REVIEW 3 major objections 3 minor 18 references
Coherent functors, powers of ideals, and asymptotic stability
T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read This paper proves that coherent functors make the associated primes, grades, and lengths of modules of the form $M/(I_1^{n_1}\cdots I_r^{n_r}N)$ stabilize or become polynomial as the exponents grow, with a sharp degree bound in the local…
desk verdict Clean completion of the coherent-functor asymptotic program for products of powers of several ideals, with sharp degree bounds; the main proofs are sound and the only real issues are presentation and two imported results worth verifying. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
A coherent functor is a covariant $R$-linear functor that fits in an exact sequence $h_K \to h_M \to F \to 0$ of Hom functors; this class includes $\operatorname{Tor}$, $\operatorname{Ext}$, and homology functors of tensor complexes. The load-bearing construction is Theorem 2.6: after lifting a presentation of $F$ to free resolutions and applying the multigraded Artin–Rees lemma, every value $F(M/I^{\underline{n}}N)$ is exhibited as $(U + I^{\underline{n}-d}V)/I^{\underline{n}-d}W$, with $U \cong F(M)$ and $W \subseteq V$. This single normal form converts questions about all exponent vectors into questions about a fixed finite module $U$ and a product-ideal filtration, which is exactly what the multigraded Hilbert polynomial and degree estimates can answer.
What would settle it
Compute the invariants for a concrete coherent functor such as $\operatorname{Tor}_1^R(k,-)$ on a local ring $R$ with two ideals $I_1, I_2$, taking $M=R$ and $N=0$: the theorem predicts that $\operatorname{Ass}_R(\operatorname{Tor}_1^R(k, R/(I_1^{n_1}I_2^{n_2})))$ stabilizes and, whenever lengths are finite, they match a polynomial in $(n_1,n_2)$ for large exponents. An explicit oscillation or a non-polynomial finite-length sequence would disprove Theorem 2.12; equivalently, checking the multigraded Artin–Rees identity on that example would locate the failure.
Extended reading notes
Core claim
The central discovery is that every coherent functor $F$ admits a stable finite presentation on the family $M/I^{\underline{n}}N$: for some exponent vector $d$, one has $F(M/I^{\underline{n}}N) \cong (U + I^{\underline{n}-d}V)/I^{\underline{n}-d}W$ for all $\underline{n} \geq d$, where $U \cong F(M)$ and $W \subseteq V$. Once this normal form is in hand, the eventual stabilization of $\operatorname{Ass}_R$ and $\operatorname{grade}$, the polynomial growth of length, and the degree bounds all follow from standard facts about multigraded modules and their Hilbert polynomials. Thus the paper reduces the entire problem to showing that coherent functors carry asymptotic stability from the ambient ring to arbitrary derived operations, uniformly in $r$ ideals.
Load-bearing premise
Everything rests on the multigraded Artin–Rees identity—intersecting with a product of ideal powers eventually just shifts by a fixed power—which the paper imports rather than proves; if that identity fails for several ideals, the reduction of $F(M/I^{\underline{n}}N)$ to a fixed quotient model collapses.
Editorial extensions
If this is right
- If $F$ is coherent and $F(M/I^{\underline{n}}N)$ has finite length for all large $\underline{n}$, then the length function is eventually a polynomial over $\mathbb{Q}$ in the $r$ variables $n_1,\ldots,n_r$.
- In the local case, the total degree of that polynomial is at most $\max\{\dim(F(M)), \ell_M(I)-r\}$, and equality holds whenever $\dim(F(M)) > \ell_M(I)-r$, giving a computable growth rate.
- For each fixed $i$, the $i$th Betti number $\beta_i^R(F(M/I^{\underline{n}}N))$ and the $i$th Bass number $\mu^i_R(F(M/I^{\underline{n}}N))$ are eventually polynomial in $\underline{n}$, with degree bounded by $\max\{0, \ell_M(I)-r\}$.
- Consequently, projective dimension and injective dimension of $F(M/I^{\underline{n}}N)$ are eventually constant for large $\underline{n}$.
- The same machinery also applies to the graded components $\mathcal{M}_{\underline{n}}$ of a finitely generated multigraded module over a Noetherian standard $\mathbb{N}^r$-graded ring, so both natural families of modules are covered.
Reading between the lines
- Because the class of coherent functors is closed under composition, the same arguments should give stability and polynomiality for iterated coherent functors, not just a single $F$; this would extend the theorem to repeated derived operations without new ideas.
