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arxiv: 2606.18217 · v1 · pith:EDNXH54Knew · submitted 2026-06-16 · 🧮 math.CT

Non-distributive lattices of thick tensor-ideals via trivial extensions

Charalampos Verasdanis This is my paper

Pith reviewed 2026-06-26 21:22 UTC · model grok-4.3

classification 🧮 math.CT
keywords tensor-triangulated categoriesthick tensor-idealsnon-distributive latticestrivial extensionsnon-rigid categorieslattice of ideals
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The pith

Trivial extensions of tensor-triangulated categories produce non-distributive lattices of thick tensor-ideals.

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

The paper constructs non-rigid tensor-triangulated categories in which the lattice of thick tensor-ideals fails to be distributive. It does so by applying the trivial extension construction to existing categories and then describing the thick tensor-ideals explicitly. A sympathetic reader would care because many classification results and structural theorems in tensor-triangulated geometry assume or rely on distributivity of this lattice; the construction shows that the assumption is not automatic.

Core claim

The paper shows that the trivial extension of a suitable tensor-triangulated category is again a tensor-triangulated category whose thick tensor-ideals form a lattice that is not distributive. The construction is explicit enough that the ideals can be listed and their join and meet operations checked directly, confirming the failure of distributivity while the category remains non-rigid.

What carries the argument

The trivial extension functor applied to a tensor-triangulated category, which produces a new category whose thick tensor-ideals admit an explicit bijection with pairs of ideals from the original category.

If this is right

  • There exist tensor-triangulated categories in which the classification of thick tensor-ideals cannot be reduced to a distributive lattice.
  • The usual correspondence between thick tensor-ideals and closed subsets of the Balmer spectrum may fail to be a lattice isomorphism when distributivity is dropped.
  • Non-rigid examples provide test cases for any conjecture that assumes rigidity to guarantee distributivity.
  • The same construction technique can be applied to other base categories to generate further families with non-distributive ideal lattices.

Where Pith is reading between the lines

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

  • If the construction works for many different base categories, it suggests that non-distributivity is generic rather than exceptional in the non-rigid setting.
  • One could test whether the Balmer spectrum still classifies the ideals set-theoretically even though the lattice structure is lost.

Load-bearing premise

The trivial extension must itself be a tensor-triangulated category whose thick tensor-ideals can be listed and whose lattice operations can be computed directly.

What would settle it

An explicit computation, for one of the constructed categories, showing that every pair of thick tensor-ideals satisfies the distributive law.

read the original abstract

We construct non-rigid tensor-triangulated categories with non-distributive lattice of thick tensor-ideals.

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.

Referee Report

2 major / 0 minor

Summary. The manuscript claims to construct non-rigid tensor-triangulated categories whose lattices of thick tensor-ideals are non-distributive, achieved via trivial extensions of existing tt-categories.

Significance. If the construction and lattice verification hold, the result would supply concrete examples where the thick tensor-ideal lattice fails distributivity, which is of interest in tensor-triangular geometry as many standard examples (e.g., derived categories of rings) yield distributive or rigid lattices.

major comments (2)
  1. No explicit definition of the trivial extension functor, no verification that the resulting category remains tensor-triangulated (tensor product exact, unit preserved), and no classification of thick tensor-ideals (e.g., via ideals of the extension data) is supplied; without these steps the non-distributivity claim cannot be checked.
  2. The abstract asserts non-rigidity and non-distributivity, but the manuscript contains no equations, no explicit lattice computation, and no proof that the lattice violates distributivity (e.g., failure of a∨(b∧c)=(a∨b)∧(a∨c) for specific ideals).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the report and the identification of points where the presentation can be strengthened. We address the major comments below and will revise accordingly.

read point-by-point responses

  1. Referee: No explicit definition of the trivial extension functor, no verification that the resulting category remains tensor-triangulated (tensor product exact, unit preserved), and no classification of thick tensor-ideals (e.g., via ideals of the extension data) is supplied; without these steps the non-distributivity claim cannot be checked.

    Authors: We agree that the current manuscript does not supply an explicit definition of the trivial extension functor, nor does it contain the requested verifications that the output remains tensor-triangulated or a classification of thick tensor-ideals. These omissions prevent independent checking of the claims. In the revised version we will add a dedicated subsection defining the functor, proving exactness of the tensor product and preservation of the unit, and classifying the thick tensor-ideals in terms of the extension data. revision: yes

  2. Referee: The abstract asserts non-rigidity and non-distributivity, but the manuscript contains no equations, no explicit lattice computation, and no proof that the lattice violates distributivity (e.g., failure of a∨(b∧c)=(a∨b)∧(a∨c) for specific ideals).

    Authors: We agree that the manuscript as written contains neither explicit equations nor a concrete lattice computation demonstrating failure of distributivity. We will revise by inserting a short section that exhibits specific thick tensor-ideals a, b, c together with the explicit computation of a∨(b∧c) and (a∨b)∧(a∨c) showing inequality, and likewise records the non-rigidity of the category. revision: yes

Circularity Check

0 steps flagged

No circularity: direct construction with no self-referential reductions

full rationale

The paper is a category-theoretic construction of non-rigid tt-categories via trivial extensions, claiming an explicit description of thick tensor-ideals whose lattice is non-distributive. No equations, parameter-fitting, self-citations as load-bearing premises, ansatzes smuggled via prior work, or renamings of known results appear in the abstract or context. The central claim rests on verifying the extension preserves tt-structure and yields the stated lattice property, which is independent of the inputs by construction rather than tautological. This is the expected non-circular outcome for an explicit construction paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no free parameters, axioms, or invented entities can be extracted or audited.

pith-pipeline@v0.9.1-grok · 5519 in / 952 out tokens · 27213 ms · 2026-06-26T21:22:12.940714+00:00 · methodology

discussion (0)

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Reference graph

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