{"id":"22da3093-c32a-4c87-a371-740265001cb6","arxiv_id":"1908.06156","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For finite groups of square-free order, the Ext and Tor groups between mark modules of the Burnside ring are explicitly described and are 2-periodic; for non-square-free order they have unbounded rank.","lead":"Working with the Burnside ring of a finite group, this paper computes the associated Ext and Tor groups for the mark modules, giving an explicit formula when the group order is square-free. A general reader might care because it is the first systematic description of these homological invariants for a central object in equivariant mathematics.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 20 is stated with no proof and the text jumps to the converse, so the central square-free periodicity claim rests on an unstated derivation from Corollary 19; Lemma 7's normalization is also underjustified. The gaps look repairable, so the result is plausible but conditional.","rationale":"The reader flagged Lemma 7 as the weakest assumption, and that is indeed a real gap: the proof of the block decomposition is not fully written. However, the most load-bearing gap for the paper's stated goal is the absence of any proof of Theorem 20, which is the square-free periodicity result that the abstract advertises as giving a complete description. In good faith, I do not see a contradiction in the mathematics: the recurrence relations, Dress's theorem, and the block decomposition are credible, and the C_p example confirms the predicted parity pattern. The missing pieces seem repairable by standard arguments, so the central claim is plausible. The paper should be treated conditionally until Theorem 20 is actually proved and Lemma 7/Proposition 8 are completed. I therefore agree with the reader's conditional verdict without endorsing the specific choice of Lemma 7 as the single weakest point.","tokens_in":10034,"tokens_out":35885,"duration_ms":373856,"concrete_test":"Write out the missing proof of Theorem 20: fix H,J and, for each p | |G|, use Corollary 19 to identify the p-primary component of Ext^l_{A(G)}(Z_H,Z_J) with Z/pZ on a fixed parity; then use the module-structure facts from Section 2 (r acts by both π_H(r) and π_J(r)) to show the direct sum over p admits an A(G)-linear isomorphism Ext^l ≅ Ext^{l+2} for every l ≥ 1. Independently, compute Ext^*_{A(S_3)}(Z_H,Z_J) for all H,J up to degree 4 using an explicit free resolution over A(S_3) and compare with the parity predictions of Corollary 19; if the parity pattern fails, the central claim is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing gap is that Theorem 20, which is the paper's central square-free periodicity statement, is never proved. The text states it and then immediately says 'In the remainder we establish the converse,' so the claimed complete description depends entirely on an implicit argument assembling the p-primary data from Corollary 19. That assembly is not formal: for each prime p dividing |G|, Corollary 19 gives the p-primary part as Z/pZ on a fixed parity, and one must show that the direct sum over p carries an A(G)-module structure for which the shift by 2 is an isomorphism. No such argument appears. A secondary but related gap is Lemma 7: the constructed element r is normalized only at a chosen representative j of each complementary class E', and the proof does not state why r vanishes mod p on every other element of E'. This is repairable because E' is a ~_p equivalence class, so every element of R is constant mod p on E', but as written Proposition 8's surjectivity, and hence the block decomposition underlying Corollaries 11, 14, and 19, is not justified. Both gaps appear fillable by standard arguments, and a check of the C_p case confirms the stated parity pattern, so the correct verdict is conditional rather than rejection.