{"id":"6ae1e740-10e5-4710-90b1-6540baf38999","arxiv_id":"2505.04520","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For smooth pure shellable toric varieties, the cellular A1-chain complex is computed explicitly in terms of homology of restriction subcomplexes Kω with Milnor-Witt K-theory coefficients, yielding MW-motivic decompositions and Chow group bases.","lead":"This paper computes cellular A1-homology for smooth toric varieties, giving explicit chain complexes and decompositions for shellable cases. It also produces an additive basis for Chow groups of general smooth toric varieties.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The η-cancellation in Lemma 3.8 is not justified: η is a zero divisor in Milnor-Witt K-theory (η(1+⟨−1⟩)=0), so the proof that φλ is a chain map, and hence Theorem 3.9, is incomplete.","rationale":"The reader's weakest assumption was Proposition 3.4, the identification of the cellular complex of XΣ with C^can_*(AZK). That is indeed a real gap: the proof only checks that the quotient map restricted to a canonical cell is an embedding, and it does not explicitly prove surjectivity of pcan on chain groups or compatibility with the boundary operators. However, this is a standard toric-topology type statement and is plausibly repairable by a detailed cell-by-cell argument. The more immediately algebraic weakness is Lemma 3.8, which the reader mentioned in passing but did not make central. The chain map φλ is the bridge between the combinatorial complex C^λ and the cellular A1-chain complex; if the η-cancellation in Lemma 3.8 is invalid, then Theorem 3.9 is not established even if Proposition 3.4 is fixed. The specific obstruction is that η is a zero divisor in KMW_*: ηh=0 with h=1+⟨−1⟩≠0. The boundary formulas in the paper explicitly involve h through ⟨−1⟩ and ε, so coefficients killed by η can arise naturally. The proof's appeal to injectivity of η on ηKMW_* does not address this, because the difference ηφ(∂)−∂φ need not lie in ηKMW_*. The proposed computation on P^2 would either expose a concrete failure or force the authors to state and prove the precise injectivity statement needed. Because the main theorem and all subsequent decompositions depend on this chain-map property, the paper remains conditionally acceptable pending a rigorous justification of Lemma 3.8.","tokens_in":33118,"tokens_out":38278,"duration_ms":389057,"concrete_test":"Recompute Lemma 3.8 directly for the shellable P^2 fan with shelling 12,23,13, writing both ∂φ([r(σ)]) and φ(η∂^{cri}[r(σ)]) explicitly as elements of ⊕_cells KMW_* using Proposition 3.6, without invoking the cancellation step. If the difference is a nonzero multiple of h=1+⟨−1⟩, the chain map fails; if the difference is zero, identify the exact coefficient submodule on which multiplication by η^{|ω|-i} is injective and prove that the difference lies in it.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 3.8's proof of the chain-map property for φω contains an unjustified cancellation of powers of η. The authors derive η^{|ω|-i+1}φ_{i-1,ω}(∂^{cri}[r]) = η^{|ω|-i}∂φ_{i,ω}([r]) and then conclude φ_{i-1,ω}(η∂^{cri}[r]) = ∂φ_{i,ω}([r]) by asserting that 'η has no zero divisors in ηKMW_* ≅ W_*'. This is not a valid cancellation in KMW_*: the defining relation η(2+η[-1])=0 gives ηh=0 with h=1+⟨−1⟩≠0, so multiplication by η, and hence by any positive power η^N, has a nonzero kernel on KMW_0. The orientation coefficients appearing in the boundary formula of Example 2.20 are ⟨−1⟩=h−1 and ε=−⟨−1⟩=1−h, so the difference ηφ(∂)−∂φ could be an h-multiple in the relevant coefficient group. The proof does not show that this difference lies in a submodule of KMW_* on which η^N is injective, and the cited isomorphism with W_* does not by itself provide such injectivity. Since φλ being a chain map is exactly what makes (C^λ_*, η∂^{cri}) compute the cellular A1-chain complex in Theorem 3.9, this gap is load-bearing: if the cancellation fails, the isomorphism of complexes, and hence Corollaries 1.2–1.4 and the surface decompositions, do not follow.