{"id":"79ecafb3-458f-4d9a-aec9-9950431d04c8","arxiv_id":"2507.20037","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Stanley-Reisner rings of a boolean complex and its barycentric subdivision are always G-equivariantly isomorphic over their shared parameter ring when the complex is Cohen-Macaulay and the field characteristic is coprime to |G|, but not in general (simplex, characteristic 2).","lead":"Two theorems settle an open question about face rings of simplicial complexes and their barycentric subdivisions: equivariant isomorphism over a common parameter subring can fail in characteristic 2, and it always exists for Cohen-Macaulay complexes when the automorphism group's order is invertible. The positive result comes with an explicit, algorithmic construction using Garsia's classic basis-transfer techniques.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.4 is the weakest load-bearing point for Theorem 1.2: its proof is a sketch citing characteristic-zero references, and the paper asserts the hypothesis is unnecessary without demonstrating it. One explicit check settles whether the counterexample stands.","rationale":"The paper's positive result has a detailed, self-contained proof chain, and the negative result is a concise counterexample. It is therefore appropriate to focus the concern on the negative result's unverified lemma. The reader's weakest_assumption identifies this same lemma; I agree. The specific gap is that Lemma 5.4 is the only place where characteristic 2 is handled by assertion rather than proof, and the references cited are either not explicitly covering this case or are not checked. The test is small and decisive: d=2 over F2 is a finite computation, and it directly validates or refutes the lemma in the simplest case of the theorem. If the lemma survives the test, the counterexample is likely correct; if not, Theorem 1.2 needs repair. No other concern rises to this level; the positive result's averaging argument and the nonconstructive existence proof are supported by detailed arguments and standard tools.","tokens_in":51785,"tokens_out":5109,"duration_ms":59236,"concrete_test":"Compute directly over F2 the A3-invariant subring of k[Sd ∆2] (d=2, n=3), using the presentation of Section 2: generators y_1,y_2,y_3,y_12,y_13,y_23,y_123, with incomparable pairs zero. Use a Gröbner basis or the linear-algebra method of Algorithm 6.11 to determine the module structure over k[θ1,θ2,θ3] = k[γ1,γ2,γ3]. Check that it is free of rank two with basis {1, \\tilde D}, where \\tilde D is the A3-orbit sum of y_3 y_23 y_123. Repeat for d=3 (A4) to confirm the pattern; if the freeness or the basis fails in either case, Lemma 5.4 is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.2's contradiction is built on Lemma 5.4, which asserts that in characteristic 2, the An-invariant subring of k[Sd ∆d] is a free k[Θ]-module of rank two with basis {1, \\tilde D}. This lemma is proved only by a sketch citing [Rei92, Theorem 4.3.5] and [GS84, Theorem 6.2], with the comment that the characteristic-zero hypothesis of [GS84] 'is not required in the proof of this claim.' That comment is not independently justified. The rank-two basis is then used to write φ(\\tilde D)=u+vD and to compute φ(\\tilde D + τ\\tilde D)=θ1...θd s, which yields the divisibility contradiction with D+τD. If the invariant subring is not free of rank two, or the specific element \\tilde D is not a basis element, the map φ need not have the stated form and the divisibility argument collapses. Because Theorem 1.2 is the paper's negative answer to Murai's question, this gap is directly load-bearing.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Murai's question whether the Stanley–Reisner ring (face ring) of a finite boolean complex and that of its barycentric subdivision are isomorphic as modules over the common parameter subring k[Θ], and whether such an isomorphism can be chosen equivariantly with respect to a group of automorphisms. The main positive result, Theorem 1.3, states that if Δ is Cohen–Macaulay over k and |G| is a unit in k for G ⊆ Aut(Δ), then a graded G-equivariant k[Θ]-module isomorphism k[Sd Δ] → k[Δ] exists and can be computed by an explicit algorithm. The main negative result, Theorem 1.2, states that for the d-simplex with d ≥ 2 over a field of characteristic 2, no Aut(Δ)-equivariant k[Θ]-module isomorphism exists. The proofs are built on a detailed generalization of Garsia's transfer method and on a linear-algebraic characterization of Cohen–Macaulayness for balanced boolean complexes.","tokens_in":52000,"tokens_out":41712,"duration_ms":479191,"significance":"If the results stand, the paper gives a definitive answer to Murai's question under the Cohen–Macaulay/coprime hypothesis and, in the modular simplex case, a counterexample. The positive theorem is constructive: it supplies an algorithm, not merely an existence statement, and the reworking of Garsia's tools at the generality of boolean complexes is a substantive contribution in its own right. The nonconstructive proof via equivariant Hilbert series and the constructive transfer argument complement each other nicely. The main caveat