- The degree bound is independent of $N$ and of the index $i$, which suggests that the leading asymptotic term of the length polynomial may be governed entirely by the pair $(F(M), \ell_M(I))$; testing this on concrete examples would be a natural numerical experiment.
- The equality condition $\dim(F(M)) > \ell_M(I)-r$ separates growth coming from the functor's target dimension from growth coming from the ambient ideal filtration, so algorithms that compute $\ell_M(I)$ could turn the theorem into explicit stabilization thresholds.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that for a coherent functor F on the category of finitely generated modules over a Noetherian ring R, the sets Ass_R(F(M/I_1^{n_1}...I_r^{n_r}N)) and grade(J, F(M/I_1^{n_1}...I_r^{n_r}N)) stabilize for large multi-indices, and that if F(M/I_1^{n_1}...I_r^{n_r}N) has finite length then its length is eventually given by a multivariable polynomial. In the local case, a degree bound max{dim F(M), ℓ_M(I)-r} is proved, with equality when dim F(M) > ℓ_M(I)-r. Applications include eventual polynomiality of Betti and Bass numbers and constancy of projective and injective dimension.
Significance. If correct, the main theorem is a substantial unification of earlier results by Brodmann, Kingsbury-Sharp, Katz-West, Kodiyalam, Theodorescu, Se, and Banda-Melkersson, covering arbitrary products of ideals and all coherent functors. The proof strategy, reducing F(M/I^nN) to a module of the form (U+I^{n-d}V)/I^{n-d}W, is elegant, and the paper includes self-contained proofs of some auxiliary reductions (Lemmas 2.4, 2.5 and Proposition 2.7). The applications to Betti and Bass numbers, with degrees independent of i and of F, are valuable. However, the manuscript currently contains a defective short exact sequence in Theorem 2.9 and an insufficiently justified strong form of the multigraded Artin-Rees lemma; both are load-bearing steps in the proof. The issues appear local and fixable, so the central claims are likely correct.
major comments (3)
- [Theorem 2.9, Eq. (2.6)] The displayed sequence 0 → ((U+I^nW) ∩ I^nV)/I^nW → (U+I^nW)/I^nW → L_n → 0 is not a short exact sequence. The map from the middle term to L_n is the natural inclusion, whose kernel is zero and whose cokernel is (U+I^nV)/(U+I^nW), not zero. Hence the stated inference that λ_R(L_n) is a polynomial once the first two modules are polynomial does not follow from this sequence. Please replace (2.6) with a correct exact sequence, for example 0 → I^nV/I^nW → L_n → U/(U∩I^nV) → 0, and adjust the degree-bound argument accordingly. This is a load-bearing error because Theorem 2.9 is used in the proof of Theorem 2.12(3).
- [Lemma 2.5 and Theorem 2.6] The identity ψ(B) ∩ I^nC' = I^{n-d}(ψ(B) ∩ I^dC') is asserted as a direct consequence of [16, 17.1.6]. The standard multigraded Artin-Rees lemma gives a componentwise sum (N ∩ I^nM = Σ_i I^{n-e_i}(N ∩ I^{e_i}M) in the usual formulation), not the displayed single-product form. The single-product form is true—it follows by choosing d larger than the degrees of a generating set of the Rees module ⊕_n (ψ(B) ∩ I^nC')—but this justification is not supplied in the manuscript. Since this identity is the mechanism that converts F(M/I^nN) into (U+I^{n-d}V)/I^{n-d}W, the proof should include the missing argument or cite a source that states precisely this strong form.
- [Theorem 2.12(1)–(2)] The proof imports [10, Prop. 3.4] to conclude that Ass_R((U+I^nV)/I^nW) stabilizes. The proposition is not stated in the manuscript, and its hypotheses are not verified against the present multigraded, non-local, arbitrary Noetherian ring setup. This is a load-bearing pillar for both the Ass and grade stabilization statements. Please quote the proposition in full, including its hypotheses, or provide a self-contained proof.
minor comments (3)
- [Title] The arXiv header shows 'ST ABILITY' instead of 'STABILITY'; please fix the spacing.
- [Theorem 2.3, proof] When claiming that ⊕_n F(M_n) is a quotient of ⊕_n h_K(M_n), the finite generation of ⊕_n h_K(M_n) over S is used implicitly; this follows by taking a presentation of K, but it should be stated for completeness.
- [Remark 2.11] The phrase 'highly generalizes' is informal; please specify in what precise sense Theorem 2.10 extends [17, Cor. 4].