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops an explicit computation of cellular A1-homology for smooth toric varieties. The authors represent a smooth toric variety XΣ as a quotient of the moment-angle complex AZK, define a subcomplex Ccan∗(AZK) of 'canonical' cells, and prove (Theorem 3.9) that for pure shellable fans the cellular A1-chain complex is quasi-isomorphic to a direct sum of critical complexes Ccri∗(Kω)⊗KMW∗ with differential η∂cri. From this they derive MW-motivic decompositions, a formula for cellular A1-homology in terms of reduced homology of the subcomplexes Kω, an additive basis for Chow groups, and explicit computations for toric surfaces including P2 and Hirzebruch surfaces.","tokens_in":33451,"tokens_out":14568,"duration_ms":154064,"significance":"If the main theorem is correct, this is a valuable contribution: it reduces a motivic computation to ordinary homology of simplicial complexes and Milnor-Witt K-theory, and it yields explicit decompositions that are not available by general motivic methods. The paper has clear strengths: the chain-level strategy is concrete, the worked examples are internally consistent with known Chow groups, and the authors correctly draw on the external foundations of Morel–Sawant and Cai–Choi rather than re-deriving them. The claims are falsifiable and the examples give useful checks. However, two load-bearing proofs are not complete as written, so the current manuscript needs revision.","major_comments":[{"comment":"The proof of the chain-map property cancels a positive power of η. From the displayed equality η^{|ω|-i+1}φ_{i-1,ω}(∂^{cri}[r]) = η^{|ω|-i}∂φ_{i,ω}([r]) the authors conclude φ_{i-1,ω}(η∂^{cri}[r]) = ∂φ_{i,ω}([r]) by asserting that 'η has no zero divisors in ηKMW_* ≅ W_*'. This cancellation is not valid in general. In KMW_0, h = 1+⟨−1⟩ ≠ 0 but ηh = 0 by the defining relation η(2+η[−1]) = 0, so multiplication by η has a nonzero kernel. The cited isomorphism with W_* does not imply injectivity of multiplication by η on KMW_{i-1} or on the submodule containing the relevant boundary terms; no degree restriction or exact-sequence argument is supplied. Since the chain map property of φλ is exactly what makes (C^λ_*, η∂^{cri}) compute the cellular A1-chain complex in Theorem 3.9, this gap is load-bearing. The lemma needs a proof that either restricts to degrees where η is injective on the relevant modules or shows that the difference of the two sides lies in an η-torsion-free submodule.","section":"Lemma 3.8"},{"comment":"The proof that pcan is an isomorphism of complexes is too terse. It asserts that a canonical cell e : G^{t_e}_m → AZK meets each Ker exp(λ)-orbit in at most one point and hence p∘e is an embedding, and then concludes that pcan is an isomorphism. This only addresses injectivity at the level of individual embeddings. The proof does not show surjectivity of pcan on chain groups: cells of C^{cell}_*(XΣ) arise from all cells of AZK modulo the Ker exp(λ)-action, not only from canonical cells. It also does not verify compatibility with the differentials, ∂pcan = pcan∂, where the differential on C^{can}_* uses the T-adjusted formula introduced just above the proposition. Since Proposition 3.4 is the bridge from the moment-angle complex to XΣ and Theorem 3.9 composes pcan∘φλ, these missing checks must be supplied.","section":"Proposition 3.4"}],"minor_comments":[{"comment":"The text says 'We now apply Corollary 2.23' but the formula used is the one stated in Corollary 2.24; please correct the cross-reference.","section":"Section 3.2, proof of Proposition 3.6"},{"comment":"The restriction complex ̲C^{can}_*(AZK) is typographically too close to C^{can}_*(AZK); please use a clearly distinct symbol, for example D^{can}_* or ̄C^{can}_*, throughout.","section":"Section 3.1"},{"comment":"The diagram for the cellular complex of a complete toric surface is hard to read: the arrows labelled d(a_i) and the direct sum decomposition of the degree-one term are not fully labelled. Please specify the source and target of each summand and draw the differentials as a standard chain-complex diagram.","section":"Section 4.1, Proposition 4.6 display"},{"comment":"The characteristic function λ is presented as a 'bordermatrix' with vertex labels above the matrix; please typeset it as an ordinary matrix with a separate row or column of labels so that the entries can be read unambiguously.","section":"Example 4.11"},{"comment":"The notation KMW_i /sslash lη appears in the Introduction before it is defined; please define /sslash in Section 2 or move the definition earlier.","section":"Introduction and Section 3.3"}],"recommendation":"major_revision","confidential_remarks":"This is a substantial computational paper with a clear main idea, and the examples suggest the main theorem is plausible. The η-cancellation in Lemma 3.8 is the most serious issue: if it cannot be repaired, Theorem 3.9 and its corollaries would not follow. The gap in Proposition 3.4 is also real but may be fillable with a more detailed orbit/cell argument. I would not recommend rejection at this stage, but the revision must contain a complete proof of the chain-map property and of the pcan isomorphism."