{"id":"0e28e03a-d8d7-450f-8c36-b3c374e176cb","arxiv_id":"2502.02122","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Stable, vertex splittable, and linear quotient monomial ideals admit minimal pruned free resolutions, unifying the Eliahou-Kervaire and Herzog-Takayama constructions.","lead":"This paper shows that a pruning method for building free resolutions of monomial ideals yields minimal resolutions for stable, vertex splittable, and linear quotient ideals. It also shows the same framework reproduces the classical Eliahou-Kervaire and Herzog-Takayama resolutions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 6 relies on an unproven transfer principle: componentwise minimal pruned resolutions in a Betti splitting do not obviously combine into a minimal pruned resolution of I, and Theorem 6.6 has a missing citation at the decisive step.","rationale":"The central claim of the paper is Theorem 6.9, and its proof, like that of Theorem 6.6, is inductive and relies on a Betti splitting I = J + K. The induction hypothesis gives minimal pruned resolutions of the pieces, but the definition of minimal pruned resolution (Definition 5.6) is tied to a specific generator order and to the iterative version of Algorithm 5.4. There is no statement or proof that minimality of the pieces implies minimality of the combined pruned resolution. The blank citation in Theorem 6.6 is a concrete symptom of this gap: the sentence needs both a citation to Lemma 6.8 for xiK and a lemma proving that the partial prunings on XJ, XK, and X' paste together without introducing units in the differential. The term-rank argument would suffice, but it requires the critical cells of Algorithm 5.4 to be exactly the union of the critical cells of the three pieces and their ranks to add via the Betti splitting formula. This is plausible and may well be true, but it is not established in the manuscript. I do not claim the theorem is false; the proposed computational test would either produce a counterexample or provide evidence that the missing lemma is derivable. The paper contains useful and verifiable material elsewhere, including the q ≤ 5 result of Theorem 3.6 and the worked examples, so the appropriate disposition is conditional acceptance while the transfer principle is either proved or refuted.","tokens_in":24145,"tokens_out":18379,"duration_ms":177675,"concrete_test":"Implement Algorithm 5.4 in Macaulay2 and run an exhaustive search over all monomial ideals with at most six generators in three variables that admit a Betti splitting I = J + K for which J, K, and J∩K each have a minimal pruned resolution (verified directly by Algorithm 3.1). For each such splitting, compute the resolution returned by Algorithm 5.4 and check whether its Betti table equals the minimal Betti table of I. A single failure would falsify the transfer principle used in Theorems 6.6 and 6.9; if no failure occurs, one should still insert the missing lemma and complete the citation in Theorem 6.6.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proofs of Theorem 6.6, Theorem 6.9, and the other inductive results of Section 6 depend on an unstated transfer principle: if I = J + K is a Betti splitting, and J, K, and J∩K each admit a minimal pruned resolution, then the resolution obtained from Algorithm 5.4 (pruning the three pieces separately inside the Taylor complex of I) is automatically a minimal pruned resolution of I. No lemma states or proves this. The gap is visible in Theorem 6.6, whose final line reads \"xiJ∩K = xiK and thus it also admits a minimal pruned resolution by .\"—the citation is missing, and even if one supplies Lemma 6.8, one still must justify that the pruned resolutions of J, K, and J∩K combine without creating degree-equal differential entries between critical cells coming from different pieces. The natural term-rank argument would work only if the critical cells of Algorithm 5.4 are exactly the disjoint union of the critical cells of the pruned resolutions of J, K, and J∩K, and if those have ranks equal to the corresponding minimal Betti numbers. This is asserted but not proved. If the transfer fails, the main theorems for vertex splittable and linear quotient ideals do not follow from the arguments given.