arxiv: 2605.10733 · v1 · submitted 2026-05-11 · 🧮 math.KT

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Equivariant Hochschild cohomology of group algebras and relative operatorname{Ext}

Andrada Pojar , Constantin-Cosmin Todea

Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:52 UTC · model grok-4.3

classification 🧮 math.KT

keywords equivariant Hochschild cohomologygroup algebrasrelative Extrelative homological algebrafinite group actionscohomology of algebrasstabilizer subgroups

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The pith

The Γ-equivariant Hochschild cohomology of Γ-algebras equals a kΓ-relative Ext group for any field k.

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

The paper establishes that Γ-equivariant Hochschild cohomology of Γ-algebras, taken with coefficients in a Γ-equivariant bimodule, is isomorphic to a relative Ext group over the group algebra kΓ. This uses the setting of relative homological algebra and holds without any restriction on the characteristic of k. Separately, the authors derive necessary conditions for the first Γ0-equivariant Hochschild cohomology of a group algebra kG to be non-trivial, where Γ0 is the stabilizer of an element of G and the characteristic p of k divides the order of G. A reader would care because the isomorphism supplies a direct bridge between two distinct computational approaches to extensions and deformations when a finite group acts on an algebra.

Core claim

For any field k the Γ-equivariant Hochschild cohomology of Γ-algebras with coefficients in a Γ-equivariant bimodule is isomorphic with some kΓ-relative Ext, in the context of relative homological algebra. In addition, necessary conditions are given under which the first Γ0-equivariant Hochschild cohomology of the group algebra kG is non-trivial when the characteristic of k divides the order of G.

What carries the argument

The isomorphism between Γ-equivariant Hochschild cohomology and kΓ-relative Ext groups supplied by relative homological algebra.

If this is right

Equivariant Hochschild cohomology computations reduce to calculations in relative Ext.
The first Γ0-equivariant Hochschild cohomology of kG is non-trivial precisely when the stated divisibility and stabilizer conditions hold.
The isomorphism applies uniformly for every field k, including those whose characteristic does not divide |G|.

Where Pith is reading between the lines

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

The identification may let techniques from relative group cohomology be imported into the study of equivariant deformations of group algebras.
Explicit verification on small examples such as cyclic groups and their automorphism actions would give a direct check of the isomorphism.
Similar isomorphisms could be sought when the acting group is infinite or when the algebra is not a group algebra.

Load-bearing premise

The groups Γ and G are finite, Γ acts on G, and for the non-triviality statement the characteristic of k divides the order of G.

What would settle it

An explicit finite group Γ acting on a finite group G together with a field k of characteristic dividing |G| where the equivariant Hochschild cohomology differs from the corresponding relative Ext would disprove the isomorphism.

read the original abstract

For a finite group $\Gamma$, acting on a finite group $G,$ we find necessary conditions for which the first $\Gamma_0$-equivariant Hochschild cohomology of the group algebra $kG$ is non-trivial, where $k$ is a field of characteristic $p$ dividing the order of $G$ and $\Gamma_0$ is the stabilizer subgroup in $\Gamma$ of some element in $G.$ For any field $k$ we show that the $\Gamma$-equivariant Hochschild cohomology of $\Gamma$-algebras with coefficients in a $\Gamma$-equivariant bimodule (Jensen, 1996) is isomorphic with some $k\Gamma$-relative $\operatorname{Ext},$ in the context of relative homological algebra.

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 gives an isomorphism from Γ-equivariant Hochschild cohomology to kΓ-relative Ext and necessary conditions for non-triviality on kG when char divides |G|.

