, and the exactness of the constructed resolution is asserted rather than shown.","tokens_in":14837,"tokens_out":4840,"would_cite":false,"duration_ms":45113,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G20","16S38","16S50"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a geodesic ghor algebra on a genus g surface, each cyclic localization has global dimension at most 2g+1, attaining equality exactly over the noetherian locus of the center.","keywords":["dimer algebra","dimer model","Jacobian algebra","quiver with potential","quiver gauge theory","geodesic ghor algebra","cyclic localization","global dimension"],"falsifier":"Take a geodesic ghor algebra on a genus $2$ surface, fix a simple module with full dimension vector whose annihilator is a maximal ideal $\\mathfrak{n}$ in $\\operatorname{Max} S$, and compute the homology of the complex (6.4) at the first projected step: if the elements $s_j - \\tilde{s}_j e_i$ fail to generate $\\mathfrak{n}$, or if the complex has nonzero homology anywhere, the projective dimension is not $5$ and the theorem fails. A cheaper check is to find a single maximal ideal in the noetherian locus where a different choice of geodesic representatives changes the length of the minimal resolution.","tokens_in":13899,"feed_emoji":"📐","tokens_out":14328,"duration_ms":115253,"temperature":0.7,"pith_summary":"This paper proves a sharp homological bound for a class of noncommutative algebras built from dimer quivers on compact surfaces. For any geodesic ghor algebra on a smooth genus $g\\ge 1$ surface, every cyclic localization at a maximal ideal of the cycle algebra has global dimension at most $2g+1$, which is also the Krull dimension of the center. The bound is an equality precisely when the localization point lies in the noetherian locus of the center; outside that locus the global dimension is strictly smaller. The result matters because it recovers, for higher-genus surfaces, the torus phenomenon in which the global dimension of a localization matches the Krull dimension of the center.","feed_headline":"Global dimension of local geodesic ghor algebras capped by 2g+1","feed_subtitle":"On a genus-g surface every cyclic localization fits the homology bound; equality marks the noetherian locus.","key_machinery":"The load-bearing object is a geodesic ghor algebra $A$ on a surface, along with its cycle algebra $S$, its center $R$, and the noetherian locus $U_{S/R}$ where $R$ and $S$ coincide locally. The argument is carried by the explicit projective complex (6.4), assembled from two pieces: a Koszul-like exterior part whose basis elements are linearly independent classes in $H_1(\\Sigma)$ (with the cycle $\\sigma$ adjoined when the simple module has full dimension vector) and whose differential uses differences $s_j - \\tilde{s}_j e_i$ attached to geodesic cycles; and a generalized part modelled on the classical torus resolution that handles homotopic paths with zero scalars. The paper's Theorem 5.7 identifies the kernel of the augmentation map as being generated by exactly these elements, which makes (6.4) a projective resolution and yields the length bound.","core_discovery":"The central claim is that the global dimension of a cyclic localization of a geodesic ghor algebra is controlled by the topology of the underlying surface. On a surface obtained by identifying opposite sides and vertices of a convex $2N$-gon, the paper proves that every cyclic localization at a maximal ideal of the cycle algebra has global dimension at most $N+1 = \\dim R = \\dim S$, with equality if and only if (for $N\\ge 3$) the maximal ideal lies in the noetherian locus $U_{S/R}$. For a smooth genus $g$ surface this becomes global dimension at most $2g+1 = \\operatorname{rank} H_1(\\Sigma)+1$. The proof builds explicit projective resolutions of simple modules: a Koszul-like complex indexed by independent classes in $H_1(\\Sigma)$, extended by a generalized version of the classical torus resolution for homotopic paths, with the first syzygy identified by the paper's Theorem 5.7. Full-dimension simple modules force the resolution to have length $N+1$ over the noetherian locus, while smaller dimension vectors shorten it to length at most $\\max\\{3,N\\}$.","pith_inferences":["If the equality statement is right, global dimension could serve as a homological definition of the noetherian locus for other nonnoetherian matrix rings that admit a depiction, not just ghor algebras.","The paper's conjecture that the strict inequality over the nonnoetherian locus is governed by a geometric height would turn the bound into a precise formula; the pinched-torus examples in Section 3 are natural test cases to compute explicitly.","The homology-indexed exterior construction suggests that algebras whose relations are governed by geodesic cycles on any surface should admit analogous resolutions, with length one plus the rank of the relevant homology group.","One could try to extend the result to surfaces with punctures or boundary, where the first homology is no longer free abelian of rank $2g$; the expected bound would involve the first Betti number and boundary data."],"forward_implications":["Every cyclic localization of a geodesic ghor algebra on a smooth genus $g$ surface is homologically finite, with global dimension at most $2g+1$.","A localization has maximal possible global