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops discrete Morse theory constructions for monomial ideals, extending the first author's earlier pruned resolution framework. The main claims are that every vertex splittable ideal, every linear quotient ideal, and several further classes admit a minimal pruned resolution; that stable ideals are vertex splittable; and that the resulting resolutions are isomorphic to the Eliahou-Kervaire and Herzog-Takayama resolutions in the relevant cases. The authors also introduce a Betti-splitting variant of the pruning algorithm, a pruned resolution for powers of squarefree monomial ideals, recursive criteria for Artinian reductions and graph edge ideals, and several corollaries for paths, trees, cycles, wheels, and complete bipartite graphs. The overall strategy is inductive: decompose an ideal as a Betti splitting, obtain minimal pruned resolutions of the pieces by induction, and then combine them inside the original Taylor complex.","tokens_in":24426,"tokens_out":5939,"duration_ms":58459,"significance":"If the main theorems are correct, the paper gives a useful unifying framework: the Eliahou-Kervaire and Herzog-Takayama resolutions appear as special cases of one pruning construction, and the new class of ideals covered by Theorem 6.12 goes beyond previous cellular-resolution results. The proof that stable ideals are vertex splittable (Theorem 6.5) is a clean structural contribution, and the worked examples, especially Example 3.7 and Example 6.11, show that the authors are attentive to characteristic dependence. The paper is written for an expert audience and the exposition of the pruning algorithms is generally clear. However, the central inductive step is not proved as a formal lemma, and one theorem is stated as an iff but proved only in one direction; these gaps are load-bearing for the main claims and need to be addressed before the results can be accepted.","major_comments":[{"comment":"The paper relies on an unstated transfer principle: if I=J+K is a Betti splitting and J, K, and J∩K each admit a minimal pruned resolution, then the pruned resolution of I obtained by applying Algorithm 5.4 to the three pieces inside the Taylor complex of I is minimal. Theorem 5.5 proves only that the matching produced by Algorithm 5.4 is a homogeneous acyclic matching and hence gives a cellular resolution; the Betti splitting formula gives the Betti numbers of I, but minimality requires additionally that no degree-equal differential entry connects critical cells coming from different pieces. The proof of Theorem 6.6 invokes this principle at the final line, which reads 'xiJ∩K = xiK and thus it also admits a minimal pruned resolution by .' with the citation missing. Even if one supplies Lemma 6.8, one still needs a proof that the critical cells of Algorithm 5.4 are exactly the disjoint union of the critical cells of the pruned resolutions of J, K, and J∩K and that their ranks equal the corresponding minimal Betti numbers. A formal lemma stating and proving this transfer is necessary before Theorems 6.6 and 6.9 follow from the given arguments.","section":"Section 6, Theorems 6.6 and 6.9; Algorithm 5.4; Theorem 5.5"},{"comment":"Theorem 6.16 is stated as an equivalence: I=J+(x1^{a1},...,xn^{an}) admits a minimal pruned resolution if and only if J does. The proof, however, contains only the forward direction: 'If I admits a minimal pruned resolution, then J also admits a minimal pruned resolution by applying Corollary 3.10 iteratively.' The converse is asserted only in the informal sentence after the proof ('In the setting of Theorem 6.16 we will say that a minimal pruned resolution ideal plus powers also has a minimal pruned resolution'). Since this theorem is announced as Theorem C and is used in Corollary 6.17, the missing converse is load-bearing and needs a proof or an explicit reference.","section":"Section 6.2.1, Theorem 6.16"},{"comment":"The proof of Theorem 6.12 is a sketch rather than a complete construction. The sentence 'to each cell of the CW-complex associated to the minimal pruned resolution of J we may attach the Taylor graph of I and perform the pruning Algorithm 3.1' describes an idea, but it does not define the resulting acyclic matching on the Taylor complex of IJ, prove that it is acyclic, or prove that the surviving cells have no degree-equal differential entries. Since Corollary 6.13 on p-Borel ideals is a direct consequence, the gap affects a stated contribution. The same criticism applies to Theorem 6.14, which is justified only by 'the same kind of arguments.'","section":"Section 6.1.3, Theorem 6.12 and Corollary 6.13"},{"comment":"Theorem 6.18 asserts that I admits a minimal pruned resolution if K and all ideals I(H_{k1,...,km}) do. The proof is a succession of assertions: that xnJ∩K 'admits a splitting' into M1+(M2+...+Ms), that the iterative process has a splitting at each step 'because of the assumption that there are no edges between the neighbors,' and that 'the result follows.' No formal statement or proof is given for the iterated splitting of xnJ∩K, for the minimal pruned resolutions of the pieces M_{k1}∩...