is that the negative theorem rests on Lemma 5.4, whose proof is currently only a sketch; the statement is plausible and the gap appears repairable, but the proof as written is incomplete.","major_comments":[{"comment":"Lemma 5.4 is the only load-bearing step in the proof of Theorem 1.2 that is not proved in the text. The lemma asserts that in characteristic 2, k[Sd Δ]^{A_n} is a free k[Θ]-module of rank two with basis {1, \\tilde D}; the proof is a sketch citing [Rei92, Theorem 4.3.5] and [GS84, Theorem 6.2], with the assertion that the characteristic-zero hypothesis of the latter is not required. The subsequent contradiction depends literally on this statement: the proof writes φ(\\tilde D) = u + vD and uses that representation, together with the freeness and basis, to derive equation (16). If the invariant subring is not free of rank two with the stated basis, or if \\tilde D is not a basis element, the divisibility argument collapses. The manuscript should give a complete proof of this lemma, or alternatively a precise citation to a statement that explicitly covers the modular case, plus a direct verification in the smallest case d = 2 over F_2. This is essential to make Theorem 1.2 fully supported.","section":"§5, Lemma 5.4"}],"minor_comments":[{"comment":"When Theorem 3.28 is used to transfer the basis b_1, ..., b_r in the construction of Φ, the text should say explicitly that the theorem is being applied with the trivial group; otherwise the unstated hypothesis that the elements be G-invariant is puzzling, since the b_j need not be G-invariant.","section":"§6.2"},{"comment":"In the maximal-shape argument of the proof of assertion 2, after choosing a chain supporting a term of G(f_j), the text should explicitly say that the chain is extended to a maximal chain so that an element of every rank is available; as written, the existence of the required stacking monomial of shape a_1(1^1)+...+a_n(1^n) is not immediate when the original chain misses some ranks.","section":"§3.3, proof of Theorem 3.28"},{"comment":"The claim that a standard monomial m not sitting under any facet to which β belongs satisfies m z_β = 0 deserves a one-sentence justification; the point is that any common upper bound of m and β would extend to a facet containing both, contradicting the hypothesis.","section":"§4, Lemma 4.6"},{"comment":"The comparison of shapes in the straightening move uses implicitly the rank identity rk(α)+rk(β)=rk(α∧β)+rk(γ); the text should state this identity explicitly, since otherwise the two shapes being compared are not visibly partitions of the same integer.","section":"§3.2, proof of Lemma 3.17"},{"comment":"The notation for the invariant basis element is inconsistent: Lemma 5.4 writes \\tilde D while the proof of Theorem 1.2 repeatedly writes pD. Please standardize the notation.","section":"§5"}],"recommendation":"major_revision","confidential_remarks":"The paper is well within the scope of the journal and the core positive theorem appears sound and valuable. The negative theorem is an important part of the claimed answer to Murai's question, and its proof currently rests on the unproved Lemma 5.4. I do not see evidence that the lemma is false, and the gap seems fixable by supplying a complete proof or a precise modular reference, so I am not recommending rejection. However, the manuscript should not appear in its current form without that step being repaired."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper is mathematically serious and the central positive result holds up: for a Cohen–Macaulay boolean complex with |G| invertible in k, it constructs an explicit G-equivariant k[Θ]-module isomorphism k[Sd Δ] → k[Δ], and the construction is genuinely new, based on a clean generalization of Garsia's basis-transfer tools. I checked the main line: the shape filtration, the transfer theorem, the linear-algebraic characterization of Cohen–Macaulayness, and the averaging argument in §6.2 are all sound. The nonconstructive proof via equivariant Hilbert series is also correct. The negative theorem for simplices in characteristic 2 answers Murai's question, assuming one lemma is right.\n\nThat lemma is Lemma 5.4, the same spot the stress-test flags. It asserts that in char 2 the An-invariants of k[Sd Δ_d] are free of rank two over k[Θ] with basis {1, \\tilde D}, and the proof is only a sketch: two references, one explicitly characteristic-zero, plus an assertion that the hypothesis is unnecessary. That assertion may be true, and likely is, but the contradiction in Theorem 1.2 is built entirely on this rank-two basis. If Lemma 5.4 fails, the counterexample collapses. This is a real gap in presentation, and it is the only one I found. It is also easily fixable: either expand the shelling argument or compute the rank directly in small dimensions; the paper itself indicates the computation is routine.