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A lower bound on levels with applications to Koszul Complexes

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Pith's one-line read For perfect complexes with power torsion homology, the paper proves level_R F ≥ dim R - dim R/I + 1 and shows the bound is optimal for Koszul complexes.

desk verdict A solid strengthening of level bounds that deserves a serious referee, but the proof has a load-bearing gap in completing a big Cohen-Macaulay algebra that needs to be justified or fixed. read the letter →

arxiv 2505.14812 v1 pith:DUOZXVGQ submitted 2025-05-20 math.AC

Antonia Kekkou This is my paper · ORCID
classification math.AC
keywords boundlowercomplexhomologykoszullevelpowertorsion
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The pith

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

The reading

Homological algebra assigns to a chain complex a number called its level: the minimum number of mapping-cone steps needed to build the complex from the base ring. This paper studies finite free complexes whose positive-degree homology is annihilated by a power of an ideal I, and whose zeroth homology has a generator killed by such a power. The main theorem says that under these conditions the level of the complex is at least one plus the difference between the Krull dimension of the ring and the dimension of the quotient ring R/I.

This improves two earlier results. A theorem of Christensen and Ferraro gave the same expression without the plus one, but for the length of the complex rather than the level. A theorem of Avramov, Iyengar, and Neeman gave a lower bound of height of I plus one; since dim R minus dim R/I is always at least the height of I, the new bound is stronger.

The paper applies the bound to Koszul complexes. For a generating set of an ideal that is a partial system of parameters, the level of the Koszul complex is exactly n + 1, so the bound cannot be improved in general. It also gives a lower bound in terms of the free rank of I/I^2, and recovers a known superheight bound with a shorter proof. The proof combines the arguments of two earlier proofs of the New Intersection Theorem and uses balanced big Cohen-Macaulay algebras.

Extended reading notes

Core claim

Theorem 3.1: if R is a commutative noetherian local ring, I an ideal, and F a finite free R-complex with H0(F) not equal to zero such that Hi(F) is I-power torsion for i ≥ 1 and a minimal generator of H0(F) is I-power torsion, then level_R F ≥ dim R - dim R/I + 1. If the paper is correct, this is the sharp lower bound for the level of such complexes, and it strengthens height-based bounds in [6] and length-based bounds in [9].

Load-bearing premise

The proof of Theorem 3.1 assumes the existence of a balanced big Cohen-Macaulay R-algebra for every local ring, and that it can be chosen m-complete and satisfying the depth inequality (2.6.1). This is the deep theorem of Andre, Bhatt, Hochster and Huneke invoked in Section 2.11 and used at the opening step of the proof. If this existence theorem failed in some mixed-characteristic local ring, the derivation of the main bound would collapse.

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Editorial analysis

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Referee Report

2 major / 4 minor

Summary. The paper proves Theorem 3.1: for a commutative noetherian local ring R, an ideal I, and a finite free R-complex F with H0(F) nonzero, if Hi(F) is I-power torsion for i≥1 and some minimal generator of H0(F) is I-power torsion, then level_R F ≥ dim R − dim R/I + 1. The proof follows the template of the Improved New Intersection Theorem proofs in [1] and [9], using balanced big Cohen-Macaulay algebras, derived m-completeness, and syzygy-level estimates. The paper then derives applications: bounds for Koszul complexes (Proposition 3.2), a free-rank bound for minimal dg-algebras and Koszul complexes (Proposition 3.5), an equality for Lech-independent sequences (Corollary 3.7), and a short proof of the superheight bound (Corollary 4.1). Examples in Section 4 show that the quantity dim R − dim R/I cannot be replaced by superheight or bigheight in general.

