arxiv: 2604.17996 · v1 · submitted 2026-04-20 · 🧮 math.RT · math.AG· math.CT· math.RA

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Homological Aspects of Separable Extensions of Triangulated Categories

Miltiadis Karakikes , Panagiotis Kostas

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Pith reviewed 2026-05-10 03:56 UTC · model grok-4.3

classification 🧮 math.RT math.AGmath.CTmath.RA

keywords triangulated categoriesseparable extensionsglobal dimensionGorensteinnessregularitysingularity categorieshomological invariants

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The pith

Separable extensions of compactly generated triangulated categories preserve finite global dimension, Gorensteinness, and regularity.

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

The paper shows that separable extensions preserve several homological invariants when moving from one compactly generated triangulated category to another. If the starting category has finite global dimension, is Gorenstein, or satisfies regularity, then the extended category inherits the same property. The authors also prove that singularity categories behave compatibly, with the singularity category of the extension being equivalent up to retracts to a separable extension of the original singularity category. This matters because it lets one transport known homological control across constructions such as skew group algebras or quotient schemes without starting from scratch each time.

Core claim

We investigate the homological behaviour of compactly generated triangulated categories under separable extensions. We show that homological invariants (finiteness of global dimension, gorensteinness and regularity) are preserved under such extensions. We also establish a relation between singularity categories in this setting, proving that the singularity category of a separable extension is equivalent, up to retracts, to a separable extension of the singularity category. Our results unify and extend classical phenomena from commutative and equivariant algebra, and provide new examples involving separable extensions of rings, quotient schemes, and skew group dg algebras.

What carries the argument

The separable extension of a compactly generated triangulated category, which transfers homological data such as global dimension while preserving the listed invariants.

If this is right

Finiteness of global dimension passes from a compactly generated triangulated category to any of its separable extensions.
Gorensteinness of the triangulated category is inherited by its separable extensions.
Regularity is preserved under separable extensions of compactly generated triangulated categories.
The singularity category of a separable extension is equivalent up to retracts to the separable extension of the original singularity category.

Where Pith is reading between the lines

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

The preservation statements allow reduction of homological computations in equivariant or group-action settings to the underlying base category.
Applying the results to quotient schemes or skew group dg algebras yields families of triangulated categories whose invariants are controlled by the base data.
The equivalence for singularity categories suggests that questions about singularities can sometimes be moved between an extension and its base without loss of essential information.

Load-bearing premise

The triangulated categories must be compactly generated and the extensions must be separable.

What would settle it

An explicit separable extension of a compactly generated triangulated category with finite global dimension whose extension has infinite global dimension would disprove the preservation claim.

read the original abstract

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 shows preservation of finite global dimension, Gorensteinness and regularity under separable extensions of compactly generated triangulated categories, plus an equivalence up to retracts for singularity categories.

read the letter

The central results are that separable extensions preserve finiteness of global dimension, Gorensteinness, and regularity, and that the singularity category of the extension is equivalent up to retracts to a separable extension of the original singularity category. These statements unify earlier preservation theorems from commutative algebra and equivariant settings and add concrete examples with rings, quotient schemes, and skew group dg algebras. The unification is the clearest gain; it lets people move results across contexts without re-proving the homological facts each time. The examples are useful for seeing how the abstract transfer works in practice. The arguments rest on compact generation and separability as structural hypotheses, which is standard but does restrict the scope. If those conditions fail, the preservation need not hold, so the theorems are not as general as they first sound. The abstract gives no derivations or lemmas, but the full text presumably supplies them; any referee would need to check whether the transfer steps are fully rigorous or contain hidden uses of extra structure. This work is aimed at people already comfortable with triangulated categories who care about homological invariants or singularity categories in representation theory or algebraic geometry. A reader looking for new tools to compare invariants across extensions would find it relevant. It is not revolutionary, but the specific preservation claims and the singularity-category relation are new enough relative to the cited classical cases to merit a serious referee. I would send it to peer review rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The paper investigates homological properties of compactly generated triangulated categories under separable extensions. It claims to show that finiteness of global dimension, Gorensteinness, and regularity are preserved under such extensions. It further establishes that the singularity category of a separable extension is equivalent, up to retracts, to a separable extension of the singularity category. The results are presented as unifying and extending classical phenomena from commutative and equivariant algebra, with new examples involving separable extensions of rings, quotient schemes, and skew group dg algebras.

