{"id":"67a4890a-aa36-4d89-bddd-b5af969c9e4a","arxiv_id":"2412.21031","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For monomial ideals with linear powers, the homological shift algebra is a finitely generated Rees module, making regularity, depth, associated primes, v-number, and Golodness of homological shift ideals eventually linear or stable.","lead":"This paper introduces the homological shift algebra, a single algebraic object that bundles the syzygy data of all powers of a monomial ideal. For ideals whose powers have linear resolutions, this object is a finitely generated module, which forces many invariants of the shifted ideals to stabilize or become linear as the power grows.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.2's 'for all i' is false as stated: for I the maximal ideal, HS_i(I^k)=0 for i≥n, and the zero ideal is not Golod.","rationale":"The central algebraic framework is credible: under the linear-powers hypothesis, Proposition 1.3 gives HS_i(I^{k+1})=I·HS_i(I^k) for all large k, so HS_i(R(I)) is a finitely generated R(I)-module, and the asymptotic consequences in Theorems 2.1 follow from standard module theory. The reader correctly identified linear powers as the key external hypothesis and noted the Golod vanishing issue as a localized error. My stress-test confirms that the Golod issue is not merely cosmetic: Theorem 4.2 as written is false for vanishing homological shift ideals, as the maximal ideal example shows. However, this falsifies only the unqualified 'all i' statement and is repaired by adding a nonvanishing condition; it does not undermine the finite-module construction. The reader's CONDITIONAL verdict is therefore appropriate, and my assessment does not move it.","tokens_in":19196,"tokens_out":36773,"duration_ms":371036,"concrete_test":"For n=2, let S=K[x,y] and I=(x,y). Compute proj dim I^k for k=1,2,3; the Koszul resolution gives pd I^k=1, hence HS_2(I^k)=0 for all k. Then compare the Poincaré series of S/0=S, namely P_S(t)=(1+t)^2, with the Serre upper bound appearing in §4, (1+t)^2/(1-t). Since the two series differ, the zero ideal is not Golod, directly falsifying the universal quantification in Theorem 4.2. This can be checked by hand or with a short Macaulay2 script (res I^k for several k and a comparison of the two series).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest_claim packages Theorem 4.2, which asserts that HS_i(I^k) is Golod for all i and all k≫0 whenever I has linear powers. This is false as stated. Take S=K[x_1,...,x_n] and I=m=(x_1,...,x_n). Then I^k=m^k has a linear resolution and proj dim m^k = n-1 for every k, so HS_i(m^k)=0 for all i≥n and all k≥1. The zero ideal is not Golod in the paper's sense: S/0=S has Poincaré series (1+t)^n, while the Serre upper bound quoted in §4 is (1+t)^n/(1-t), so equality fails for n≥1. Thus Theorem 4.2 cannot hold for the vanishing components. The proof applies Theorem 4.3 to M_k=HS_i(I^k); for vanishing components this would conclude that the zero ideal is Golod, and this is precisely where the Massey-operation criterion degenerates because the positive Koszul homology is empty. This does not damage the finite-generation theorem or the asymptotic statements for the nonzero range, but the abstract's unqualified 'for all i' must be restricted to values of i for which HS_i(I^k)≠0 for all k≫0, e.g., 0≤i≤limsup_k pd I^k.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces the i-th homological shift algebra HS_i(R(I)) = \\bigoplus_{k\\ge 1} HS_i(I^k) of a monomial ideal I and studies it as a bigraded module over the Rees algebra R(I). The central structural theorem is that, when I has linear powers, HS_i(R(I)) is finitely generated over R(I) (Theorem 1.4); from this the authors deduce eventual stabilization or eventual linearity of depth, associated primes, regularity, and v-number of HS_i(I^k) (Theorem 2.1). The paper also identifies several families of ideals whose homological shift ideals have linear resolutions for all powers or eventually (Theorem 3.9), gives a counterexample to a natural question, and proves an asymptotic Golodness statement (Theorem 4.2). The core finite-generation argument is credible, but the Golod theorem is over-stated as written because it includes indices for which HS_i(I^k) = 0 and the zero ideal is not