Circularity Check
No circular derivation: Theorem 2.12 is assembled from external theorems and independently defined invariants, with no fitted input renamed as a prediction.
full rationale
The paper's derivation chain is external throughout. The central structural reduction, Theorem 2.6, converts a coherent functor presentation through Yoneda's lemma and the multigraded Artin-Rees lemma into the module family (U + I^{n-d}V)/I^{n-d}W; the cited Artin-Rees identity [16, 17.1.6] is a standard theorem and is not the paper's own conclusion. Theorem 2.12 then obtains stabilization of Ass and grade from [10, Prop. 3.4] and polynomial behavior from [9, Thm. 4.1], [16, Thm. 17.4.2], and [5, Thm. 3.2]. The target invariants—Ass, grade, and length—are independently defined and are not used to define the functor or the ideals, so there is no self-definitional reduction. The degree bound involves dim(F(M)) and the separately defined multigraded dimension ell_M(I); neither quantity is fitted against the eventual polynomial lambda_R(F(M/I^n N)), so no fitted input is being called a prediction. The self-citations [6] and [7] are to published technical results: [6, Lem. 3.4(ii)] supplies stabilization of annihilators inside Theorem 2.9, and [7] is mentioned only as a concise proof of Katz-West's result; neither assumes the main theorem, and neither is the sole justification of the central claim. A possible correctness concern—whether [10, Prop. 3.4] is proved in the full multigraded generality used here—would be a missing-verification issue, not circularity, because the proposition is not restated as the paper's conclusion and no equation in the paper reduces the target statement to itself. Accordingly, no circular step is exhibited, and the honest finding is that the derivation is self-contained against external, published inputs.
Assumptions & free parameters
assumptions (7)
- domain assumption R is a Noetherian ring, ideals are arbitrary, modules are finitely generated
- domain assumption F is a coherent functor, i.e. there is an exact sequence h_L → h_K → F → 0
- standard math Multigraded Artin-Rees lemma (Swanson-Huneke [16, 17.1.6])
- standard math Ass and grade stabilization for multigraded modules (West [18, Thm 3.4(i), Cor 3.9(i)])
- standard math Ass stabilization for (U+I^nV)/I^nW is taken from Katz-West [10, Prop. 3.4]
- domain assumption Finite length hypothesis for the polynomial part
- standard math Hilbert polynomial theorem for multigraded modules (Herrmann et al. [9, Thm 4.1])
Cite this review
Pith. "Pith review of Coherent functors, powers of ideals, and asymptotic stability." pith.science (2026). https://pith.science/paper/V3RSZYWQ
@misc{pith2026250600529,
author = {Pith},
title = {Pith review of: Coherent functors, powers of ideals, and asymptotic stability},
year = {2026},
howpublished = {\url{https://pith.science/paper/V3RSZYWQ}},
note = {Machine review of arXiv:2506.00529}
}
abstract
Let $R$ be a Noetherian ring, $I_1,\ldots,I_r$ be ideals of $R$, and $N\subseteq M$ be finitely generated $R$-modules. Let $S = \bigoplus_{\underline{n} \in \mathbb{N}^r} S_{\underline{n}}$ be a Noetherian standard $\mathbb{N}^r$-graded ring with $S_{\underline{0}} = R$, and $\mathcal{M} $ be a finitely generated $\mathbb{Z}^r$-graded $S$-module. For $ \underline{n} = (n_1,\dots,n_r) \in \mathbb{N}^r$, set $G_{\underline{n}} := \mathcal{M}_{\underline{n}}$ or $G_{\underline{n}} := M/{\bf I}^{\underline{n}} N$, where ${\bf I}^{\underline{n}} = I_1^{n_1} \cdots I_r^{n_r}$. Suppose $F$ is a coherent functor on the category of finitely generated $R$-modules. We prove that the set $\rm{Ass}_R \big(F(G_{\underline{n}}) \big)$ of associate primes and $\rm{grade}\big(J, F(G_{\underline{n}})\big)$ stabilize for all $\underline{n} \gg 0$, where $J$ is a non-zero ideal of $R$. Furthermore, if the length $\lambda_R(F(G_{\underline{n}}))$ is finite for all $\underline{n} \gg 0$, then there exists a polynomial $P$ in $r$ variables over $\mathbb{Q}$ such that $\lambda_R(F(G_{\underline{n}})) = P(\underline{n})$ for all $\underline{n}\gg 0$. When $R$ is a local ring, and $G_{\underline{n}} = M/{\bf I}^{\underline{n}} N$, we give a sharp upper bound of the total degree of $P$. As applications, when $R$ is a local ring, we show that for each fixed $i \geq 0$, the $i$th Betti number $\beta_i^R(F(G_{\underline{n}}))$ and Bass number $\mu^i_R(F(G_{\underline{n}}))$ are given by polynomials in $\underline{n}$ for all $\underline{n} \gg 0$. Thus, in particular, the projective dimension $\rm{pd}_R(F(G_{\underline{n}}))$ (resp., injective dimension $\rm{id}_R(F(G_{\underline{n}}))$) is constant for all $\underline{n}\gg 0$.