∩M_{km}, or for the transfer of these minimalities to I. The corollaries on paths, trees, cycles, wheels, and complete bipartite graphs depend on this theorem and therefore inherit the gap.","section":"Section 6.2.2, Theorem 6.18"},{"comment":"The proof of Theorem 6.5 says that 'G(I) is the disjoint union of G(xiJ) and G(K)' when I=xiJ+K is the xi-partition of a stable ideal. This needs a short justification: one must rule out a minimal generator of xiJ being divisible by a minimal generator of K, and vice versa. The claim is likely true for stable ideals, but it is not proved in the text. Since Theorem 6.5 feeds directly into vertex splittability and hence into Theorem 6.6, the point should be made explicit.","section":"Section 6.1.1, Theorem 6.5"}],"minor_comments":[{"comment":"The blank citation 'by .' at the end of the proof must be completed; this is likely the intended reference to Lemma 6.8 or to an earlier minimality result.","section":"Theorem 6.6"},{"comment":"The sentence 'This approach will be used in Section .' has a blank section number; it should be filled in, likely with a reference to Section 5.","section":"Remark 3.3(v)"},{"comment":"The phrase 'thetraedral curves' appears to be a typo; it should probably read 'toroidal curves' or the intended geometric term.","section":"Page 25, Example 6.4"},{"comment":"The text says 'It is proved in [AMFRG20] that, applying a partial pruning algorithm to XJ∩K, we obtain X′'; a precise proposition number would help the reader verify the claim.","section":"Section 5, after Equation (5.0.1)"},{"comment":"The phrase 'the iterative version of Theorem 5.5' is not formalized; it would be useful to state the recursive algorithm and prove termination before defining minimality through it.","section":"Definition 5.6"},{"comment":"The isomorphisms to the Eliahou-Kervaire and Herzog-Takayama resolutions are stated without proof. If the intended argument is uniqueness of minimal free resolutions up to isomorphism, that should be said explicitly, together with the relevant cellular structures from the literature.","section":"Corollaries 6.7 and 6.10"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern lands: the unproven transfer principle for Algorithm 5.4 is exactly the point where the main inductive theorems depend on something not stated as a lemma. I do not think this is a fatal flaw; the construction is plausible and the missing pieces are of a kind that can be supplied. The missing citation in Theorem 6.6 and the one-direction proof in Theorem 6.16 are best read as signs of an unfinished manuscript rather than as evidence of a false claim. The paper is a reasonable fit for math.AC, provided the authors add the missing formal lemma and fill in the sketchier proofs of Theorems 6.12 and 6.18. I have no concerns about the citation pattern beyond the usual level of self-citation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper has a real new result—stable ideals are vertex splittable (Thm 6.5)—and a genuinely useful framing: pruned resolutions plus Betti splittings unify Eliahou-Kervaire and Herzog-Takayama. But the main theorems for vertex splittable and linear quotient ideals are not proven as written. The induction in Theorems 6.6 and 6.9 assumes that if J, K, and J∩K each admit a minimal pruned resolution, then the pruning of the Taylor resolution of I via Algorithm 5.4 is automatically minimal. That transfer is not a formal consequence of the Betti splitting formula alone—you need to know the critical cells of the combined matching are exactly the union of the critical cells of the three pieces, and that no new degree-equal differentials appear between cells from different pieces. Neither is shown. The missing citation at the end of Theorem 6.6 ('also admits a minimal pruned resolution by .') is not a typo; it is the load-bearing step.\n\nTheorem 6.12 is also a sketch—'attach the Taylor graph to each cell' is an idea, not a proof. These are fixable: a lemma stating and proving the transfer would do it. But until that lemma exists, the main theorems do not follow from the arguments given.\n\nCredit where due: the paper is well-organized, the literature coverage is good, and the concrete corollaries for graphs (paths, cycles, wheels, complete bipartite) are nice payoffs. Example 6.11 (Bolognini's example) is a useful sanity check on the limits of the method. The self-citation to [AMFRG20] is legitimate—the pruned construction is the foundation, and this paper extends it.