\n\nThe citation pattern is fine; some folklore lemmas are cited to the first author's thesis, which is acceptable, and the paper even corrects an ambiguity in Garsia's original algorithm, a sign of careful work. So I am not skeptical of the main line. The paper deserves a serious referee, and I would ask the referee to verify Lemma 5.4 and its characteristic-2 assertion. If that checks out, both theorems stand.","headline":"Positive theorem is solid; negative theorem hinges on a compressed Lemma 5.4 that deserves verification before publication.","tokens_in":52517,"tokens_out":3149,"would_cite":true,"duration_ms":37533,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13F55","05E18","05E40","13C14","13A50","13C70","05E45"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a Cohen–Macaulay complex, the face ring and its barycentric subdivision ring are equivariantly isomorphic over the parameter subring when the automorphism-group order is invertible in the field; a simplex in characteristic 2 fails.","keywords":["Stanley–Reisner ring","barycentric subdivision","boolean complex","Cohen–Macaulay","equivariant module isomorphism","Garsia transfer","shape grading","parameter subring"],"falsifier":"Carry out a direct Gröbner-basis or module computation over $\\mathbb{F}_2$ of the $A_n$-invariants of $k[\\operatorname{Sd}\\Delta_d]$ for the smallest simplex $d=2$ ($n=3$): if the invariants are not a free rank-two $k[\\Theta]$-module with basis $\\{1,\\tilde D\\}$, Lemma 5.4 is false and Theorem 1.2 loses its foundation; if they are, the counterexample stands. Running Algorithm 6.11 on a balanced boolean complex over a field whose characteristic divides $|\\operatorname{Aut}(\\Delta)|$ would additionally test whether the coprime hypothesis marks the exact boundary of the positive result.","tokens_in":51578,"feed_emoji":"🔷","tokens_out":9612,"duration_ms":100914,"temperature":0.7,"pith_summary":"The paper addresses S. Murai's question whether the Stanley–Reisner ring $k[\\Delta]$ of a finite complex and that of its barycentric subdivision $k[\\operatorname{Sd}\\Delta]$ are isomorphic as modules over the parameter subring they share, and whether the isomorphism can be made equivariant under the complex's automorphisms; the setting is boolean complexes, a mild generalization of simplicial complexes in which two faces may meet along several faces at once. The answer is negative in general: for a simplex of dimension at least $2$ over a field of characteristic $2$, the full symmetric group admits no equivariant isomorphism, even though the two rings are non-equivariantly isomorphic. The answer is positive when the complex is Cohen–Macaulay over the field and the characteristic is coprime to the order of the automorphism group: a graded equivariant isomorphism exists, and the paper constructs it by an explicit algorithm. A sympathetic reader should care because this turns a conjectural structural equivalence into a computable one in the well-behaved case, while isolating the characteristic obstruction to symmetry.","feed_headline":"Face rings match their subdivisions when group size is invertible","feed_subtitle":"Cohen–Macaulay complexes get an explicit equivariant isomorphism over the shared parameter ring.","key_machinery":"The load-bearing object is the Garsia transfer $G\\colon k[\\operatorname{Sd}\\Delta]\\to k[\\Delta]$, which sends each generator $y_\\alpha$ to $x_\\alpha$ and extends multiplicatively on standard monomials; it is a graded, $\\operatorname{Aut}(\\Delta)$-equivariant $k$-linear isomorphism that need not be a ring or module map. Its power comes from the shape grading of $k[\\operatorname{Sd}\\Delta]$ by partitions and the compatible shape filtration of $k[\\Delta]$ by dominance order (Lemma 3.17 and Proposition 3.18), under which the transfer is a 'homomorphism in the top shape': products of standard monomials move to shapes no larger than the sum of the factors, with equality only when the factors stack up. Theorem 3.28 uses this to transfer generation and linear independence from $k[\\operatorname{Sd}\\Delta]$ to $k[\\Delta]$, and a separate averaging lemma (Proposition 6.10) converts any shape-filtered isomorphism that is equivariant in the top shape into a fully $G$-equivariant isomorphism when $|G|$ is invertible in $k$. Section 4's subspace-arrangement characterization of Cohen–Macaulayness drives Algorithm 6.11, which constructs the required shape-homogeneous basis.","core_discovery":"The paper's central claim is Theorem 1.3: for a finite boolean complex $\\Delta$ that is Cohen–Macaulay over $k$, and a group $G$ of automorphisms whose order is a unit in $k$, there exists a graded, $G$-equivariant isomorphism $k[\\operatorname{Sd}\\Delta]\\to k[\\Delta]$ of modules over the common parameter subring $k[\\Theta]$, together with an algorithm that computes it. The complementary Theorem 1.2 says that for a $d$-simplex with $d\\ge 2$ over a field of characteristic $2$, no such equivariant isomorphism exists for the full automorphism group $S_n$; this shows the equivariant version of Murai's question fails without the coprime hypothesis. The positive construction takes a shape-homogeneous basis of $k[\\operatorname{Sd}\\Delta]$ over $k[\\Theta]$, transfers it by the Garsia map to a basis of $k[\\Delta]$, forms the resulting module isomorphism, and then averages over $G$, with the shape filtration guaranteeing that averaging remains invertible.","pith_inferences":["The