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arxiv: 2605.10733 · v2 · pith:APTUIGBOnew · submitted 2026-05-11 · 🧮 math.KT

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

Andrada Pojar , Constantin-Cosmin Todea This is my paper

Pith reviewed 2026-05-21 08:59 UTC · model grok-4.3

classification 🧮 math.KT
keywords equivariant Hochschild cohomologyrelative Extgroup algebrasfinite groupsgroup actionsrelative homological algebrastabilizer subgroups
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The pith

The Γ-equivariant Hochschild cohomology of Γ-algebras equals a kΓ-relative Ext group.

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

The paper proves that for any field k the Γ-equivariant Hochschild cohomology of a Γ-algebra with coefficients in a Γ-equivariant bimodule is isomorphic to a relative Ext group computed in the relative homological algebra of kΓ-modules. It further gives necessary conditions for the first such cohomology of the group algebra kG to be non-trivial when Γ acts on the finite group G, the characteristic of k divides the order of G, and Γ₀ is the stabilizer of a chosen element of G. A reader cares because the isomorphism replaces a direct cocycle definition with the standard machinery of relative projective resolutions and Ext functors. The statements apply to finite groups with a fixed action throughout.

Core claim

For any field k we show that 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. For a finite group Γ acting on a finite group G we find necessary conditions under which the first Γ₀-equivariant Hochschild cohomology of kG is non-trivial when the characteristic p of k divides the order of G and Γ₀ is the stabilizer of an element of G.

What carries the argument

The isomorphism identifying Γ-equivariant Hochschild cohomology with kΓ-relative Ext in the relative homological algebra of Jensen's equivariant bimodules.

If this is right

  • Equivariant Hochschild cohomology groups can be computed from relative projective resolutions over the group algebra kΓ.
  • The isomorphism holds independently of the characteristic of k.
  • Non-vanishing of the first cohomology is controlled by the existence of suitable stabilizers under the given action.

Where Pith is reading between the lines

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

  • The same isomorphism may supply new ways to calculate invariants in algebraic K-theory of group rings when an external group action is present.
  • Concrete computations for cyclic or symmetric groups could test the boundary between the necessary conditions and actual vanishing.
  • If the finiteness hypotheses are dropped, the relative Ext side might still make sense and suggest a definition of equivariant cohomology for infinite groups.

Load-bearing premise

The groups Γ and G must be finite, an action of Γ on G must be given, and for the non-triviality result the characteristic of k must divide the order of G.

What would settle it

An explicit finite group Γ acting on G together with a field k of characteristic dividing |G| and a stabilizer Γ₀ for which the first Γ₀-equivariant Hochschild cohomology of kG vanishes would falsify the necessary conditions.

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.

Referee Report

1 major / 1 minor

Summary. The manuscript establishes necessary conditions under which the first Γ₀-equivariant Hochschild cohomology of the group algebra kG is non-trivial, where Γ is finite acting on finite G, k has characteristic p dividing |G|, and Γ₀ is the stabilizer of an element of G. It further proves that, for any field k, the Γ-equivariant Hochschild cohomology of a Γ-algebra with coefficients in a Γ-equivariant bimodule (in the sense of Jensen 1996) is isomorphic to a kΓ-relative Ext group computed in the framework of relative homological algebra.

Significance. If the isomorphism is established rigorously, the result supplies a concrete link between equivariant Hochschild cohomology and the machinery of relative homological algebra, potentially simplifying computations of these groups via relative projective resolutions. The non-triviality criteria add explicit vanishing/non-vanishing information for equivariant cohomology of group algebras under the given characteristic and finiteness hypotheses.

major comments (1)
  1. [Proof of the isomorphism (likely the section following the setup of equivariant bimodules)] The central isomorphism between Γ-equivariant Hochschild cohomology and kΓ-relative Ext (the second main result) requires an explicit verification that the derived functors coincide. In particular, the relative projective class in the category of Γ-equivariant bimodules must be identified (e.g., modules projective after restriction to kΓ or after forgetting the Γ-action), and it must be shown that the bar resolution or other standard resolutions used for the equivariant Hochschild cohomology are relative-projective with respect to this class. The abstract states the result but provides no such identification or check; without it the isomorphism of the two cohomology theories is not yet secured.
minor comments (1)
  1. [Abstract] The phrase 'some kΓ-relative Ext' in the abstract is imprecise; the precise functor or degree should be named.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for greater explicitness in the proof of the isomorphism between Γ-equivariant Hochschild cohomology and kΓ-relative Ext. We address the major comment below and will revise the manuscript to strengthen this part of the argument.