dimension $2g+1$ exactly when its point lies in the noetherian locus, so the global dimension functions as a detector of that locus.","The $g=1$ (torus) case reproduces the known formula $\\operatorname{gldim} A_m = 3 = \\dim R$ for noetherian dimer algebras on a torus.","The explicit resolution gives a practical way to compute projective dimensions of simple modules: length $N+1$ for full-support simples and length at most $\\max\\{3,N\\}$ otherwise.","The equality $\\operatorname{gldim} A_n = \\dim R = \\dim S$ ties homological dimension to the rank of the first homology group plus one, making the bound topological rather than accidental."],"supporting_citations":[{"why":"supplies the definition of geodesic ghor algebras, the equality ker η = ker τ, and the homological criterion [p]=[q] iff p = qσ^ℓ used throughout the proof.","marker":"[2]"},{"why":"introduces the depiction R⊂S, the noetherian locus U_{S/R}, and geometric height, all of which the equality statement depends on.","marker":"[4]"},{"why":"introduces cyclic localizations of matrix algebras, the localization procedure whose global dimension is bounded in the theorem.","marker":"[5]"},{"why":"provides the noetherian criteria for dimer algebras and the torus result R=S that motivate the higher-genus generalization.","marker":"[6]"},{"why":"develops the central geometry of nonnoetherian dimer algebras and is used for the behaviour of cyclic localizations.","marker":"[7]"},{"why":"introduces ghor algebras as the quotient kQ/ker η whose localizations are studied here.","marker":"[8]"},{"why":"gives the original projective resolution for torus dimer algebras that the new complex generalizes.","marker":"[9]"}],"fun_headline_variants":["Global dimension of local ghor algebras: cap at 2g+1","Cyclic localizations: global dimension ≤ 2g+1 on genus-g surfaces","Ghor algebra bound: 2g+1 for local dimensions on genus-g","Local ghor algebras: dimension capped by 2g+1","Genus-g surfaces cap ghor localization dimension at 2g+1"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof leans on the assertion, stated rather than demonstrated, that the geodesic cycles chosen to represent a homology basis generate the maximal ideal of the localization and form a regular sequence there, which is what makes the constructed complex a genuine projective resolution.","fun_headline_variants_meta":{"raw":{"variants":["Global dimension of local ghor algebras: cap at 2g+1","Cyclic localizations: global dimension ≤ 2g+1 on genus-g surfaces","Ghor algebra bound: 2g+1 for local dimensions on genus-g","Local ghor algebras: dimension capped by 2g+1","Genus-g surfaces cap ghor localization dimension at 2g+1"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000957,"raw_usage":{"total_tokens":4037,"prompt_tokens":861,"completion_tokens":3176,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":477,"completion_tokens_details":{"reasoning_tokens":3075}},"tokens_in":477,"tokens_out":3176,"duration_ms":18383,"temperature":1.0,"reasoning_tokens":3075,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T22:40:42.636105+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a geodesic ghor algebra on a genus $2$ surface, fix a simple module with full dimension vector whose annihilator is a maximal ideal $\\mathfrak{n}$ in $\\operatorname{Max} S$, and compute the homology of the complex (6.4) at the first projected step: if the elements $s_j - \\tilde{s}_j e_i$ fail to generate $\\mathfrak{n}$, or if the complex has nonzero homology anywhere, the projective dimension is not $5$ and the theorem fails. A cheaper check is to find a single maximal ideal in the noetherian locus where a different choice of geodesic representatives changes the length of the minimal resolution.","supporting_citations":[{"cited_title":"A generalization of cancellative dimer algebras to hyperbolic surfaces","cited_arxiv_id":"2101.11512","evidence_quote":"supplies the definition of geodesic ghor algebras, the equality ker η = ker τ, and the homological criterion [p]=[q] iff p = qσ^ℓ used throughout the proof."},{"cited_title":"Nonnoetherian geometry","cited_arxiv_id":null,"evidence_quote":"introduces the depiction R⊂S, the noetherian locus U_{S/R}, and geometric height, all of which the equality statement depends on."},{"cited_title":"Nonnoetherian homotopy dimer algebras and noncommutative crepant resolutions","cited_arxiv_id":null,"evidence_quote":"introduces cyclic localizations of matrix algebras, the localization procedure whose global dimension is bounded in the theorem."},{"cited_title":"Noetherian criteria for dimer algebras","cited_arxiv_id":null,"evidence_quote":"provides the noetherian criteria for dimer algebras and the torus result R=S that motivate the higher-genus generalization."},{"cited_title":"On the central geometry of nonnoetherian dimer algebras","cited_arxiv_id":null,"evidence_quote":"develops the central geometry of nonnoetherian dimer algebras and is used for the behaviour of cyclic localizations."},{"cited_title":"Dimer algebras, ghor algebras, and cyclic contractions","cited_arxiv_id":null,"evidence_quote":"introduces ghor algebras as the quotient kQ/ker η whose localizations are studied here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the original projective resolution for torus dimer algebras that the new complex generalizes."}],"review_version":1}