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, here's my read of arXiv:2505.04520.\n\nThe first genuinely new thing is Theorem 3.9 and its corollaries: a computation of cellular A1-homology and MW-motivic decompositions for smooth pure shellable toric varieties, plus an additive basis for Chow groups. The authors adapt the real toric machinery of Cai–Choi and the cubical cells of Morel–Sawant, and the explicit chain-level formulas are useful. The worked examples — P2, Hirzebruch surfaces, complete toric surfaces, and the non-shellable/non-pure cases — are consistent with known Chow and motivic computations, which is a good sign.\n\nThe soft spots are in the proof of the main theorem. First, Proposition 3.4: the identification C^can_*(AZK) ≅ C^cell_*(X_Σ) is asserted with a one-paragraph argument that checks the orbit-intersection condition and then claims an isomorphism of complexes. Surjectivity of pcan on chain groups and compatibility with the differential are not demonstrated. That is a load-bearing step.\n\nSecond, and more seriously, Lemma 3.8 contains an invalid cancellation. The authors derive η^{|ω|-i+1} φ(∂r) = η^{|ω|-i} ∂φ(r) and then cancel η^{|ω|-i}, citing 'η has no zero divisors in ηKMW_*'. But multiplication by η is not injective in KMW_*: ηh = 0 with h = 1+⟨−1⟩ ≠ 0. The term ηφ(∂) − ∂φ(r) need not lie in the image of η, so injectivity on that submodule is not established. The stress-test note is right: as written, the chain-map property of φλ, and hence Theorem 3.9, does not follow. This is not a cosmetic issue; the cancellation is exactly what identifies the complex C^λ_* with the cellular complex.\n\nIf the authors can repair Lemma 3.8 — perhaps by showing the relevant coefficients land in a free submodule of KMW_* on which η^N is injective, or by replacing the cancellation with a direct argument — the paper would be a solid contribution. As it stands, the main theorem is plausible but unproven.\n\nWho should read it? People working on A1-homology of toric varieties will want to know these formulas, even if they treat Theorem 3.9 as conditional. I would send it to a serious referee, but with a request to check Lemma 3.8 and Proposition 3.4 closely.","headline":"The main theorem is plausible and the concrete computations are valuable, but the proof has a serious gap in Lemma 3.8 (invalid η-cancellation) and an under-proved Proposition 3.4.","tokens_in":34014,"tokens_out":7414,"would_cite":false,"duration_ms":70110,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14M25","14F42"],"pacs":[],"model":"deepseek-v4-flash","headline":"For smooth pure shellable toric varieties, the cellular $\\mathbb{A}^1$-chain complex is isomorphic to a sum over mod-2 row sets of critical subcomplexes tensored with Milnor-Witt K-theory, with differential $\\eta\\partial^{\\mathrm{cri}}$.","keywords":["cellular A1-homology","toric varieties","Milnor-Witt K-theory","shellable simplicial complexes","moment-angle complex","motivic decomposition","Chow group","cubical cells"],"falsifier":"Compute the boundary of a canonical cell in a smooth pure shellable fan whose facet is shared by two maximal cones, using the paper's action formula (Proposition 2.23). If the resulting expression contains any non-canonical cell, or if two canonical cells lie in one $\\mathrm{Ker}(\\exp(\\lambda))$-orbit, then Proposition 3.4 fails and the Theorem 3.9 decomposition collapses; this can be checked directly from the fan data alone.","tokens_in":32894,"feed_emoji":"🧩","tokens_out":12559,"duration_ms":110559,"temperature":0.7,"pith_summary":"The paper tries to show that the cellular $\\mathbb{A}^1$-homology of a smooth toric variety—the analogue of cellular homology that computes motivic and Chow-Witt invariants—can be reduced to finite combinatorial data. For the main case, a smooth toric variety whose fan is pure and shellable, it claims the entire cellular $\\mathbb{A}^1$-chain complex is isomorphic to a sum, over mod-2 row sets of the characteristic map, of critical subcomplexes tensored with Milnor-Witt K-theory, with a differential twisted by the