Golod.","tokens_in":19427,"tokens_out":18101,"duration_ms":182821,"significance":"If the Golod statement is corrected, the paper's framework is a valuable organizing tool: finite generation of the homological shift algebra over the Rees algebra turns a whole family {HS_i(I^k)} into one finitely generated module, which simultaneously explains several otherwise unrelated asymptotic regularities. The explicit treatments of two-variable monomial ideals, principal and c-bounded Borel ideals, complete intersections, and the negative answer to Question 3.1 via Example 3.4 are useful concrete contributions. The paper contains no fitted parameters and does not reduce its main claims to input definitions; the finite-generation argument is coherent and the Rees-module point of view is well motivated. The unqualified 'for all i' in the abstract and in Theorem 4.2 is a genuine error, but it is local and repairable by restricting to indices for which HS_i(I^k) is nonzero for all large k.","major_comments":[{"comment":"Theorem 4.2 is false as stated. Let S = K[x_1,\\ldots,x_n] and I = (x_1,\\ldots,x_n). Then I has linear powers, but for every i \\ge n and every k we have HS_i(I^k) = 0, and the zero ideal is not Golod under the Serre-bound definition given in Section 4: P_{S/0}(t) = P_S(t) = (1+t)^n, while the quoted upper bound is (1+t)^n/(1-t), so equality fails. The same issue appears in Theorem 4.3, whose hypothesis 'M_k is a proper ideal' also admits the zero module. The statements should be restricted to indices i (and components k) for which HS_i(I^k) \\neq 0 for all k \\gg 0; with that restriction the Massey-operation argument appears to prove the intended result. The abstract's unqualified 'all i' must be corrected accordingly, and Theorem 2.1(c)-(e) should carry the same nonvanishing caveat so that regularity and v-number are always defined.","section":"Theorem 4.2 / Theorem 4.3, Section 4"},{"comment":"The displayed regularity formula has an extra d_m term. For the principal case m=1, i=0, the paper's formula gives reg HS_0(I^k) = d_1 k + d_1, while HS_0(I^k) = I^k has regularity d_1 k. The proof's final computation yields d_m k + \\sum_{j=1}^{m-1} d_j + \\sum_{j=1}^{i} d_{m-j} - (m-1), so the first sum over j=1,\\ldots,m should be j=1,\\ldots,m-1. As printed, the formula contradicts the proof and the elementary one-variable example.","section":"Proposition 2.7"},{"comment":"The condition 'm \\in Ass^\\infty_i(I) if and only if m > 0' is vacuous as stated, because m denotes the number of generators of I_{a,b} and is always positive. It is also false for principal ideals: for I = (x^a y^b), the associated primes of every power are (x) and (y), not (x,y). The intended condition is presumably m > 1; the proof, which cites [21, Proposition 5.1], should be checked against this correction.","section":"Proposition 2.3(c3)"}],"minor_comments":[{"comment":"The sentence 'If HS_2(I^k) were to have linear resolution, then by Corollary 3.2 the ideal J:u would be generated by variables' appears to misname the reference: Corollary 3.2 concerns HS_1 of ideals whose powers have linear quotients, while the criterion used here is Corollary 3.3.","section":"Example 3.4"},{"comment":"The symbol m is overloaded: it denotes both the maximal ideal (x,y) and the number of generators of I_{a,b} in Section 2.1, and it is reused for the number of generators in Theorem 3.9(a). This makes statements such as Proposition 2.3(c3) needlessly confusing even apart from the mathematical typo; a different letter should be used for one of the two objects.","section":"Sections 2.1 and 3.9(a)"},{"comment":"The displayed formula for the limit of depth appears to have a typesetting problem in the denominator: the expression 'dim HSi(R(I)) R(I)(*,1) * HSi(R(I))' should be typeset as the dimension of a quotient by the submodule R(I)_{(*,1)} * HS_i(R(I)), or equivalently of a colon module, rather than as a product of two dimensions.","section":"Proposition 2.2"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about the zero ideal is valid and should be fixed: Theorem 4.2 needs a nonvanishing-component hypothesis, and the same correction should be propagated to the abstract and to Theorem 2.1 where regularity of the zero ideal is involved. The core finite-generation theorem and the asymptotic consequences for nonzero components