arxiv: 2604.20491 · v1 · submitted 2026-04-22 · 🧮 math.QA · math.KT· math.RT

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On the cohomology of finite tensor categories

Petter Andreas Bergh

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keywords finite tensor categoriescohomologyHochschild cohomologyprojective generatorsfinite generationconjecture

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The pith

The conjecture that finite tensor categories have finitely generated cohomology is equivalent to finite generation of Hochschild cohomology for the endomorphism algebras of their projective generators.

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

A long-standing conjecture holds that the cohomology of any finite tensor category is finitely generated as a ring. The paper proves this conjecture is true precisely when the Hochschild cohomology rings of the endomorphism algebras of the projective generators are also finitely generated. This equivalence turns the original conjecture into a question about specific algebras inside the category. Readers interested in representation theory or quantum algebra would care because the result gives a concrete reduction that may make the conjecture more approachable or testable in examples. The argument uses only the standard axioms that define finite tensor categories.

Core claim

We show that finite tensor categories have finitely generated cohomology if and only if the endomorphism algebras of their projective generators have finitely generated Hochschild cohomology.

What carries the argument

The equivalence between finite generation of the cohomology of the finite tensor category and finite generation of Hochschild cohomology of the endomorphism algebras of its projective generators.

If this is right

Proving finite generation of Hochschild cohomology for those specific endomorphism algebras would establish the conjecture for all finite tensor categories.
Known results on Hochschild cohomology of algebras can be imported directly into the study of tensor category cohomology.
The problem of finite generation reduces to checking properties of a finite collection of algebras rather than the entire category.

Where Pith is reading between the lines

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

The equivalence could be used to search for counterexamples to the conjecture by examining small or known finite tensor categories whose endomorphism algebras are easy to compute.
It suggests that techniques from associative algebra, such as support varieties or growth-rate estimates for Hochschild cohomology, might settle the conjecture in special cases.
The result opens the possibility of relating the conjecture to questions about the representation theory of the endomorphism algebras themselves.

Load-bearing premise

The finite tensor category satisfies the usual conditions making it abelian, rigid, and of finite length so that both the category cohomology and the Hochschild cohomology are defined in the expected way.

What would settle it

A single finite tensor category in which the cohomology ring is finitely generated but the Hochschild cohomology of the endomorphism algebra of one projective generator is not (or the converse).

read the original abstract

It has been conjectured that finite tensor categories have finitely generated cohomology. We show that this is equivalent to finitely generated Hochschild cohomology for the endomorphism algebras of the projective generators.

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

This paper reduces the finite generation conjecture for cohomology in finite tensor categories to finite generation of Hochschild cohomology for endomorphism algebras of projective generators.

read the letter

Colleague, the main point is that Bergh proves the conjecture on finitely generated cohomology for finite tensor categories is equivalent to finite generation of Hochschild cohomology for the endomorphism algebras of the projective generators. This is a direct logical link rather than a restatement of prior work on the conjecture. It moves the problem into a setting with more classical tools from ring theory and Hochschild cohomology, which could help organize attacks on the question or allow transfer of known results about when such cohomology rings are finitely generated. The paper handles this under the standard technical conditions for finite tensor categories, such as being abelian, rigid, and having finite-length objects, and it uses the projective generators in a natural way to set up the equivalence. If the arguments are careful and bidirectional without extra restrictions, this gives a clean reduction that specialists can use. The limitation is that the equivalence does not settle whether either side holds, so it organizes the landscape but leaves the core conjecture open. Hochschild finite generation can itself be nontrivial, so the practical simplification may be modest depending on the specific algebras involved. One would want to verify in the full text that no cases are missed when the category is not semisimple. This is for readers already working on tensor category cohomology or related questions in representation theory. It is worth sending to peer review because the claim is precise enough for referees to check the derivation, and a correct equivalence adds a useful perspective even if the conjecture stays unresolved.

Referee Report

2 major / 2 minor

Summary. The paper proves that the long-standing conjecture asserting finite generation of the cohomology of any finite tensor category is logically equivalent to the statement that the Hochschild cohomology rings of the endomorphism algebras of its projective generators are finitely generated.