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Ext and Tor groups between the mark modules Z_H over the Burnside ring A(G) of a finite group G. It first develops the framework of B-rings, i.e. subrings of the ghost ring Gh(I), and proves a structural theorem modulo a prime p: the mod-p quotient R = R/pR decomposes as a direct sum of local k-algebras whose simple modules correspond to equivalence classes of the relation defined by p | d(i,j). Two recurrences relate the p-ranks of Ext^l_R(Z_i,Z_j) and Tor^R_l(Z_i,Z_j) to the dimensions of Ext/Tor over the mod-p algebra, and Corollary 11 derives the clean identity z_l = a_{l+1}. Applying this to the Burnside ring via the embedding into the ghost ring and Dress's congruence criterion, the paper claims that for square-free |G| all Ext and Tor groups are periodic in l with period 2, with p-primary parts equal to Z/pZ on a parity determined by the ∼_p-class structure (Corollary 19 and Theorem 20). For non-square-free |G|, it claims that Ext and Tor have unbounded rank (Theorem 23), proved using Gustafson's theorem that A(G)⊗F_p is not symmetric and Gulliksen's theorem on the homology of local rings.","tokens_in":10265,"tokens_out":12662,"duration_ms":114622,"significance":"If the missing proof of Theorem 20 is supplied, the paper would give a complete, explicit computation of Ext and Tor between mark modules for all finite groups of square-free order, and a sharp unboundedness result in the complementary case. The B-ring framework and the two-term recurrences are elegant and of independent interest; Corollary 11 (z_l = a_{l+1}) is a particularly clean structural fact. The use of Dress's characterization to read the p-divisibility of d(H,J) directly from the subgroup lattice is a strong idea, and the non-square-free argument correctly imports Gustafson's and Gulliksen's theorems. The claims are stated with full precision and are falsifiable.","major_comments":[{"comment":"The central square-free periodicity theorem is stated without proof. After the statement the text reads 'In the remainder we establish the converse,' and the proof never returns to Theorem 20. The author should supply a proof that the p-primary pattern of Corollary 19 assembles into an A(G)-module isomorphism Ext^l_{A(G)}(Z_H,Z_J) ≅ Ext^{l+2}_{A(G)}(Z_H,Z_J) for each l ≥ 1. This requires checking that the A(G)-module structure on each non-zero Z/pZ summand (given by the mark homomorphism at H or at J) is the same at l and at l+2, and that the direct sum over primes p of the p-primary parts is a direct sum of A(G)-submodules. Without this argument the abstract's 'complete description' in the square-free case is not established.","section":"Theorem 20"},{"comment":"The proof of Lemma 7 constructs r as the product over the complementary equivalence classes E' of elements r_{E'} that are normalized only at a chosen representative j of E': the proof ensures r_{E'}(j) = 0 and r_{E'}(i) ≡ 1 mod p, but it does not justify that r_{E'}(j') ≡ 0 mod p for all remaining j' ∈ E'. This follows from the definition of ∼_p, because every element of R is constant modulo p on each equivalence class, so the vanishing at j propagates to all of E'. Since this vanishing is needed for the surjectivity of θ in Proposition 8, and hence for the block decomposition used in Corollaries 11, 14, and 19, the one-line justification should be added.","section":"Lemma 7"}],"minor_comments":[{"comment":"In the converse direction of Proposition 3, the claim that the coordinate projections of S' land in Z would benefit from a one-sentence justification: a subring of Q that is finitely generated as a Z-module is contained in Z, and this applies factorwise.","section":"Proposition 3"},{"comment":"Theorem 23 proves unbounded rank only for Ext, whereas the abstract also promises unbounded Tor. The author should add the one-line deduction from Corollary 11 (z_l = a_{l+1}) so that the stated result matches the abstract.","section":"Theorem 23"},{"comment":"The term 'rank' is used for the p-rank (dimension over F_p) when defining a_l and z_l, while the same word elsewhere refers to the Z-rank of finite abelian groups; clarifying this would avoid ambiguity.","section":"B-rings section"},{"comment":"There are minor typographical issues: the affiliation line reads 'York YO10 5D D' with an extra space, and the displayed long exact sequence after the short exact sequence (†) is poorly aligned. These do not affect the mathematics.","section":"Affiliation and diagrams"}],"recommendation":"major_revision","confidential_remarks":"The two gaps identified are both repairable within the scope of the manuscript: Theorem 20 needs a proof that assembles Corollary 19, and Lemma 7 needs one sentence invoking constancy of elements of R modulo p on each ∼_p class. If these are supplied, the paper is likely publishable. The external theorems (Gustafson, Gulliksen) are used appropriately, and I see no indication of any deeper flaw."