\n\nIf I were refereeing, I'd ask for a proof of the transfer lemma, a filled-in citation, and a real proof of Theorem 6.12. The claim that linear quotient ideals have minimal pruned resolutions is plausible and probably true; the unification is worth having. I'd send it to review despite the gaps.","headline":"A worthwhile unification of known minimal resolutions, but the main inductive step in Section 6 depends on an unproven transfer principle and a missing citation.","tokens_in":24969,"tokens_out":3007,"would_cite":true,"duration_ms":29654,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13D02","05E40","13F55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that every monomial ideal with linear quotients admits a minimal pruned resolution, unifying two classical explicit resolutions.","keywords":["monomial ideals","free resolutions","discrete Morse theory","pruned resolutions","Betti splittings","linear quotients","stable ideals","edge ideals"],"falsifier":"Run Algorithm 5.4 on a small vertex splittable ideal I = xiJ + K whose pieces J, K, and xiJ∩K individually admit minimal pruned resolutions, and inspect the surviving cells of equal multidegree: if any two surviving cells are connected in the Morse graph by a path with nonzero coefficient, Theorem 6.6 fails. For the linear quotient claim, the same check on any stable ideal, for instance the edge ideal of a complete bipartite graph, would settle it; the paper's Example 6.11 already shows the analogous statement fails for componentwise linear ideals.","tokens_in":1879,"feed_emoji":"✂️","tokens_out":1759,"duration_ms":63207,"temperature":0.7,"pith_summary":"The paper establishes that a broad class of monomial ideals admits a minimal free resolution obtained by a simple \"pruning\" of the Taylor resolution. The main results are that stable ideals are vertex splittable and that every monomial ideal with linear quotients has an ordering of its generators for which the pruned resolution, formed with the help of Betti splittings, is minimal. If the arguments hold, this unifies two classical constructions, the Eliahou-Kervaire resolution for stable ideals and the Herzog-Takayama resolution for linear quotient ideals, as instances of one Morse-theoretic procedure. The paper also gives recursive criteria that reduce the minimality question for a given ideal to that of a smaller subideal, with concrete applications to edge ideals of graphs.","feed_headline":"Morse pruning yields minimal resolutions for linear-quotient ideals","feed_subtitle":"One Morse-theoretic pruning recipe unifies Eliahou-Kervaire, Herzog-Takayama, and graph edge-ideal resolutions.","key_machinery":"The load-bearing object is the pruned resolution: start with the Taylor simplicial complex whose cells are labelled by the lcms of sets of generators, then prune edges in the coordinate directions (Algorithm 3.1) to obtain a homogeneous acyclic matching, and finally pass to the Morse complex of critical cells. The Betti-splitting variant (Algorithm 5.4) prunes inside the subcomplexes for J, for K, and for the join-type complex representing J∩K separately, so that a Betti splitting I = J + K is compatible with the Morse reduction in the sense that the pruned Betti numbers of I are the sum of those of J, K, and J∩K with a homological shift. Minimality is decided by whether two surviving cells of equal multidegree are connected by a path with nonzero coefficient in the Morse differential. The recursion is carried by the Betti splitting identity β(I) = β(J) + β(K) + β(J∩K) shifted one degree.","core_discovery":"The central discovery is that discrete Morse pruning of the Taylor resolution can be made minimal precisely when the ideal can be decomposed recursively by Betti splittings. Theorem 6.9 asserts that for any monomial ideal with linear quotients there exists a generator order such that the pruning Algorithm 3.1 or its Betti-splitting refinement Algorithm 5.4 leaves no two surviving cells of the same multidegree connected by a nonzero differential path, so the resulting Morse complex is a minimal free resolution. Theorem 6.5 proves that stable ideals are vertex splittable, and Theorem 6.6 shows that vertex splittable ideals admit minimal pruned resolutions. Corollaries identify the pruned resolution with the Eliahou-Kervaire resolution for stable ideals and with the Herzog-Takayama resolution for linear quotient ideals. The paper further proves that an ideal obtained by adjoining high variable powers admits a minimal pruned resolution if and only if the original ideal does, and it gives graph-theoretic reduction criteria for edge ideals.","pith_inferences":["If the transfer principle behind Algorithm 5.4 is made fully explicit, the same pruning and splitting machinery becomes a constructive recursive algorithm that computes a minimal free resolution for any ideal admitting a Betti splitting tree, not only the named classes.","The graph-theoretic reduction theorem suggests a testable extension to other splitting-closed families, such as cover ideals of chordal graphs, which are known to be vertex splittable.","Example 3.7 shows that the plain coordinate-order