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arxiv: 2606.02377 · v1 · pith:YEXRCODGnew · submitted 2026-06-01 · 🧮 math.AG · math.AC

Localizing subcategories for algebraic stacks

Pat Lank This is my paper

Pith reviewed 2026-06-28 12:53 UTC · model grok-4.3

classification 🧮 math.AG math.AC
keywords algebraic stackslocalizing subcategoriesderived categoriesquasi-coherent cohomologytensor localizingdescentsmooth presentationsunderlying topology
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The pith

⊗-localizing subcategories of the derived category on suitable algebraic stacks correspond exactly to subsets of the underlying topological space.

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

The paper proves a descent principle that transfers the classification of tensor-localizing subcategories from schemes to algebraic stacks along smooth presentations. It applies Balmer-Mathew descendability to show that these subcategories of the derived category of quasi-coherent complexes are determined by which points they contain in the stack's topology. A reader cares because this reduces an algebraic question about categories to a purely topological one for a wide class of stacks. The result extends known classifications on schemes by showing the same support-based description holds after descent.

Core claim

Using a descent principle for ⊗-localizing subcategories along smooth presentations and a notion of descendability due to Balmer and Mathew, the ⊗-localizing subcategories of the derived category of complexes with quasi-coherent cohomology on suitable algebraic stacks are classified in terms of subsets of the stack's underlying topology.

What carries the argument

Descent principle for ⊗-localizing subcategories along smooth presentations, relying on Balmer-Mathew descendability to reduce the stack case to the scheme case.

If this is right

  • The classification on stacks reduces directly to the known classification on schemes via the smooth presentation.
  • Every ⊗-localizing subcategory is determined by its support in the topological space of the stack.
  • The result applies whenever the stack satisfies the conditions that make descendability hold along the presentation.
  • Localizing subcategories can now be studied by examining only the topological data of the stack.

Where Pith is reading between the lines

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

  • Similar descent arguments might classify localizing subcategories in other tensor-triangulated categories built from stacks.
  • The approach could be tested by verifying the bijection explicitly on concrete stacks such as weighted projective stacks.
  • It connects the problem to existing support classifications in algebraic geometry without requiring new invariants.

Load-bearing premise

The stacks must admit smooth presentations to which Balmer-Mathew descendability applies and must satisfy the suitability conditions such as quasi-compactness with affine diagonal.

What would settle it

A concrete counterexample would be a suitable algebraic stack together with a ⊗-localizing subcategory whose support is not a subset of the underlying topology or for which the descent map fails to be bijective.

read the original abstract

We establish a descent principle for $\otimes$-localizing subcategories along smooth presentations using a notion of descendability due to Balmer and Mathew. This allows us to classify $\otimes$-localizing subcategories of the derived category of complexes with quasi-coherent cohomology on suitable algebraic stacks in terms of subsets of its underlying topology.

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

0 major / 2 minor

Summary. The manuscript establishes a descent principle for ⊗-localizing subcategories along smooth presentations of algebraic stacks, employing the descendability notion introduced by Balmer and Mathew. This principle is then used to classify the ⊗-localizing subcategories of the derived category of complexes with quasi-coherent cohomology on suitable algebraic stacks in terms of subsets of the underlying topology.

Significance. If the result holds, the work extends tensor-triangular classification theorems from schemes to algebraic stacks via a standard descent argument. This provides a topological description of localizing subcategories that may facilitate further study of derived categories on stacks. The approach builds directly on prior results without introducing new free parameters or ad-hoc axioms.

minor comments (2)
  1. [Abstract] The abstract and introduction should explicitly list the precise conditions that make a stack 'suitable' for the classification (e.g., quasi-compactness, affine diagonal, or the existence of a smooth presentation to which Balmer-Mathew descendability applies), as these conditions are load-bearing for the descent step.
  2. Notation for the derived category (e.g., D_qc(X) or similar) and the precise meaning of 'underlying topology' should be fixed consistently from the first appearance onward.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript, including the recommendation for minor revision. The report contains no major comments requiring point-by-point response.

Circularity Check

0 steps flagged

No circularity: classification extends external Balmer-Mathew descendability

full rationale

The paper invokes Balmer-Mathew descendability as an external input to establish a descent principle for ⊗-localizing subcategories along smooth presentations of suitable algebraic stacks. The resulting classification in terms of subsets of the underlying topology follows directly from this application without any self-definitional reduction, fitted parameters renamed as predictions, or load-bearing self-citations. The suitability conditions on the stacks are stated explicitly to enable the external descent, and no equations or steps in the derivation chain collapse to the paper's own inputs by construction. This is a standard extension of prior tensor-triangular geometry results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard background in derived categories and tensor-triangular geometry; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard properties of derived categories, tensor products, and localizing subcategories in algebraic geometry hold.
    Invoked implicitly to set up the classification and descent.

pith-pipeline@v0.9.1-grok · 5556 in / 1095 out tokens · 23142 ms · 2026-06-28T12:53:42.360165+00:00 · methodology

discussion (0)

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Reference graph

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