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe takeaway: the paper gives a clean B-ring framework for Ext and Tor between mark modules, and the main square-free description is probably correct, but the paper does not actually prove its central periodicity theorem (Theorem 20) — it states it and then moves on to the converse. The other gap, Lemma 7, is smaller: a missing one-sentence justification. Both look fillable.\n\nWhat is genuinely new: the recurrence a_{l+1}=b_l-a_l and z_l = a_{l+1} relating p-ranks of Ext/Tor to the mod-p block algebra, plus the use of Dress's theorem to identify the equivalence classes with O^p(H) conjugacy. The consequence — square-free order gives a full description of Ext^l and Tor_l, alternating between 0 and Z/pZ depending on parity and whether the two conjugacy classes form a 2-element block — is a nice structural result. The non-square-free unboundedness via Gustafson and Gulliksen is also well argued and gives a clean contrast.\n\nSoft spots, in order. First, Theorem 20 is announced with no proof; the text immediately says 'In the remainder we establish the converse.' The intended argument must assemble the p-primary decompositions from Corollary 19 and show the shift by 2 is an A(G)-module isomorphism. That is not formal: the module action is via the mark homomorphisms, and the parity pattern in Corollary 19 gives it for each prime, but you have to verify compatibility across primes and with the A(G)-action. This is the main gap and it is load-bearing. Second, Lemma 7: the product r_E' is normalized only at a chosen representative j of each complementary class, and it doesn't state why it vanishes on the rest of E'. That follows immediately from the definition of d(i,j) (all elements of a block are congruent mod p), so this is a one-line fix, but as written Proposition 8's surjectivity is not fully justified. Third, the notation O^p(H) is written as Op(H), and with subscripts floating it looks like O_p(H); that's a minor readability issue, not substance.\n\nI checked the C_p example myself and the parity pattern from Corollary 14 is right. The recurrences are derived, not fitted. The citations look appropriate; no circularity.\n\nOverall: this is a credible paper with one real missing proof and one near-miss. Readers working on Burnside rings or equivariant cohomology will want the result. A serious referee should see it — the author needs to fill the gap in Theorem 20 and clarify Lemma 7 before it is published. I would accept it for peer review with that expectation.","headline":"Useful B-ring framework and believable claims, but the central square-free periodicity theorem is stated without proof and the block-decomposition gap in Lemma 7 is a missing one-liner.","tokens_in":10785,"tokens_out":3633,"would_cite":true,"duration_ms":36425,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16E30","19A22","16E40"],"pacs":[],"model":"deepseek-v4-flash","headline":"For square-free group orders, Ext between mark modules is 2-periodic; non-square-free orders give unbounded rank.","keywords":["Burnside ring","mark homomorphism","Ext and Tor","B-ring","ghost ring","square-free order","periodic cohomology","mod-p block decomposition"],"falsifier":"Work with $G=S_3$ and the two non-conjugate subgroups of order $2$; compute $\\mathrm{Ext}^1$ and $\\mathrm{Ext}^3$ between the corresponding mark modules and check whether they are isomorphic as $A(G)$-modules with the predicted $\\mathbb{Z}/2\\mathbb{Z}$ value. A mismatch in rank or module structure would refute Theorem 20. Alternatively, test Lemma 7 directly on a B-ring in which a $\\sim_p$-class has more than one member: if the constructed product does not vanish modulo $p$ on an unselected member of a complementary class, Proposition 8 would need another proof.","tokens_in":9795,"feed_emoji":"🔁","tokens_out":10320,"duration_ms":93069,"temperature":0.7,"pith_summary":"The Burnside ring $A(G)$ organizes the isomorphism classes of finite $G$-sets, and for each subgroup $H$ the mark module $\\mathbb{Z}_H$ is the integers with $A(G)$ acting through the number of $H$-fixed points. The paper proves that when $|G|$ is square-free, the homological algebra of these modules is completely determined: for every $H,J$ and every $l \\geq 1$, $\\mathrm{Ext}^l_{A(G)}(\\mathbb{Z}_H,\\mathbb{Z}_J) \\cong \\mathrm{Ext}^{l+2}_{A(G)}(\\mathbb{Z}_H,\\mathbb{Z}_J)$, and the same two-step periodicity governs $\\mathrm{Tor}$. Concretely, each prime $p$ dividing $|G|$ contributes a single copy of $\\mathbb{Z}/p\\mathbb{Z}$, appearing only in degrees of one fixed parity, and all other $p$-parts vanish; which parity occurs is read from whether $H$ and $J$ form a two-element equivalence class under agreement modulo $p$. The paper also establishes the converse: if $|G|$ is not square-free, some $\\mathrm{Ext}$ and $\\mathrm{Tor}$ groups have unbounded rank as $l$ grows, so no finite table can describe them. The payoff is a closed formula for an entire family of finite groups where explicit computations are otherwise intractable.","feed_headline":"Square-free group orders make Ext groups 2-periodic","feed_subtitle":"A complete description of Ext and Tor for Burnside-ring modules, with non-square-free orders giving unbounded rank.","key_machinery":"The argument is carried by B-rings: subrings $R$ of a product of copies of $\\mathbb{Z}$ (the ghost ring) with the property that any two distinct coordinates can be separated by an element of $R$ that is nonzero in one coordinate and zero in the other. The Burnside ring embeds as a B-ring through its mark homomorphisms. For a prime $p$, two coordinates $i,j$ are related by $i \\sim_p j$ when every element of $R$ takes congruent values at $i$ and $j$ modulo $p$; these classes are the blocks of the mod-$p$ algebra. The main computational tool is the long exact sequence coming from multiplication by $p$ on $0 \\to \\mathbb{Z}_j \\to \\mathbb{Z}_j \\to k_j \\to 0$, which yields recurrences $a_{l+1}=b_l-a_l$ and $z_l=y_{l+1}-z_{l+1}$ on the $p$-ranks of Ext and Tor. When $|G|$ is square-free, each $\\sim_p$-class has size at most two, the relevant mod-$p$ block is a two-dimensional local algebra, and the recurrences force the period-2 pattern; conversely, when $p^2$ divides $|G|$, a standard criterion on local rings implies unbounded dimensions in the mod-$p$ algebra, which carry back to unbounded ranks.","core_discovery":"The central discovery is that the Ext and Tor groups between mark modules of a Burnside ring are 2-periodic exactly when the group order is square-free. Theorem 20 states that for $|G|$ square-free, $\\mathrm{Ext}^l_{A(G)}(\\mathbb{Z}_H,\\mathbb{Z}_J) \\cong \\mathrm{Ext}^{l+2}_{A(G)}(\\mathbb{Z}_H,\\mathbb{Z}_J)$ as $A(G)$-modules for all $H,J$ and all $l \\geq 1$; the comparison in Corollary 11 then transfers the periodicity to $\\mathrm{Tor}$. Corollary 19 sharpens this to a complete computation: the $p$-primary part is $\\mathbb{Z}/p\\mathbb{Z}$ precisely when the conjugacy classes of $H$ and $J$ form a two-element class under the relation of agreeing modulo $p$, with even-degree self-Ext and odd-degree mixed Ext, and zero in all other degrees. When $|G|$ is not square-free, the paper proves the opposite behavior: there exist $H,J$ for which the groups $\\mathrm{Ext}^l_{A(G)}(\\mathbb{Z}_H,\\mathbb{Z}_J)$ and $\\mathrm{Tor}^{A(G)}_l(\\mathbb{Z}_H,\\mathbb{Z}_J)$ have unbounded rank.","pith_inferences":["I would conjecture that the period-2 pattern holds for any B-ring whose pairwise differences $d(i,j)$ are all square-free, not only for Burnside rings; the Burnside-ring proof uses the group only to control the size of the $\\sim_p$ classes, so the framework is directly testable on other subrings of ghost rings.","If the missing vanishing in Lemma 7 cannot be supplied, the square-free description might still survive for groups in which each $\\sim_p$ class is 'generated' by a single subgroup; failures would first appear in groups with larger classes, and one could hunt for them by explicit computation in small B-rings.","The unboundedness half raises a quantitative question the paper leaves open: for a fixed non-square-free $G$, how fast does the rank of $\\mathrm{Ext}^l$ grow, and is the growth rate determined by the radical filtration of the mod-$p$ Burnside algebra?"],"forward_implications":["For every square-free group $G$, all Ext and Tor groups between mark modules are determined by a parity check: the computation reduces to counting two-element $\\sim_p$ classes rather than resolving modules.","The $p$-torsion in the square-free case is always exactly $\\mathbb{Z}/p\\mathbb{Z}$; no $p^2$ or higher torsion appears.","For non-square-free orders, the unbounded-rank result