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The homological shift algebra of a monomial ideal

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Pith's one-line read 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…

desk verdict A genuinely useful new construction with sound asymptotic theorems, but the Golod statement overreaches as written and two displayed formulas have typos. read the letter →

arxiv 2412.21031 v3 pith:O2XFD44S submitted 2024-12-30 math.AC math.CO

Antonino Ficarra , Ayesha Asloob Qureshi This is my paper · ORCID
classification math.ACmath.CO MSC 13F2013F5505C7005E40
keywords monomialidealshomologicalshiftReesalgebralinearpowerssyzygiesCastelnuovo-Mumfordregularityv-numberGolod
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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 $k$.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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 $k$.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

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.

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 (3)
  1. [Theorem 4.2 / Theorem 4.3, Section 4] 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.
  2. [Proposition 2.7] 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.
  3. [Proposition 2.3(c3)] 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.
minor comments (3)
  1. [Example 3.4] 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.
  2. [Sections 2.1 and 3.9(a)] 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.
  3. [Proposition 2.2] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity; central derivation is self-contained; only minor non-central self-citations.

full rationale

The paper's central derivation chain is not circular. Theorem 1.4 is proved from Proposition 1.3, whose proof uses only the linear-resolution hypothesis and an exact-sequence/Tor argument; the opposite inclusion is Theorem 1.2, proved from finite generation of Tor_i^S(K, R(I)) over the associated graded ring F_m(I). No equation in this chain is assumed into the definition of HS_i(I^k); the module structure is derived, not imposed. Theorem 2.1 then follows by standard asymptotic results for finitely generated modules over the Rees algebra (McAdam-Eakin, Herzog-Hibi, Trung-Wang, Fiorindo-Ghosh), not by restating the conclusion. Theorem 4.2 is a special case of Theorem 4.3, whose Massey-operation proof is self-contained and generalizes Herzog-Welker-Yassemi; the finite generation input comes from Theorem 1.4. The cited self-references ([12], [13], [14], [19], [21]) support auxiliary classifications and lemmas (e.g., Theorem 3.9(c),(e), Proposition 2.4, Corollary 3.2) and are not the source of the main results. The skeptical objection to Theorem 4.2 concerns the zero-ideal convention for Golodness when HS_i(I^k) vanishes, which is a correctness/edge-case issue, not a reduction of the claim to its input. Therefore no circular step is exhibited; the paper is essentially self-contained against external benchmarks, with only minor non-load-bearing self-citations.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper is a pure mathematics contribution with no empirical free parameters. The main load-bearing input is the linear-powers hypothesis, which is a condition on the ideal rather than an unexplained axiom. All other inputs are standard theorems from the graded-algebra and homological literature, several of which are self-cited but used as tools rather than as the target result.

assumptions (4)
  • domain assumption I has linear powers, i.e., I^k has a linear resolution for every k ≥ 1.
    Central hypothesis of Theorems 1.4, 2.1, and 4.2. Example 1.1 shows this condition is not automatic for monomial ideals.
  • standard math Standard asymptotic facts for finitely generated graded modules over standard graded algebras, including results from [36], [28], and [39].
    Used in Theorem 2.1 to convert finite generation of the homological shift algebra into stabilization and linearity of invariants.
  • standard math Descriptions of homological shift ideals from [30] and [33], including HSn−1(I) = x1···xn · soc(I).
    Used in Proposition 1.5, Example 1.1, and the computations in Section 2.
  • standard math Golod criterion via Massey operations from [25], together with the extension in Theorem 4.3 of a result from [34].
    Underpins the asymptotic Golodness result in Theorem 4.2.

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Pith. "Pith review of The homological shift algebra of a monomial ideal." pith.science (2026). https://pith.science/paper/O2XFD44S

@misc{pith2026241221031,
  author       = {Pith},
  title        = {Pith review of: The homological shift algebra of a monomial ideal},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O2XFD44S}},
  note         = {Machine review of arXiv:2412.21031}
}
abstract

Let $S=K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$, and let $I\subset S$ be a monomial ideal. In this paper, we introduce the $i$th \textit{homological shift algebras} $\text{HS}_i(\mathcal{R}(I))=\bigoplus_{k\ge1}\text{HS}_i(I^k)$ of $I$. If $I$ has linear powers, these algebras have the structure of a finitely generated bigraded module over the Rees algebra $\mathcal{R}(I)$ of $I$. Hence, many invariants of $\text{HS}_i(I^k)$, such as depth, associated primes, regularity, and the $\text{v}$-number, exhibit well behaved asymptotic behavior. We determine several families of monomial ideals $I$ for which $\text{HS}_i(I^k)$ has linear resolution for all $k\gg0$. Finally, we show that $\text{HS}_i(I^k)$ is Golod for all monomial ideals $I\subset S$ with linear powers and all $k\gg0$.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On the homological shifts of cover ideals of Cohen-Macaulay graphs

    math.AC 2025-06 conditional novelty 8.0 of 10

    For every k≥2, a Cohen-Macaulay very well-covered whiskered bipartite graph has non-linear HS_k, and the paper identifies classes where HS_k does have linear quotients.

  2. Homological shift ideals of weighted oriented graphs

    math.AC 2026-08 conditional novelty 6.0 of 10

    For weighted oriented graphs, the homological shift ideals have linear quotients exactly when the underlying tree is a star or broom and the weighted oriented graph avoids D1,D2,D5,D6,D8 as induced subgraphs.

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