pruning can fail to be minimal for ideals with six generators in characteristic zero, so the linear quotient theorem likely depends on choosing the Betti-splitting order; a computational search could determine exactly where the plain version breaks.","A natural generalization to explore is whether every ideal with a Betti splitting tree whose leaves are monomial prime ideals admits a minimal pruned resolution, which would extend the reach of the method beyond stable and linear quotient ideals."],"forward_implications":["Every stable monomial ideal and every linear quotient monomial ideal has a minimal free resolution supported on a CW-complex, obtained by pruning the Taylor resolution.","The Eliahou-Kervaire and Herzog-Takayama resolutions are isomorphic to the pruned resolution, so two classical constructions are unified in one framework.","An ideal of the form J plus high powers of the variables admits a minimal pruned resolution if and only if J does; in particular, lexsegment-plus-powers and stable-plus-powers ideals admit such resolutions.","Edge ideals of paths, trees, forests, cycles, wheels, and complete bipartite graphs all admit minimal pruned resolutions.","The minimality of a pruned resolution can be reduced recursively to that of smaller subideals via Betti splittings, giving an iterative method that works as long as a splitting tree exists."],"supporting_citations":[{"why":"Supplies the original pruned resolution algorithm and the notion of minimal pruned resolution that Section 3 reviews.","marker":"[AMFRG20]"},{"why":"Introduces Betti splittings and the recursive Betti-number formula that the paper's Algorithm 5.4 is built on.","marker":"[FHVT09]"},{"why":"Provides the Betti splitting criterion for componentwise linear ideals used in the proof of Theorem 6.9.","marker":"[Bol16]"},{"why":"Defines stable ideals and gives the Eliahou-Kervaire resolution that Corollary 6.7 identifies with the pruned resolution.","marker":"[EK90]"},{"why":"Constructs the Herzog-Takayama resolution for linear quotient ideals that Corollary 6.10 identifies with the pruned resolution.","marker":"[HT02]"},{"why":"Provides discrete Morse theory for cellular resolutions and the minimality criterion for Morse differentials used throughout.","marker":"[BW02]"},{"why":"Introduces vertex splittable ideals and their splitting properties, which underpin Theorems 6.5 and 6.6.","marker":"[MKA16]"},{"why":"Provides algebraic discrete Morse theory and the p-Borel framework that input into Corollary 6.13 and the iterative pruning viewpoint.","marker":"[JW09]"},{"why":"Defines the smaller simplicial complex L^r_q for powers of squarefree monomial ideals that Section 4 uses for its pruned power resolution.","marker":"[CEKF+24]"}],"fun_headline_variants":["Morse + Betti splittings yield minimal free resolutions","Morse pruning unifies classic monomial resolutions","Betti splittings make Morse resolutions minimal","One Morse recipe yields minimal resolutions for linear quotients","Minimal free resolutions via Morse pruning and Betti splittings"],"cache_read_input_tokens":27136,"weakest_assumption_plain":"The argument relies on the transfer assumption that if each piece J, K, and J∩K of a Betti splitting I = J + K has a minimal pruned resolution obtained by pruning inside the Taylor complex of I, then the whole ideal I has one too.","fun_headline_variants_meta":{"raw":{"variants":["Morse + Betti splittings yield minimal free resolutions","Morse pruning unifies classic monomial resolutions","Betti splittings make Morse resolutions minimal","One Morse recipe yields minimal resolutions for linear quotients","Minimal free resolutions via Morse pruning and Betti splittings"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000799,"raw_usage":{"total_tokens":3462,"prompt_tokens":841,"completion_tokens":2621,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":457,"completion_tokens_details":{"reasoning_tokens":2545}},"tokens_in":457,"tokens_out":2621,"duration_ms":18907,"temperature":1.0,"reasoning_tokens":2545,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T13:19:13.786297+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run Algorithm 5.4 on a small vertex splittable ideal I = xiJ + K whose pieces J, K, and xiJ∩K individually admit minimal pruned resolutions, and inspect the surviving cells of equal multidegree: if any two surviving cells are connected in the Morse graph by a path with nonzero coefficient, Theorem 6.6 fails. For the linear quotient claim, the same check on any stable ideal, for instance the edge ideal of a complete bipartite graph, would settle it; the paper's Example 6.11 already shows the analogous statement fails for componentwise linear ideals.","supporting_citations":[],"review_version":1}