rules out any finite description of the full family, so the square-free case is not just one example but the boundary of tractability.","Because the periodicity is an isomorphism of $A(G)$-modules and not merely of abelian groups, the multiplicative action of the Burnside ring is also periodic, which is additional structure for applications."],"supporting_citations":[{"why":"Supplies the criterion that mark homomorphisms agree modulo $p$ exactly when the $p$-cores are conjugate, which identifies the $\\sim_p$ classes used in the full square-free description.","marker":"[2]"},{"why":"Supplies the theorem that the mod-$p$ Burnside algebra is not symmetric when $p^2$ divides $|G|$, the starting point of the unbounded-rank direction.","marker":"[5]"},{"why":"Supplies the criterion that the dimensions of $\\mathrm{Tor}^S_l(k,k)$ are bounded only under a dimension condition, which forces unbounded ranks for the non-symmetric block.","marker":"[3]"},{"why":"Supplies the standard identification of Tor and Ext of the residue field over a local algebra, used throughout the recurrence arguments.","marker":"[4]"},{"why":"Supplies the block-theoretic decomposition of the mod-$p$ algebra into indecomposable local summands that structures the whole analysis.","marker":"[1]"}],"fun_headline_variants":["Square-free Burnside: Ext and Tor fully computed; otherwise unbounded","Burnside Ext: 2-periodic for square-free, unbounded otherwise","Ext groups of Burnside rings: periodic or unbounded","Burnside rings: square-free gives 2-periodic Ext, else unbounded"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The block decomposition that drives the whole proof assumes that for each prime $p$ and each equivalence class of subgroups agreeing modulo $p$, some element of $A(G)$ is congruent to $1$ mod $p$ on the entire class and to $0$ mod $p$ on every other class; the construction given verifies the zero condition only at one chosen subgroup in each complementary class, not at all of them.","fun_headline_variants_meta":{"raw":{"variants":["Square-free Burnside: Ext and Tor fully computed; otherwise unbounded","Burnside Ext: 2-periodic for square-free, unbounded otherwise","Ext groups of Burnside rings: periodic or unbounded","Burnside rings: square-free gives 2-periodic Ext, else unbounded"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000847,"raw_usage":{"total_tokens":3716,"prompt_tokens":1006,"completion_tokens":2710,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":622,"completion_tokens_details":{"reasoning_tokens":2631}},"tokens_in":622,"tokens_out":2710,"duration_ms":17615,"temperature":1.0,"reasoning_tokens":2631,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:58:06.645345+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Work with $G=S_3$ and the two non-conjugate subgroups of order $2$; compute $\\mathrm{Ext}^1$ and $\\mathrm{Ext}^3$ between the corresponding mark modules and check whether they are isomorphic as $A(G)$-modules with the predicted $\\mathbb{Z}/2\\mathbb{Z}$ value. A mismatch in rank or module structure would refute Theorem 20. Alternatively, test Lemma 7 directly on a B-ring in which a $\\sim_p$-class has more than one member: if the constructed product does not vanish modulo $p$ on an unselected member of a complementary class, Proposition 8 would need another proof.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the criterion that mark homomorphisms agree modulo $p$ exactly when the $p$-cores are conjugate, which identifies the $\\sim_p$ classes used in the full square-free description."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the theorem that the mod-$p$ Burnside algebra is not symmetric when $p^2$ divides $|G|$, the starting point of the unbounded-rank direction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the criterion that the dimensions of $\\mathrm{Tor}^S_l(k,k)$ are bounded only under a dimension condition, which forces unbounded ranks for the non-symmetric block."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the standard identification of Tor and Ext of the residue field over a local algebra, used throughout the recurrence arguments."},{"cited_title":"Auslander, I","cited_arxiv_id":null,"evidence_quote":"Supplies the block-theoretic decomposition of the mod-$p$ algebra into indecomposable local summands that structures the whole analysis."}],"review_version":1}