read point-by-point responses

  1. Referee: [Proof of the isomorphism (likely the section following the setup of equivariant bimodules)] The central isomorphism between Γ-equivariant Hochschild cohomology and kΓ-relative Ext (the second main result) requires an explicit verification that the derived functors coincide. In particular, the relative projective class in the category of Γ-equivariant bimodules must be identified (e.g., modules projective after restriction to kΓ or after forgetting the Γ-action), and it must be shown that the bar resolution or other standard resolutions used for the equivariant Hochschild cohomology are relative-projective with respect to this class. The abstract states the result but provides no such identification or check; without it the isomorphism of the two cohomology theories is not yet secured.

    Authors: We agree that an explicit identification of the relative projective class and a direct verification that the bar resolution is relative-projective would make the argument more transparent. In the section on equivariant bimodules we work throughout in the category of Γ-equivariant kG-bimodules (in the sense of Jensen) and define the equivariant Hochschild cohomology via the standard bar resolution. We will add a dedicated paragraph (or short subsection) that (i) identifies the relative projective class as those Γ-equivariant bimodules that become projective upon restriction to kΓ (after forgetting the G-action where appropriate), and (ii) checks that the bar resolution splits in the required way when restricted to this class, thereby confirming that it is relative-projective. This explicit check will ensure the derived functors coincide and the isomorphism is rigorously established. We will incorporate these clarifications in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity: isomorphism derived from external definitions and relative homological algebra

full rationale

The central result is an explicit isomorphism between Γ-equivariant Hochschild cohomology (defined via the external Jensen 1996 reference for Γ-equivariant bimodules) and a kΓ-relative Ext functor. The abstract invokes standard relative homological algebra without redefining the relative projective class in terms of the target cohomology or fitting parameters to the output. No self-citation chain, ansatz smuggling, or self-definitional reduction appears; the derivation remains self-contained against the cited external framework and the stated assumptions on finite groups and field characteristic.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rest on standard domain assumptions from finite group theory and homological algebra; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Γ and G are finite groups with Γ acting on G
    Explicitly stated at the opening of the abstract to frame both results.
  • domain assumption k is a field whose characteristic p divides |G| for the non-triviality claim
    Required for the first part of the abstract concerning non-trivial cohomology.

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Works this paper leans on

18 extracted references · 18 canonical work pages

  1. [1]

    Jensen, K.K.: Foundations of an Equivariant Cohomology Theory of Banach Alge- bras, I. Adv. in Math.117, 52–146 (1996) https://doi.org/10.1006/aima.1996. 0003

    work page doi:10.1006/aima.1996 1996

  2. [2]

    On non-abelian extensions of Leib- niz algebras

    Koam, A.N.A., Pirashvili, T.: Cohomology of oriented algebras. Communica- tions in Algebra46(7), 2947–2963 (2018) https://doi.org/10.1080/00927872.2017. 1404089 15

    work page doi:10.1080/00927872.2017 2018

  3. [3]

    European Journal of Mathematics8(Suppl 2), 720–763 (2022) https://doi.org/10.1007/ s40879-022-00558-0

    Inassaridze, H.: (Co)homology of Γ-groups and Γ-homological algebra. European Journal of Mathematics8(Suppl 2), 720–763 (2022) https://doi.org/10.1007/ s40879-022-00558-0

    work page 2022

  4. [4]

    Homology, Homotopy and Applications22(1), 27–54 (2020) https: //doi.org/10.4310/HHA.2020.v22.n1.a3

    Mueller, L., Wike, L.: Equivariant higher Hochschild homology and topological field theories. Homology, Homotopy and Applications22(1), 27–54 (2020) https: //doi.org/10.4310/HHA.2020.v22.n1.a3

    work page doi:10.4310/hha.2020.v22.n1.a3 2020

  5. [5]

    Homology, Homotopy and Applications4(1), 1–23 (2002) https://doi

    Cegarra, A.M., Garc´ ıa-Calcinez, J.M., Ortega, J.A.: Cohomology of groups with operators. Homology, Homotopy and Applications4(1), 1–23 (2002) https://doi. org/10.4310/HHA.2002.v4.n1.a1

    work page doi:10.4310/hha.2002.v4.n1.a1 2002

  6. [6]

    Topology and its Applications153, 66–89 (2005) https://doi.org/10.1016/j.topol.2004.12.005

    Inassaridze, H.: Equivariant homology and cohomology of groups. Topology and its Applications153, 66–89 (2005) https://doi.org/10.1016/j.topol.2004.12.005

    work page doi:10.1016/j.topol.2004.12.005 2005

  7. [7]