appear sound, and the proposed repair is local rather than a change of approach. I would therefore treat this as a major revision rather than a rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper introduces the homological shift algebra HSi(R(I)) = ⊕_{k≥1} HSi(I^k) and shows that if I has linear powers, it is a finitely generated bigraded module over the Rees algebra. That is a genuinely useful idea: it turns the study of all homological shifts into one module, and it yields clean asymptotic statements for regularity, depth, associated primes, and the v-number in Theorem 2.1. The finite-generation proof is coherent, and the applications to principal Borel ideals and complete intersections are nice. The main construction is new, and the self-citations are used as lemmas rather than as the substance of the results. This is the real contribution and I think it holds up.\n\nThe soft spots are localized but real. First, Theorem 4.2 says HSi(I^k) is Golod for all i. For i beyond the projective dimension, HSi(I^k)=0. The paper's Serre bound as written (sum starting at i=0) makes the zero ideal not Golod, so the theorem is false under the paper's own definition. The standard bound has a -1 or starts at i=1, under which the zero ideal is Golod, but as written the statement needs fixing. Either correct the bound or restrict to i with HSi(I^k) ≠ 0 for all k≫0. Second, Proposition 2.7's displayed regularity formula has an extra d_m term; the proof gives d_m k + ∑_{j=1}^{m-1} d_j + ∑_{j=1}^i d_{m-j} - (m-1), not the displayed expression with an additional ∑_{j=1}^m d_j. Third, Proposition 2.3(c3) says m ∈ Ass∞ iff m > 0, which is trivially true; presumably it should be m ≥ 3, matching the depth cases.\n\nNone of these damage the central argument. The Rees-module framework is a solid contribution and the asymptotic theorems follow from standard module theory. The paper deserves a serious referee; I'd send it to review, with a request to correct the Golod formulation and the two formula typos.","headline":"A genuinely useful new construction with sound asymptotic theorems, but the Golod statement overreaches as written and two displayed formulas have typos.","tokens_in":20010,"tokens_out":7656,"would_cite":true,"duration_ms":68838,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13F20","13F55","05C70","05E40"],"pacs":[],"model":"deepseek-v4-flash","headline":"If every power of a monomial ideal has a linear resolution, then the homological shift ideals of all powers form one finitely generated module over the Rees algebra, forcing regularity, depth, associated primes, v-numbers, and Golodness…","keywords":["monomial ideals","homological shift ideals","Rees algebra","linear powers","syzygies","Castelnuovo-Mumford regularity","v-number","Golod ideals"],"falsifier":"Take any monomial ideal with linear powers, compute $\\mathrm{HS}_i(I^k)$ for increasing $k$, and check whether the equality $\\mathrm{HS}_i(I^{k+1})=I\\cdot\\mathrm{HS}_i(I^k)$ holds for all sufficiently large $k$ and whether $\\mathrm{Ass}(\\mathrm{HS}_i(I^k))$ stabilizes; the paper predicts both, so a single linear-powers ideal violating either for arbitrarily large $k$ would refute the main theorem.","tokens_in":18950,"feed_emoji":"📐","tokens_out":17990,"duration_ms":145927,"temperature":0.7,"pith_summary":"The paper introduces, for each $i$, the $i$th homological shift algebra of a monomial ideal $I$: the direct sum over all powers $I^k$ of the homological shift ideals $\\mathrm{HS}_i(I^k)$, which record the multigraded degrees appearing in the $i$th syzygies of $I^k$. The central claim is that if $I$ has linear powers — every power $I^k$ has a linear resolution — then this direct sum is a finitely generated bigraded module over the Rees algebra $\\mathcal{R}(I)=\\bigoplus_{k\\ge0}I^k$. From that structural fact, the paper derives that for every $i$ the associated primes of $\\mathrm{HS}_i(I^k)$ stabilize, the depth of $S/\\mathrm{HS}_i(I^k)$ stabilizes, and the Castelnuovo–Mumford regularity and the $v$-number of $\\mathrm{HS}_i(I^k)$ are eventually linear in $k$. It also proves $\\mathrm{HS}_i(I^k)$ is Golod for all $k\\gg0$ and identifies several families for which $\\mathrm{HS}_i(I^k)$ has a linear resolution for every $i$ and $k$.","feed_headline":"One finite module controls homological shifts of