averaging deformation is phrased for barycentric subdivisions, but the same argument would turn any shape-filtered, top-shape-equivariant module isomorphism between two graded $G$-rings with equal Hilbert series into a fully equivariant one whenever $|G|$ is invertible.","For a fixed complex whose full automorphism group has order divisible by the characteristic, the positive theorem still applies to each subgroup of order coprime to the characteristic, so the symmetry that survives is precisely the symmetry whose order is invertible in the field.","Algorithm 6.11 could serve as a combinatorial Cohen–Macaulayness test for balanced boolean complexes: its termination behavior gives a certificate, which may be implemented and compared with the homology criterion of Reisner–Munkres."],"forward_implications":["In the Cohen–Macaulay coprime setting, the face ring and the barycentric-subdivision ring carry the same $G$-equivariant structure over the parameter ring, so all equivariant Betti numbers over $k[\\Theta]$ coincide.","The algorithm makes the isomorphism computable from a shape-homogeneous basis; the Gröbner-free linear-algebraic route only requires facet-vector row reductions.","The simplex counterexample in characteristic $2$ shows the equivariant version fails for a Cohen–Macaulay complex when the characteristic divides $|\\operatorname{Aut}(\\Delta)|$, so the unit-order hypothesis in Theorem 1.3 cannot simply be dropped.","The non-equivariant answer to Murai's question and Adams–Reiner's original Betti-number conjecture remain plausibly true in full generality, as the paper notes."],"supporting_citations":[{"why":"Introduces the transfer and basis-transfer method and the linear-algebraic Cohen–Macaulayness characterization that Sections 3 and 4 generalize.","marker":"[Gar80]"},{"why":"Extends the transfer to permutation invariants and supplies the rank-two invariant-subring computations that Lemmas 5.2–5.4 draw on.","marker":"[GS84]"},{"why":"Poses the Betti-number conjecture and identifies the universal and colorful parameters forming the common parameter subring; Section 6.1's nonconstructive proof follows its ideas.","marker":"[AR23]"},{"why":"Supplies the shelling result for quotients of Coxeter complexes cited in the proof sketch of Lemma 5.4.","marker":"[Rei92]"},{"why":"Shows depth of face rings of simplicial posets is preserved under barycentric subdivision, underpinning the Cohen–Macaulay passage between $\\Delta$ and $\\operatorname{Sd}\\Delta$.","marker":"[Duv97]"},{"why":"Gives the squarefree Gröbner degeneration result that frames the ASL relationship between $k[\\Delta]$ and $k[\\operatorname{Sd}\\Delta]$.","marker":"[CV20]"},{"why":"Vasconcelos's theorem converts the surjective equivariant map in Lemma 6.2 into an isomorphism under the Cohen–Macaulay hypothesis.","marker":"[Vas69]"},{"why":"Provides the equivariant Hilbert series facts used in the nonconstructive existence proof.","marker":"[BRSW11]"}],"fun_headline_variants":["Face rings match subdivisions when group order is invertible","Explicit equivariant isomorphism for face rings under coprime group orders","Subdivision and original face rings isomorphic when automorphism group is invertible","Boolean complex face rings survive subdivision when group size is a unit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The counterexample rests on the assertion that in characteristic $2$ the alternating-group invariant subring $k[\\operatorname{Sd}\\Delta]^{A_n}$ is a free module of rank two over the parameter subring with basis $\\{1,\\tilde D\\}$; the paper supplies only a proof sketch for this step, and if the assertion failed, the contradiction behind Theorem 1.2 would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Face rings match subdivisions when group order is invertible","Explicit equivariant isomorphism for face rings under coprime group orders","Subdivision and original face rings isomorphic when automorphism group is invertible","Boolean complex face rings survive subdivision when group size is a unit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000977,"raw_usage":{"total_tokens":4141,"prompt_tokens":929,"completion_tokens":3212,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":545,"completion_tokens_details":{"reasoning_tokens":3150}},"tokens_in":545,"tokens_out":3212,"duration_ms":25118,"temperature":1.0,"reasoning_tokens":3150,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T13:51:51.824310+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Carry out a direct Gröbner-basis or module computation over $\\mathbb{F}_2$ of the $A_n$-invariants of $k[\\operatorname{Sd}\\Delta_d]$ for the smallest simplex $d=2$ ($n=3$): if the invariants are not a free rank-two $k[\\Theta]$-module with basis $\\{1,\\tilde D\\}$, Lemma 5.4 is false and Theorem 1.2 loses its foundation; if they are, the counterexample stands. Running Algorithm 6.11 on a balanced boolean complex over a field whose characteristic divides $|\\operatorname{Aut}(\\Delta)|$ would additionally test whether the coprime hypothesis marks the exact boundary of the positive result.","supporting_citations":[],"review_version":1}