read the letter

The punchline is that the paper shows the Γ-equivariant Hochschild cohomology of Γ-algebras with equivariant coefficients is isomorphic to a kΓ-relative Ext, and it gives necessary conditions for the first cohomology of kG to be non-trivial when the characteristic divides the group order and Γ0 is a stabilizer. This extends Jensen's 1996 result to the equivariant case in a direct way. The isomorphism uses relative homological algebra to connect the two, which is a standard but useful move. The necessary conditions for non-triviality are the fresher contribution, as they specify when the cohomology does not vanish for these group algebras under the given finite group action and characteristic assumption. That part feels concrete and tied to the stabilizer. The main soft spot is the general isomorphism. It assumes or needs to prove that the equivariant Hochschild complex is relative projective in the appropriate category, or that a comparison theorem applies without further restrictions on the action or the algebra. If the proofs do not verify this step explicitly, the identification could require extra hypotheses. The non-triviality conditions seem less affected by this and stand more independently. Readers working in equivariant homological algebra or the K-theory of group algebras will find this relevant. It is a specialized piece that organizes tools rather than opening broad new directions, so it suits experts who already know the Jensen framework and want to see it adapted to equivariant bimodules. A general algebraist might skip it. I would send this to peer review. The claims are specific enough to be checked, and the extension plus conditions add something to the literature even if the relative projectivity needs careful confirmation in the text.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes two results. For finite groups Γ acting on finite G, with field k of characteristic p dividing |G|, it derives necessary conditions under which the first Γ₀-equivariant Hochschild cohomology of the group algebra kG (with Γ₀ the stabilizer of an element of G) is non-trivial. Separately, for arbitrary field k, it proves that the Γ-equivariant Hochschild cohomology of a Γ-algebra with coefficients in a Γ-equivariant bimodule is isomorphic to a kΓ-relative Ext group in the sense of relative homological algebra (citing Jensen 1996).

Significance. If the isomorphism holds as stated, it supplies a direct translation between equivariant Hochschild cohomology and relative Ext, which may facilitate computations via the machinery of relative homological algebra. The non-triviality conditions are concrete and could be tested on explicit group actions, potentially informing work on deformations or cohomology of group algebras. The paper works within standard definitions and does not introduce free parameters or ad-hoc axioms.

minor comments (3)

The abstract and introduction should explicitly state the precise category in which the relative projectives are taken when asserting the isomorphism to kΓ-relative Ext; this would clarify that the standard equivariant bar resolution is indeed relative projective without extra hypotheses on the action.
In the section deriving the necessary conditions for non-triviality, include a brief remark on whether the conditions are also sufficient or provide a counter-example where they hold but the cohomology vanishes.
Notation for the stabilizer Γ₀ and the bimodule coefficients should be introduced once and used consistently; a small table summarizing the hypotheses (finiteness of Γ and G, characteristic p, etc.) would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the two main results, and recommendation for minor revision. The referee correctly identifies the necessary conditions for non-trivial first Γ₀-equivariant Hochschild cohomology and the isomorphism with kΓ-relative Ext. No specific major comments or criticisms were raised.

Circularity Check

0 steps flagged

No significant circularity; isomorphism is a derived theorem

full rationale

The paper's central claim is a theorem establishing an isomorphism between the Γ-equivariant Hochschild cohomology (defined via the external reference to Jensen 1996) of Γ-algebras with coefficients in a Γ-equivariant bimodule and a kΓ-relative Ext group in relative homological algebra. This identification is presented as a result to be proven rather than a definitional equivalence or fitted prediction. The secondary result on necessary conditions for non-triviality of the first cohomology group for kG (under finiteness, characteristic p dividing |G|, and stabilizer Γ₀) is obtained by direct computation from the given group action and module assumptions, without reducing to self-referential inputs or self-citations. No load-bearing steps invoke self-citations, ansatzes smuggled via prior work, or renaming of known results; the derivation relies on standard comparison theorems in relative homological algebra applied to the equivariant bar resolution. The paper is self-contained against external benchmarks for its claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies entirely on standard definitions and properties of group algebras, Hochschild cohomology, and relative homological algebra; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)

standard math Standard properties of finite group algebras over a field and the definition of equivariant Hochschild cohomology from Jensen 1996
The claims presuppose the usual functorial properties of group algebras and the existence of the equivariant cohomology theory as previously defined.

pith-pipeline@v0.9.0 · 5428 in / 1261 out tokens · 42806 ms · 2026-05-12T04:52:32.301205+00:00 · methodology

discussion (0)

Reference graph

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