monomial ideal powers","feed_subtitle":"When every power has a linear resolution, one module governs depths, regularities, and v-numbers of homological shifts.","key_machinery":"The working object is the homological shift algebra, assembled from homological shift ideals: for a monomial ideal $I$, $\\mathrm{HS}_i(I)$ is the monomial ideal generated by all monomials $x^a$ with $\\beta_{i,a}(I)\\neq0$, and $\\mathrm{HS}_i(\\mathcal{R}(I))=\\bigoplus_{k\\ge1}\\mathrm{HS}_i(I^k)$. The load-bearing identity is the eventual equality $\\mathrm{HS}_i(I^{k+1})=I\\cdot\\mathrm{HS}_i(I^k)$: one inclusion is proved for arbitrary monomial ideals (Theorem 1.2), and the reverse inclusion follows when powers have linear resolutions (Proposition 1.3). This equality is what makes $\\mathrm{HS}_i(\\mathcal{R}(I))$ a finitely generated bigraded module over the Rees algebra $\\mathcal{R}(I)=\\bigoplus_{k\\ge0}I^k$. Once finiteness is established, the paper imports standard asymptotic theorems for graded modules over a standard graded ring to get stabilization of associated primes and depth, eventual linearity of regularity and $v$-number, and the Golod criterion used in Theorem 4.3.","core_discovery":"The central discovery is that the homological shift ideals of all powers of a monomial ideal can be studied as a single algebraic object. For fixed $i$, the $i$th homological shift algebra is $\\mathrm{HS}_i(\\mathcal{R}(I))=\\bigoplus_{k\\ge1}\\mathrm{HS}_i(I^k)$, bigraded by internal degree and by $k$. The paper proves that if $I$ has linear powers, then $\\mathrm{HS}_i(I^{k+1})=I\\cdot\\mathrm{HS}_i(I^k)$ for all $k\\gg0$: the inclusion $\\mathrm{HS}_i(I^{k+1})\\subseteq I\\cdot\\mathrm{HS}_i(I^k)$ holds for every monomial ideal, and the reverse inclusion uses the linear-resolution hypothesis. Consequently $\\mathrm{HS}_i(\\mathcal{R}(I))$ is a finitely generated bigraded module over the Rees algebra $\\mathcal{R}(I)$ (Theorem 1.4). This module-theoretic fact immediately yields the asymptotic conclusions (Theorem 2.1): $\\mathrm{Ass}(\\mathrm{HS}_i(I^k))$ stabilizes, $\\operatorname{depth} S/\\mathrm{HS}_i(I^k)$ stabilizes, and $\\operatorname{reg}\\mathrm{HS}_i(I^k)$ and $v(\\mathrm{HS}_i(I^k))$ are eventually linear functions of $k$. The paper also proves that $\\mathrm{HS}_i(I^k)$ is Golod for all $i$ and all $k\\gg0$ (Theorem 4.2) and exhibits families — principal Borel ideals, $c$-bounded principal Borel ideals, cover ideals of whisker graphs, Hibi ideals, and two-variable monomial ideals with linear resolution — for which $\\mathrm{HS}_i(I^k)$ has a linear resolution for all $i$ and $k$.","pith_inferences":["The proofs use the eventual equality $\\mathrm{HS}_i(I^{k+1})=I\\cdot\\mathrm{HS}_i(I^k)$ more directly than the full linear-powers hypothesis, so the same stabilization and linearity conclusions should hold for ideals with only eventually linear powers, or even for any monomial ideal satisfying this equality for large $k$.","The finite-generation theorem is non-constructive: it guarantees some $k_0$ with $\\mathrm{HS}_i(I^k)=I^{k-k_0}\\mathrm{HS}_i(I^{k_0})$ for $k\\ge k_0$, but gives no bound. Finding explicit stabilization bounds for the families in Section 3 would turn the asymptotic statements into effective predictions.","Theorem 4.3 applies to any finitely generated module over the Rees algebra whose graded pieces are proper ideals, so the Golod conclusion is not a special feature of homological shift ideals; the same argument would yield Golodness for other natural ideal filtrations with finite generation."],"forward_implications":["For any monomial ideal with linear powers, the sets $\\mathrm{Ass}(\\mathrm{HS}_i(I^k))$ stabilize and the depth of $S/\\mathrm{HS}_i(I^k)$ is eventually constant, for every $i$.","The Castelnuovo–Mumford regularity of $\\mathrm{HS}_i(I^k)$ is an eventually linear function of $k$; in cases where $\\mathrm{HS}_i(I^k)$ has a linear resolution, it equals $\\alpha(I)k+i$.","The $v$-number of $\\mathrm{HS}_i(I^k)$, and each $v_p$-number for an eventually associated prime $p$, are eventually linear functions of $k$.","For every $i$ and all $k\\gg0$, the ideal $\\mathrm{HS}_i(I^k)$ is Golod, meaning the Poincaré series of its quotient ring attains Serre's upper bound.","For several families (principal Borel ideals, $c$-bounded principal Borel