Significance. If the main theorem is correct, it is a genuine strengthening of the height-based bound in [6, Theorem 4.2] and of the length-based bound in [9, Theorem 2.2], and the examples in Section 4 show that the new invariant is sharp and incomparable with bigheight and superheight. The proof is a coherent synthesis of existing machinery rather than a fundamentally new method, but the resulting bound is natural and the applications to Koszul complexes are useful. The paper is clearly written and the central derivation is transparent, which makes it easy to check the main steps.

major comments (2)
  1. [Section 3, proof of Theorem 3.1, first paragraph and equations (2.6.1)–(2.6.2)] The proof asserts that completing a balanced big Cohen-Macaulay R-algebra with respect to m yields an m-complete big Cohen-Macaulay R-algebra, and then applies (2.6.1) and (2.6.2) to this S. This step is load-bearing but is not justified by the cited existence theorems. The completion of a non-Noetherian algebra need not preserve balancedness, and the subsequent applications require S to be derived m-complete, not merely classically m-adically complete. Please provide a proof or a precise citation for the existence of an m-complete balanced big Cohen-Macaulay algebra with the derived-completeness property used in (2.6.1)–(2.6.2).
  2. [Section 3.4, Proposition 3.5 and its proof] The sentence "It is easy to see that H1(g⊗k) is an isomorphism" is false without an additional minimality hypothesis on x. For example, take R = k[[t]], I = (t), and x = (t,0); then the map H1(g⊗k): k^2 → Tor_1^R(R/I,k) ≅ k has rank one and is not an isomorphism. As written, the proof of Proposition 3.5 does not establish the stated bound for an arbitrary generating set x. The statement should require that x is a minimal generating set of I, or the proof must be modified to reduce to a minimal generating subsequence and control the additional Koszul factors.
minor comments (4)
  1. [Abstract and Introduction] The abstract says "with power torsion homology" and "a power torsion minimal generator"; the word "I-" is missing, and this makes the hypotheses ambiguous on first reading.
  2. [Section 3, proof of Theorem 3.1] The sentence "The proof of Theorem 2.2 in [9] shows that I ⊆ p" is terse; spelling out the supporting argument or citing the exact statement in [9] would improve readability.
  3. [Section 3.4, Proposition 3.5] The notation "TorR(R/I, k)" and "TorR(R/I, k) ≅ B ⊗_k Λ" is ambiguous; it should be written as Tor^R or Tor^R_* to indicate the graded Tor algebra.
  4. [Section 4.3] In the diagram for the factorization through K(x2,...,xn; R), the middle row is visually confusing; labeling the Koszul complex and its differential explicitly would make the example easier to follow.
Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The main theorem rests on standard derived-category depth and level facts and on the existence of balanced big Cohen-Macaulay algebras. No numerical parameter is fitted and no new object is postulated. The only non-elementary input is the big Cohen-Macaulay existence theorem.

assumptions (5)
  • domain assumption Existence of balanced big Cohen-Macaulay algebras over arbitrary local rings
    The proof of Theorem 3.1 takes a balanced big Cohen-Macaulay R-algebra S and completes it. This is Andre's theorem extending Hochster-Huneke and Bhatt, a deep external result about all local rings.
  • standard math Depth inequality depth_R M ≤ depth_R(I, M) + dim R/I for derived m-complete complexes
    Used to convert the depth of the completed big Cohen-Macaulay algebra into the dimension term in Theorem 3.1.
  • standard math Level base change inequality level_R M ≥ level_S(M tensor_R^L S)
    Used in Theorem 3.1 and Corollary 4.1 to transfer level lower bounds along ring maps.
  • standard math Level criterion for non-projective nth syzygies
    Essential at the end of Theorem 3.1's proof to deduce level_S(F tensor_R S) ≥ n - s + 1 from non-projectivity of a syzygy.
  • standard math The structure theorem for Tor algebras with a free exterior factor of rank equal to the free rank of I/I^2
    Used to prove the free-rank lower bound of Proposition 3.5.

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Pith. "Pith review of A lower bound on levels with applications to Koszul Complexes." pith.science (2026). https://pith.science/paper/DUOZXVGQ

@misc{pith2026250514812,
  author       = {Pith},
  title        = {Pith review of: A lower bound on levels with applications to Koszul Complexes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DUOZXVGQ}},
  note         = {Machine review of arXiv:2505.14812}
}
read the original abstract

In this paper, we establish a lower bound on the level of a perfect complex with power torsion homology on positive degrees and a power torsion minimal generator for zero homology. Examples are provided to demonstrate that the bound is optimal. This result is applied to improve existing lower bounds on the level of a Koszul complex on various classes of sequences.

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