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arxiv: 2606.09398 · v1 · pith:VLEAEDASnew · submitted 2026-06-08 · 🧮 math.CT · math.AC· math.AG

Colocalizing subcategories on differentially graded algebras

Leovigildo Alonso , Ana Jerem\'ias , Eduardo Loureiro This is my paper

Pith reviewed 2026-06-27 14:06 UTC · model grok-4.3

classification 🧮 math.CT math.ACmath.AG
keywords differential graded algebrasderived categorieslocalizing subcategoriescolocalizing subcategoriesspectrum of a ringNeeman theorems
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The pith

If the derived category of a bounded non-positive commutative DG-algebra is generated by residue field modules, then localizing and colocalizing subcategories biject with subsets of Spec(H^0(A)).

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

The paper shows that for a bounded non-positive commutative differential graded algebra A, the generation of its derived category D(A) by the DG-modules of residue fields of the ring H^0(A) implies that localizing subcategories of D(A) correspond one-to-one with subsets of the prime spectrum of H^0(A), and the same holds for colocalizing subcategories. This classification uses the spectrum data to label the subcategories in a way that mirrors geometric support. A sympathetic reader cares because the result recovers the known classification for ordinary Noetherian rings as a special case while extending it to DG-algebras that satisfy the generation hypothesis.

Core claim

If D(A) is generated by the DG-modules corresponding to the residue fields of the ordinary ring H^0(A) then its localizing subcategories and its colocalizing subcategories are in bijection with the subsets of Spec(H^0(A)).

What carries the argument

The generation condition on D(A) by residue field DG-modules, which carries the bijection to spectrum subsets.

Load-bearing premise

That D(A) is generated by the DG-modules corresponding to the residue fields of H^0(A).

What would settle it

A bounded non-positive commutative DG-algebra A where D(A) is generated by the residue field modules yet the correspondence between localizing subcategories and subsets of Spec(H^0(A)) fails to be bijective.

read the original abstract

Let $A$ be a bounded non positive commutative differential graded algebra $A$. Let $\mathbf{D}(A)$ its derived category of DG-modules. If $\mathbf{D}(A)$ is generated by the DG-modules corresponding to the residue fields of the ordinary ring $H^0(A)$ then its localizing subcategories and its colocalizing subcategories are in bijection with the subsets of $\textrm{Spec}(H^0(A))$. These results generalize well-known theorems by A. Neeman (from 1992 and 2011, respectively), because any Noetherian ring satisfies this condition.

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

1 major / 0 minor

Summary. The paper claims that if A is a bounded non-positive commutative differential graded algebra such that D(A) is generated by the DG-modules corresponding to the residue fields of the ordinary ring H^0(A), then the localizing subcategories and colocalizing subcategories of D(A) are in bijection with the subsets of Spec(H^0(A)). This is presented as a generalization of Neeman's theorems from 1992 and 2011, which are recovered when A is an ordinary Noetherian ring.

Significance. If the result holds, it extends the classification of localizing and colocalizing subcategories from ordinary rings to the DG-algebra setting under an explicit generation hypothesis that is known to hold for Noetherian rings, providing a uniform statement for both types of subcategories.

major comments (1)
  1. The central claim is conditional on the generation hypothesis stated in the abstract, but the full derivation of the bijection (including any use of the bounded non-positive hypothesis on A) cannot be verified because the proof is unavailable in the provided manuscript text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We address the major comment below.

read point-by-point responses

  1. Referee: The central claim is conditional on the generation hypothesis stated in the abstract, but the full derivation of the bijection (including any use of the bounded non-positive hypothesis on A) cannot be verified because the proof is unavailable in the provided manuscript text.

    Authors: We agree that the version provided to the referee lacked the full proof. The revised manuscript will include the complete derivation of the bijection (for both localizing and colocalizing subcategories), explicitly using the generation hypothesis together with the bounded non-positive hypothesis on A (via the relevant lemmas establishing properties of residue fields and the action on Spec(H^0(A))). revision: yes

Circularity Check

0 steps flagged

No significant circularity; result is a conditional generalization of independent prior theorems

full rationale

The paper states a theorem conditional on the explicit assumption that D(A) is generated by residue-field DG-modules over H^0(A), notes that this assumption holds for any Noetherian ring (recovering Neeman's 1992/2011 results), and frames the bijections with subsets of Spec(H^0(A)) as a direct generalization. No self-definitional steps, fitted quantities renamed as predictions, load-bearing self-citations, or imported uniqueness theorems appear in the stated claim or its relation to the classical results. The derivation chain is self-contained against the external benchmark of Neeman's theorems and the stated generation hypothesis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract; the paper relies on standard axioms of triangulated categories and derived categories of DG-modules. No free parameters or invented entities are indicated.

axioms (1)
  • standard math The derived category D(A) of DG-modules over a commutative DG-algebra carries the standard triangulated structure with coproducts and products.
    Invoked implicitly as background for the statements about localizing and colocalizing subcategories.

pith-pipeline@v0.9.1-grok · 5632 in / 1161 out tokens · 32052 ms · 2026-06-27T14:06:41.899385+00:00 · methodology

discussion (0)

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Reference graph

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