element $\\eta$. If true, this turns the motivic computation into ordinary homology of explicit subcomplexes, and it yields a decomposition of the Milnor-Witt motive into shifted $\\tilde{\\mathbb{Z}}/l\\eta$ summands. The paper also claims an additive basis for the Chow group of any smooth toric variety, indexed by facets and their minimal new faces, and gives explicit formulas for toric surfaces.","feed_headline":"A1-homology of shellable toric varieties is pure combinatorics","feed_subtitle":"For shellable fans, the A1-chain complex splits into ordinary homology of subcomplexes and Milnor-Witt K-theory.","key_machinery":"The carrying object is the oriented cubical-cell presentation of the cellular $\\mathbb{A}^1$-chain complex. Cells of $\\mathbb{A}^n$ are products of the two types $\\{x_i=1\\}$ and $\\{x_i\\neq 0\\}$; gluing them along the moment-angle complex $AZ_K$ and applying the action of the torus kernel $\\mathrm{Ker}(\\exp(\\lambda))$ yields canonical cells, one per orbit. The central identity is the action formula of Proposition 2.23, which expresses the change of orientation under a group section as coefficients in $\\mathbb{Z}\\eta^r$, $\\mathbb{Z}+\\mathbb{Z}h$, or $\\mathbb{Z}[-1]^r$. For shellable fans, a regular expanding sequence selects critical facets $r(\\sigma)=\\sigma\\cap\\omega$, and the complex $C^\\lambda_* = \\bigoplus_{\\omega\\in\\mathrm{row}\\lambda} C^{\\mathrm{cri}}_*(K_\\omega)\\otimes K^{\\mathrm{MW}}_*$ with differential $\\eta\\partial^{\\mathrm{cri}}$ is shown in Theorem 3.9 to be isomorphic to the canonical complex and quasi-isomorphic to $C^{\\mathrm{cell}}_*(X_\\Sigma)$.","core_discovery":"On its own terms, the paper's central discovery is Theorem 3.9: for a smooth pure shellable toric variety $X_\\Sigma$, written as a fan $\\Sigma=(K,\\lambda)$, the cellular $\\mathbb{A}^1$-chain complex $C^{\\mathrm{cell}}_*(X_\\Sigma)$ is quasi-isomorphic to $C^\\lambda_* = \\bigoplus_{\\omega\\in\\mathrm{row}\\lambda} C^{\\mathrm{cri}}_*(K_\\omega)\\otimes K^{\\mathrm{MW}}_*$, with differential $\\eta\\partial^{\\mathrm{cri}}$. Here $\\mathrm{row}\\lambda$ is the set of mod-2 row sets of the characteristic map, $K_\\omega$ is the subcomplex of the simplicial complex $K$ cut out by $\\omega$, $C^{\\mathrm{cri}}_*$ is the critical complex generated by facets whose restriction equals themselves, and $\\eta$ is the degree $-1$ generator of Milnor-Witt K-theory. This reduces a motivic computation to ordinary homology of simplicial complexes: each summand contributes Milnor-Witt K-theory shifted by the size of the restriction. From it the paper derives the Milnor-Witt motivic decomposition $\\tilde{M}(X_\\Sigma)\\cong\\bigoplus_{l\\in\\mathbb{N}}\\bigoplus_{\\sigma\\in B(l)}\\tilde{\\mathbb{Z}}/l\\eta(|r(\\sigma)|)[2|r(\\sigma)|]$, and for general smooth toric varieties an additive basis of the Chow group indexed by facets and their minimal new faces. The paper notes that for non-shellable or non-pure fans this decomposition fails, and gives two surface examples.","pith_inferences":["If Theorem 3.9 is correct, the same combinatorial recipe should compute cellular $\\mathbb{A}^1$-homology for any smooth toric variety that admits a regular expanding sequence, including non-complete fans; the paper states the pure shellable case, but the local input is the same.","The appearance of ordinary homology of the $K_\\omega$ suggests a stratum interpretation: cellular $\\mathbb{A}^1$-homology is assembled from the ordinary homology of subcomplexes cut out by mod-2 linear functions, in analogy with Borel-style descriptions of torus actions.","The Section 4.2 examples indicate that a general decomposition for arbitrary smooth toric varieties would need extra summands indexed by higher Chow groups, not just by critical facets, since the cycle class map to ordinary cohomology fails to be surjective there."],"forward_implications":["For every smooth pure shellable toric variety, cellular $\\mathbb{A}^1$-homology is computable from ordinary homology of the subcomplexes $K_\\omega$: $H^{\\mathrm{cell}}_i(X_\\Sigma)=\\bigoplus_{l\\in\\mathbb{N}}\\bigoplus_{B(l)_{i-1}}K^{\\mathrm{MW}}_i/l\\eta\\;\\oplus\\;\\bigoplus_{l\\in\\mathbb{N}_+}\\bigoplus_{B(l)_{i-2}}l\\eta K^{\\mathrm{MW}}_i$.","In the derived category of strictly $\\mathbb{A}^1$-invariant sheaves, the chain complex splits into shifts of $K^{\\mathrm{MW}}/l\\eta$, giving the stated Milnor-Witt motivic decomposition; after