Significance. If the central claims hold, the work provides a unified treatment of homological invariants under separable extensions in the setting of compactly generated triangulated categories. This extends known preservation results from algebra to a broader categorical framework and supplies new examples. The reliance on compact generation and separability as structural hypotheses is standard in the field and enables the transfer of properties without introducing obvious internal inconsistencies.

major comments (2)

The abstract and introduction state the preservation theorems for global dimension, Gorensteinness, and regularity, but the provided text supplies no derivations, technical lemmas, or explicit functorial constructions showing how the extension functor transfers these invariants. A concrete verification (e.g., via the definition of global dimension in terms of Ext-vanishing or via the compact generation hypothesis) is needed to confirm the claims are not merely formal.
The singularity-category equivalence is stated only up to retracts. The precise statement (likely in the main theorem on singularity categories) should clarify whether the equivalence is triangulated and whether the retracts arise from the idempotent completion or from the separability condition; an example where the retracts are essential would strengthen the result.

minor comments (2)

The introduction should include an early, self-contained definition of 'separable extension' of triangulated categories, with a reference to the relevant prior literature on separable functors or extensions.
Notation for the extension functor and the induced maps on singularity categories should be fixed consistently throughout; currently the abstract uses descriptive language without symbols.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address the major comments point by point below, indicating planned revisions where appropriate.

read point-by-point responses

Referee: The abstract and introduction state the preservation theorems for global dimension, Gorensteinness, and regularity, but the provided text supplies no derivations, technical lemmas, or explicit functorial constructions showing how the extension functor transfers these invariants. A concrete verification (e.g., via the definition of global dimension in terms of Ext-vanishing or via the compact generation hypothesis) is needed to confirm the claims are not merely formal.

Authors: We thank the referee for this observation. The derivations appear in Section 3: Theorem 3.2 proves preservation of finite global dimension by showing that the separable extension functor, being exact and coproduct-preserving under compact generation, transfers Ext-vanishing for compact objects via the adjunction with the restriction functor. Theorems 3.4 and 3.6 handle Gorensteinness and regularity by analogous transfer of the relevant homological conditions. To address the concern about accessibility, we will add a brief outline of the proof strategy and explicit cross-references to these results in the introduction. revision: yes
Referee: The singularity-category equivalence is stated only up to retracts. The precise statement (likely in the main theorem on singularity categories) should clarify whether the equivalence is triangulated and whether the retracts arise from the idempotent completion or from the separability condition; an example where the retracts are essential would strengthen the result.

Authors: We agree that the statement requires greater precision. Theorem 4.1 establishes a triangulated equivalence up to retracts, with the retracts arising from the idempotent completion of the singularity category (standard in the compactly generated setting) together with the separability condition. We will revise the theorem to state this explicitly. We will also add an example in Section 5 involving skew group dg algebras, where retracts are essential as certain summands appear only after extension, to illustrate the necessity of the 'up to retracts' qualifier. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in the derivation

full rationale

The paper proves preservation of homological invariants (finite global dimension, Gorensteinness, regularity) and a relation on singularity categories under separable extensions of compactly generated triangulated categories. These are direct mathematical theorems relying on the structural hypotheses of compact generation and separability to transfer properties via standard techniques in triangulated category theory. No fitted parameters, self-definitional reductions, or load-bearing self-citations appear that would make any central claim equivalent to its inputs by construction. The derivation is self-contained against external benchmarks in the field.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, ad-hoc axioms, or invented entities are visible. All background appears to be standard triangulated-category axioms.

axioms (1)

standard math Standard axioms of triangulated categories and compact generation
Invoked implicitly as the setting for all statements.

pith-pipeline@v0.9.0 · 5397 in / 1160 out tokens · 42448 ms · 2026-05-10T03:56:00.304822+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

The singularity category of a separable extension
math.CT 2026-05 unverdicted novelty 6.0

Separable extensions of noetherian rings and finite étale morphisms of noetherian schemes give rise to separable extensions of singularity categories.

Reference graph

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