ideals, cover ideals of whisker graphs, Hibi ideals, and two-variable monomial ideals with linear resolution), $\\mathrm{HS}_i(I^k)$ has a linear resolution for all $i$ and $k$, so the eventual linearity is explicit."],"supporting_citations":[{"why":"Defines homological shift ideals and supplies the socle formula $\\mathrm{HS}_{n-1}(I)=x_1\\cdots x_n\\,\\mathrm{soc}(I)$ used throughout.","marker":"[30]"},{"why":"Provides the asymptotic identity for Tor over the fiber ring and the v-number asymptotics that power Theorem 2.1.","marker":"[21]"},{"why":"Shows associated primes of a finitely generated graded module over a standard graded ring stabilize, used in Theorem 2.1(a).","marker":"[36]"},{"why":"Shows depth of graded components of a finitely generated graded module stabilizes, used in Theorem 2.1(b).","marker":"[28]"},{"why":"Shows regularity of $I^kM$ is eventually linear for a finitely generated module $M$, used in Theorem 2.1(c).","marker":"[39]"},{"why":"Shows the v-number and v_p-number of $I^kM$ are eventually linear, used in Theorem 2.1(d)-(e).","marker":"[24]"},{"why":"The result on Golodness of powers that Theorem 4.3 generalizes to arbitrary finitely generated modules over the Rees algebra.","marker":"[34]"}],"fun_headline_variants":["One module packs all homological shifts of monomial ideal powers","Single graded module controls asymptotic invariants of ideal powers","One Rees module encodes all homological shifts of I^k","Homological shift algebra: one module for all powers","Finite module over Rees algebra governs depth and regularity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every power $I^k$ has a linear resolution, with all syzygies concentrated in a single degree, since this is what makes the homological shift ideals of consecutive powers satisfy $\\mathrm{HS}_i(I^{k+1})=I\\cdot\\mathrm{HS}_i(I^k)$ for large $k$ and hence makes the direct sum a finitely generated module over the Rees algebra.","fun_headline_variants_meta":{"raw":{"variants":["One module packs all homological shifts of monomial ideal powers","Single graded module controls asymptotic invariants of ideal powers","One Rees module encodes all homological shifts of I^k","Homological shift algebra: one module for all powers","Finite module over Rees algebra governs depth and regularity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000641,"raw_usage":{"total_tokens":3046,"prompt_tokens":1138,"completion_tokens":1908,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":754,"completion_tokens_details":{"reasoning_tokens":1829}},"tokens_in":754,"tokens_out":1908,"duration_ms":13514,"temperature":1.0,"reasoning_tokens":1829,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:08:19.154829+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any monomial ideal with linear powers, compute $\\mathrm{HS}_i(I^k)$ for increasing $k$, and check whether the equality $\\mathrm{HS}_i(I^{k+1})=I\\cdot\\mathrm{HS}_i(I^k)$ holds for all sufficiently large $k$ and whether $\\mathrm{Ass}(\\mathrm{HS}_i(I^k))$ stabilizes; the paper predicts both, so a single linear-powers ideal violating either for arbitrarily large $k$ would refute the main theorem.","supporting_citations":[{"cited_title":"Herzog, S","cited_arxiv_id":null,"evidence_quote":"Defines homological shift ideals and supplies the socle formula $\\mathrm{HS}_{n-1}(I)=x_1\\cdots x_n\\,\\mathrm{soc}(I)$ used throughout."},{"cited_title":"McAdam, P","cited_arxiv_id":null,"evidence_quote":"Shows associated primes of a finitely generated graded module over a standard graded ring stabilize, used in Theorem 2.1(a)."},{"cited_title":"Herzog, T","cited_arxiv_id":null,"evidence_quote":"Shows depth of graded components of a finitely generated graded module stabilizes, used in Theorem 2.1(b)."},{"cited_title":"Trung, H.-J","cited_arxiv_id":null,"evidence_quote":"Shows regularity of $I^kM$ is eventually linear for a finitely generated module $M$, used in Theorem 2.1(c)."},{"cited_title":"Fiorindo, D","cited_arxiv_id":null,"evidence_quote":"Shows the v-number and v_p-number of $I^kM$ are eventually linear, used in Theorem 2.1(d)-(e)."},{"cited_title":"Herzog, V","cited_arxiv_id":null,"evidence_quote":"The result on Golodness of powers that Theorem 4.3 generalizes to arbitrary finitely generated modules over the Rees algebra."}],"review_version":1}