inverting $\\eta$, only the $l\\neq 1$ summands remain.","In the ordinary motivic category, a pure shellable smooth toric variety is Tate, with $M(X_\\Sigma)\\cong\\bigoplus_{\\sigma\\in K_{\\max}}\\mathbb{Z}(|r(\\sigma)|)[2|r(\\sigma)|]$, so the Chow group has an additive basis indexed by facets and their restrictions.","For complete toric surfaces the complex reduces to $K^{\\mathrm{MW}}_2\\to(K^{\\mathrm{MW}}_1)^{\\oplus l-3}\\oplus\\mathbb{Z}$ with differential $(a_\\Sigma)\\epsilon\\eta$, so the parity of the gcd of the self-intersection numbers determines orientable versus non-orientable behavior in the real realization.","For fans that are not pure or not shellable, Corollary 3.11 does not hold; the paper's two surface examples show extra summands survive, so any extension to arbitrary fans must be more subtle."],"supporting_citations":[{"why":"Defines cellular $\\mathbb{A}^1$-homology and the oriented cubical-cell chain complex that the paper computes with.","marker":"[16]"},{"why":"Supplies the quotient presentation $X_\\Sigma\\cong\\mathrm{Coker}\\,\\exp(\\lambda)\\times(AZ_K/\\mathrm{Ker}\\,\\exp(\\lambda))$ that underlies the canonical-subcomplex identification.","marker":"[7, Theorem 2.1]"},{"why":"Provides the shelling, regular expanding sequence, and critical-complex machinery adapted here to the $\\mathbb{A}^1$ setting.","marker":"[3]"},{"why":"Establishes the strictly $\\mathbb{A}^1$-invariant sheaf framework and the identification $\\mathbb{Z}^{\\mathbb{A}^1}[\\mathbb{G}_m^{\\wedge n}]\\cong K^{\\mathrm{MW}}_n$.","marker":"[15]"},{"why":"Defines the Milnor-Witt motive category and the cones $\\tilde{\\mathbb{Z}}/l\\eta$ used in the motivic decomposition corollaries.","marker":"[2]"},{"why":"Gives the Chow group basis that the paper generalizes to non-complete and non-pure fans.","marker":"[11]"}],"fun_headline_variants":["Shellable toric A1-homology decomposes into subcomplex homologies","A1-homology of shellable fans splits via ordinary homology","Toric A1-homology: shellable case reduces to ordinary homology","Shellable fans give A1-homology a combinatorial decomposition"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on Proposition 3.4's identification of the cellular chain complex of $X_\\Sigma$ with the canonical subcomplex of the moment-angle complex; the paper does not fully prove the orbit-intersection and cellular-structure compatibility on which that identification depends.","fun_headline_variants_meta":{"raw":{"variants":["Shellable toric A1-homology decomposes into subcomplex homologies","A1-homology of shellable fans splits via ordinary homology","Toric A1-homology: shellable case reduces to ordinary homology","Shellable fans give A1-homology a combinatorial decomposition"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002301,"raw_usage":{"total_tokens":8881,"prompt_tokens":947,"completion_tokens":7934,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":563,"completion_tokens_details":{"reasoning_tokens":7855}},"tokens_in":563,"tokens_out":7934,"duration_ms":49259,"temperature":1.0,"reasoning_tokens":7855,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:26:15.605704+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the boundary of a canonical cell in a smooth pure shellable fan whose facet is shared by two maximal cones, using the paper's action formula (Proposition 2.23). If the resulting expression contains any non-canonical cell, or if two canonical cells lie in one $\\mathrm{Ker}(\\exp(\\lambda))$-orbit, then Proposition 3.4 fails and the Theorem 3.9 decomposition collapses; this can be checked directly from the fan data alone.","supporting_citations":[{"cited_title":"Cellular A1-homology and the motivic version of Matsumoto’s theorem","cited_arxiv_id":null,"evidence_quote":"Defines cellular $\\mathbb{A}^1$-homology and the oriented cubical-cell chain complex that the paper computes with."},{"cited_title":"Integral cohomology groups of real t oric manifolds and small covers","cited_arxiv_id":null,"evidence_quote":"Provides the shelling, regular expanding sequence, and critical-complex machinery adapted here to the $\\mathbb{A}^1$ setting."},{"cited_title":"A1-Algebraic Topology over a Field , volume 2052","cited_arxiv_id":null,"evidence_quote":"Establishes the strictly $\\mathbb{A}^1$-invariant sheaf framework and the identification $\\mathbb{Z}^{\\mathbb{A}^1}[\\mathbb{G}_m^{\\wedge n}]\\cong K^{\\mathrm{MW}}_n$."}],"review_version":1}