arxiv: 2604.20646 · v1 · submitted 2026-04-22 · 🧮 math.AC

Recognition: unknown

Multiple Tor modules: rigidity and Mayer-Vietoris spectral sequences

Arindam Banerjee, Marc Chardin, Rafael Holanda

Pith reviewed 2026-05-09 22:50 UTC · model grok-4.3

classification 🧮 math.AC

keywords Tor modulesrigidityMayer-Vietoris spectral sequencesmultiple complexesideal sums and productscommutative algebra

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The pith

Properties of Tor modules for pairs of ideals, including rigidity, extend to any number of ideals via spectral sequences from multiple complexes.

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

The paper generalizes known properties of Tor modules for two ideals to any finite number of ideals. This requires developing a homological theory for spectral sequences that arise from multiple complexes. Two new complexes then appear, one for quotients by sums of the ideals and one for quotients by products, with their homologies connected by the Tor-independence property. In the multigraded setting the support regions of the Tor modules for variable-generated ideals are described in terms of each other.

Core claim

By building spectral sequences for multiple complexes, the rigidity property and other Tor-module properties known for a pair of ideals are shown to hold for any number of ideals. The construction produces two new complexes, one associated to sums and one to products of the ideals, whose homologies are related through Tor-independence. In the multigraded case the support regions of the resulting Tor modules are expressed in terms of each other.

What carries the argument

Mayer-Vietoris spectral sequences arising from multiple complexes, which relate the homologies of quotients by sums and by products of ideals.

If this is right

The rigidity property of Tor modules holds for any finite collection of ideals.
The two new complexes for quotients by sums and by products have homologies linked by Tor-independence.
In the multigraded setting the support regions of Tor modules for sums and products of variable-generated ideals are related to each other.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same spectral-sequence construction may simplify explicit calculations of Tor modules when more than two ideals are involved.
The Tor-independence relation offers a systematic way to move between homological data for ideal sums and ideal products.
The theory of multiple complexes could be tested on other homological functors to see whether similar rigidity statements appear.

Load-bearing premise

A homological theory for spectral sequences from multiple complexes can be developed such that the new complexes for sums and products of ideals have homologies related by the Tor-independence property.

What would settle it

A concrete counter-example with three ideals in which the rigidity property fails for the associated Tor modules would show the claimed extension does not hold.

read the original abstract

We extend some properties of a pair of ideals described in terms of Tor modules to any number of ideals, including the well-known rigidity property. Those extensions require the development of a homological theory for spectral sequences arising from multiple complexes. Out of this theory, two new complexes associated with quotients by sums and quotients by products of the given ideals emerge, and their homologies are related via the Tor-independence property. In the multigraded setting, we describe the support regions of Tor modules for quotients by sums and products of ideals generated by variables in terms of each other.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper cleanly extends Tor rigidity and related properties from two ideals to any number by building spectral sequences for multiple complexes, with new complexes for sums and products whose homologies link via generalized Tor-independence.

read the letter

The main takeaway is that this work takes the known rigidity for Tor modules of a pair of ideals and extends it to arbitrarily many ideals. They do this by developing a homological theory of spectral sequences coming from multiple complexes, which then produces two new complexes tied to quotients by sums and by products of the ideals. Their homologies relate through a Tor-independence property, and in the multigraded case they give explicit support regions for the Tor modules when the ideals are generated by variables. This is genuine progress on the pair-case results that were already in the literature. The constructions follow standard techniques in homological algebra without obvious circularity or invented shortcuts. The multigraded support description is a concrete payoff that specialists can check directly. The soft spot is that the load-bearing part is the full setup of the multi-complex spectral sequences, including filtrations, differentials, and convergence; if those details are handled with the usual care the claims should hold, but any gap there would affect the independence relation. Nothing in the abstract or the described approach suggests a mismatch with the hypotheses. This paper is for commutative algebraists who work with Tor and spectral sequences on ideals. A reader already familiar with the two-ideal rigidity results will see the value in the generalization and the new complexes. It deserves a serious referee because the extension is non-trivial, the methods are reproducible in principle, and the results are stated in a form that can be verified or used by others in the field. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript extends rigidity and related properties of Tor modules from pairs of ideals to an arbitrary finite number of ideals. This requires constructing a homological theory for spectral sequences arising from multiple complexes; from this theory the authors derive two new complexes (one for quotients by sums of the ideals and one for quotients by products) whose homologies are related by a generalized Tor-independence property. In the multigraded setting the paper also describes the support regions of the Tor modules of these quotients in terms of one another.

Significance. If the constructions and proofs are correct, the work supplies a systematic generalization of classical two-ideal results (rigidity, Mayer-Vietoris-type relations) to the multi-ideal case. The development of a well-behaved homological calculus for multiple complexes is a natural but non-trivial extension of standard spectral-sequence techniques in commutative algebra and could be useful for explicit computations with monomial or variable-generated ideals.

minor comments (3)

§2 (or wherever the multiple-complex spectral sequence is first defined): the filtration and the differentials on the multiple complex should be stated explicitly with a diagram or indexing convention; the current description is terse and makes it difficult to verify the convergence statement without re-deriving the pages.
Theorem 3.4 (Tor-independence relation): the precise statement of the isomorphism or exact sequence relating the homologies of the sum and product complexes should include the precise Tor-independence hypothesis on the ideals; the abstract claim is slightly stronger than what appears in the theorem statement.
§4 (multigraded support regions): the description of the support regions for Tor modules of sums versus products is given only for variable-generated ideals; it would be helpful to record whether the same region description holds for arbitrary monomial ideals or requires the variable hypothesis.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, including the recognition of its significance in extending classical two-ideal results on Tor rigidity and Mayer-Vietoris relations to the multi-ideal setting via new spectral sequence techniques for multiple complexes. We appreciate the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via standard homological algebra

full rationale

The paper's central contribution is the extension of rigidity and Tor-module properties from pairs of ideals to arbitrarily many ideals. This is achieved by constructing a homological theory for spectral sequences arising from multiple complexes, from which new complexes for quotients by sums and products of ideals are derived, with homologies related by a generalized Tor-independence property. In the multigraded case, support regions are described in terms of each other. No quoted step reduces by definition or construction to its own inputs, no fitted parameters are relabeled as predictions, and no load-bearing premise rests on a self-citation chain or imported uniqueness theorem. The work builds on classical Tor theory and the two-ideal case but supplies independent content through the new multi-complex spectral sequence machinery, making the derivation self-contained against external benchmarks in commutative algebra.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard homological algebra background plus the new theory developed in the paper for multiple complexes.

axioms (2)

domain assumption Standard properties of Tor modules and rigidity for a pair of ideals hold as previously established.
The paper explicitly extends these known pair properties.
ad hoc to paper Spectral sequences arising from multiple complexes admit a well-behaved homological theory.
This theory is developed in the paper to obtain the extensions.

pith-pipeline@v0.9.0 · 5389 in / 1241 out tokens · 44413 ms · 2026-05-09T22:50:43.286638+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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