Reference graph
Works this paper leans on
-
[1]
Maurice Auslander. Coherent functors. In Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) , pages 189–231. Springer-Verlag New York, Inc., New York, 1966. 2
work page 1965
-
[2]
Coherent functors and asymptotic stability
Adson Banda and Leif Melkersson. Coherent functors and asymptotic stability. J. Algebra, 522:1–10, 2019. 2, 8
work page 2019
- [3]
- [4]
-
[5]
J. Bruce Fields. Lengths of Tors determined by killing powers of ideals in a local ring. J. Algebra , 247(1):104–133,
-
[6]
Asymptotic linear bounds of Castelnuovo-Mumford regularity in multigraded modules
Dipankar Ghosh. Asymptotic linear bounds of Castelnuovo-Mumford regularity in multigraded modules. J. Algebra, 445:103–114, 2016. 6
work page 2016
-
[7]
Dipankar Ghosh and Tony J. Puthenpurakal. A short proof of a result of Katz and West. J. Commut. Algebra , 11(2):237–240, 2019. 1
work page 2019
-
[8]
Robin Hartshorne. Coherent functors. Adv. Math., 140(1):44–94, 1998. 2
work page 1998
Show all 18 references
-
[9]
Reduction numbers and multiplicities of multi- graded structures
Manfred Herrmann, Eero Hyry, J¨ urgen Ribbe, and Zhongming Tang. Reduction numbers and multiplicities of multi- graded structures. J. Algebra, 197(2):311–341, 1997. 6
1997
-
[10]
A linear function associated to asymptotic prime divisors
Daniel Katz and Eric West. A linear function associated to asymptotic prime divisors. Proc. Amer. Math. Soc. , 132(6):1589–1597, 2004. 1, 7, 8
2004
-
[11]
Kingsbury and Rodney Y
Alan K. Kingsbury and Rodney Y. Sharp. Asymptotic behaviour of certain sets of prime ideals. Proc. Amer. Math. Soc., 124(6):1703–1711, 1996. 1, 5
1996
-
[12]
Homological invariants of powers of an ideal
Vijay Kodiyalam. Homological invariants of powers of an ideal. Proc. Am. Math. Soc. , 118(3):757–764, 1993. 1, 2, 8
1993
-
[13]
Asymptotic prime ideals related to derived functors
Leif Melkersson and Peter Schenzel. Asymptotic prime ideals related to derived functors. Proc. Am. Math. Soc. , 117(4):935–938, 1993. 1
1993
-
[14]
Joseph J. Rotman. An introduction to homological algebra . Universitext. Berlin: Springer, 2nd ed. edition, 2009. 4
2009
-
[15]
Covariant functors and asymptotic stability
Tony Se. Covariant functors and asymptotic stability. J. Algebra, 484:247–264, 2017. 2, 3, 4
2017
-
[16]
Integral closure of ideals, rings, and modules , volume 336 of Lond
Irena Swanson and Craig Huneke. Integral closure of ideals, rings, and modules , volume 336 of Lond. Math. Soc. Lect. Note Ser. Cambridge: Cambridge University Press, 2006. 4, 5
2006
-
[17]
Derived functors and Hilbert polynomials
Emanoil Theodorescu. Derived functors and Hilbert polynomials. Math. Proc. Cambridge Philos. Soc. , 132(1):75–88,
-
[18]
Primes associated to multigraded modules
Eric West. Primes associated to multigraded modules. J. Algebra, 271(2):427–453, 2004. 1, 3 Department of Algebra, F aculty of Mathematics and Physics, Charles University in Prague, Sokolovsk ´a 83, 186 75 Praha, Czech Republic Email address : souvik.dey@matfyz.cuni.cz URL: ht...
2004
Reviewed August 7, 2026 · model on record in the stance chip above.
Discussion (0). Sign in to comment.