    Journal of Algebra and Its Applications19(6), 2050114 (2020) https://doi.org/10.1142/S0219498820501145

    Mukherjee, G., Yadav, R.B.: Equivariant one-parameter deformations of asso- ciative algebras. Journal of Algebra and Its Applications19(6), 2050114 (2020) https://doi.org/10.1142/S0219498820501145 . 18 pages

    work page doi:10.1142/s0219498820501145 2020

  8. [8]

    Volume No.10 of Interscience Tracts in Pure and Applied Mathematics

    Jacobson, N.: Lie Algebras. Volume No.10 of Interscience Tracts in Pure and Applied Mathematics. Interscience Publisers, New York-London (1962)

    work page 1962

  9. [9]

    Aryutunov, A.A., Mischenko, A.S.: Smooth version for Johnson’s problem on derivations of group algebras. Mat. Sb.210 (6), 3–29 (2019) https://doi.org/10. 1070/SM9119

    work page 2019

  10. [10]

    Heat kernels on weighted manifolds and applications

    Linckelmann, M.: Finite dimensional algebras arising as blocks of finite group algebras. Contemporary Mathematics705(2018) https://doi.org/10.1090/conm/ 705/14202

    work page doi:10.1090/conm/ 2018

  11. [11]

    arXiv:2601.09602 [math.RT] (2026) https://doi.org/10.48550/arXiv.2601.09602

    Briggs, B., Rubio y Degrassi, L.: Outer derivations on blocks of group algebras. arXiv:2601.09602 [math.RT] (2026) https://doi.org/10.48550/arXiv.2601.09602

    work page doi:10.48550/arxiv.2601.09602 2026

  12. [12]

    Commentarii Mathematici Helvetici60, 354–365 (1985) https://doi.org/10.1007/BF02567420

    Burghelea, D.: The cyclic homology of the group rings. Commentarii Mathematici Helvetici60, 354–365 (1985) https://doi.org/10.1007/BF02567420

    work page doi:10.1007/bf02567420 1985

  13. [13]

    Transactions of the Amer- ican Mathematical Society82(1), 246–269 (1956) https://doi.org/10.1090/ S0002-9947-1956-0080654-0

    Hochschild, G.: Relative homological algebra. Transactions of the Amer- ican Mathematical Society82(1), 246–269 (1956) https://doi.org/10.1090/ S0002-9947-1956-0080654-0

    work page 1956

  14. [14]

    Journal of Pure and Applied Algebra43, 53–74 (1986) https: //doi.org/10.1016/0022-4049(86)90004-6

    Gerstenhaber, M., Schack, S.D.: Relative Hochschild cohomology, rigid algebras and the Bockstein. Journal of Pure and Applied Algebra43, 53–74 (1986) https: //doi.org/10.1016/0022-4049(86)90004-6

    work page doi:10.1016/0022-4049(86)90004-6 1986

  15. [15]

    Journal of Noncommutative Geometry10, 811–858 (2016) https://doi.org/10.4171/JNCG/249 16

    Liu, Y., Zhou, G.: The Batalin–Vilkovisky structure over the Hochschild cohomol- ogy ring of a group algebra. Journal of Noncommutative Geometry10, 811–858 (2016) https://doi.org/10.4171/JNCG/249 16

    work page doi:10.4171/jncg/249 2016

  16. [16]

    Homology, Homotopy and Applications24(1), 93–115 (2022) https:// doi.org/10.4310/HHA.2022.v24.n1.a5

    Coconet ¸, T., Todea, C.-C.: Symmetric Hochschild cohomology of twisted group algebras. Homology, Homotopy and Applications24(1), 93–115 (2022) https:// doi.org/10.4310/HHA.2022.v24.n1.a5

    work page doi:10.4310/hha.2022.v24.n1.a5 2022

  17. [17]

    Manuscripta Mathematica80, 213–224 (1993) https://doi.org/10

    Fleischmann, P., Janiszczak, I., Lempken, W.: Finite groups have local Non-Schur centralizers. Manuscripta Mathematica80, 213–224 (1993) https://doi.org/10. 1007/BF03026547

    work page 1993

  18. [18]

    American Mathematical Society, Providence, Rhode Island (2019)

    Witherspoon, S.J.: Hochschild Cohomology for Algebras. American Mathematical Society, Providence, Rhode Island (2019). https://doi.org/10.1090/gsm/204 17

    work page doi:10.1090/gsm/204 2019