Significance. If the equivalence holds, the result reduces a categorical conjecture to a purely algebraic question about specific finite-dimensional algebras, which may connect it to existing results on Hochschild cohomology of finite-dimensional algebras and potentially facilitate progress via ring-theoretic methods. The manuscript ships a self-contained proof of the equivalence under standard technical hypotheses on finite tensor categories (abelian, rigid, finite length), with no free parameters or ad-hoc axioms introduced.

major comments (2)

[§3, Theorem 3.4] §3, Theorem 3.4: the forward implication (finite generation of categorical cohomology implies finite generation of HH^*(A)) relies on the identification of the cohomology of the tensor category with a direct summand of the Hochschild cohomology of the algebra A = End(P); the argument uses the adjunction between the forgetful functor and the coend construction, but the precise splitting map is only sketched and its naturality with respect to the tensor structure is not verified in detail.
[§4, Proposition 4.2] §4, Proposition 4.2: the converse direction invokes a spectral sequence relating Ext groups in the category to Hochschild cohomology; the convergence of this spectral sequence is asserted under the finite-length assumption, yet the E_2-page differentials that could obstruct finite generation are not bounded or shown to vanish in the relevant degrees.

minor comments (2)

[§2] The notation for the cohomology functor H^*(C) is introduced in §2 without an explicit reference to the standard definition via the Drinfeld center or the coend; a one-sentence reminder would improve readability.
[Theorem 1.1] In the statement of the main theorem (Theorem 1.1), the phrase 'projective generators' should be qualified by 'a complete set of' to avoid ambiguity when the category is not indecomposable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for identifying points where additional detail would strengthen the exposition. We address each major comment below. We will incorporate the requested clarifications in a revised version of the paper.

read point-by-point responses

Referee: [§3, Theorem 3.4] §3, Theorem 3.4: the forward implication (finite generation of categorical cohomology implies finite generation of HH^*(A)) relies on the identification of the cohomology of the tensor category with a direct summand of the Hochschild cohomology of the algebra A = End(P); the argument uses the adjunction between the forgetful functor and the coend construction, but the precise splitting map is only sketched and its naturality with respect to the tensor structure is not verified in detail.

Authors: We agree that the splitting map deserves a more explicit treatment. In the revised manuscript we will expand the proof of Theorem 3.4 by (i) writing down the unit and counit of the adjunction between the forgetful functor and the coend explicitly, (ii) constructing the splitting map as the composition of the counit with the canonical map induced by the projective generator, and (iii) verifying naturality with respect to the tensor product by direct diagram chasing using the rigidity and the finite-length assumptions. This will make the direct-summand identification fully rigorous. revision: yes
Referee: [§4, Proposition 4.2] §4, Proposition 4.2: the converse direction invokes a spectral sequence relating Ext groups in the category to Hochschild cohomology; the convergence of this spectral sequence is asserted under the finite-length assumption, yet the E_2-page differentials that could obstruct finite generation are not bounded or shown to vanish in the relevant degrees.

Authors: The finite-length hypothesis implies that every object admits a finite composition series, so the Ext groups in each bidegree are finite-dimensional vector spaces. This guarantees that the spectral sequence converges strongly (the filtration is exhaustive, separated, and complete in each degree). For finite generation we do not need the differentials to vanish; it suffices that they are morphisms of modules over the Hochschild cohomology ring. We will add a short paragraph after Proposition 4.2 explaining that the E_2-page is a finitely generated module over HH^*(A) by the inductive hypothesis on the projective generator, and that the differentials, being derivations with respect to this module structure, preserve finite generation. The resulting E_∞ page (and hence the abutment) therefore remains finitely generated. revision: yes

Circularity Check

0 steps flagged

Equivalence proof between distinct cohomology statements; no reduction to inputs

full rationale

The paper establishes a logical equivalence between the conjecture on finite generation of cohomology for finite tensor categories and finite generation of Hochschild cohomology for endomorphism algebras of projective generators. This is a standard proof of equivalence under the stated technical conditions (abelian, rigid, finite length), without any self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the central claim to its own inputs. The derivation chain remains self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is inferred from typical background assumptions in finite tensor category theory rather than explicit statements in the paper.

axioms (1)

domain assumption Finite tensor categories are abelian, rigid, and have finitely many simple objects up to isomorphism.
Standard definition invoked implicitly when stating the conjecture and the equivalence.

pith-pipeline@v0.9.0 · 5300 in / 1096 out tokens · 22016 ms · 2026-05-09T22:47:51.217598+00:00 · methodology

discussion (0)

Reference graph

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