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arxiv: 2606.30513 · v1 · pith:ZX5PRHCPnew · submitted 2026-06-29 · 🧮 math.RA

On the Cohomology of Cyclic Associative Algebras

Hassan Alhussein This is my paper

Pith reviewed 2026-06-30 02:59 UTC · model grok-4.3

classification 🧮 math.RA
keywords cyclic associative algebrascohomologyHochschild cohomologycyclic cohomologyalgebra extensionsdifferential formsuniversal derivation
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The pith

The cohomology H^•_cyc of cyclic associative algebras classifies their extensions by cyclic bimodules and lies between cyclic and Hochschild cohomology.

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

This paper defines a cohomology theory H^•_cyc(A, M) for cyclic associative algebras by taking only those Hochschild cochains that obey an additional cyclic compatibility condition coming from the algebra's defining identity. It shows that the second cohomology group classifies extensions of the algebra by a suitable bimodule, constructs the universal differential graded algebra of forms over A, and proves that for trivial coefficients the new groups sit inside the usual Hochschild groups and contain the cyclic cohomology groups. A reader would care because the construction isolates the part of Hochschild theory that respects the cyclic symmetry, offering a tool for studying deformations that preserve that symmetry.

Core claim

H^2_cyc(A, M) classifies cyclic associative extensions of A by a cyclic bimodule M; (Ω^•_F(A), d) is the universal cyclic differential graded algebra over A; and there are natural inclusions HC^n(A) ↪ H^n_cyc(A, F) ↪ HH^n(A, F) for trivial coefficients.

What carries the argument

The cyclic subcomplex of the Hochschild cochain complex obtained by restricting to cochains satisfying the cyclic compatibility condition derived from the identity (xy)z = x(yz) = y(zx)

If this is right

  • If the second cohomology classifies extensions, then infinitesimal deformations of cyclic associative algebras are controlled by this group.
  • The universal cyclic differential graded algebra allows any cyclic associative algebra to generate a larger structure with a differential satisfying the cyclic property.
  • The natural inclusions position the new theory strictly between Connes cyclic cohomology and Hochschild cohomology.
  • For trivial coefficients the inclusions hold, allowing transfer of computations from one theory to the others.

Where Pith is reading between the lines

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

  • This framework could be used to compute explicit cohomology groups for known examples of cyclic associative algebras by leveraging existing Hochschild calculations.
  • The universal property might lead to constructions of cyclic DG algebras in other algebraic settings beyond the paper's scope.
  • Potential links to operadic cohomology or other non-associative structures remain to be investigated as natural extensions.

Load-bearing premise

The cyclic compatibility condition derived from the defining identity produces a subcomplex of the Hochschild complex whose cohomology groups satisfy the stated classification and inclusion properties.

What would settle it

An explicit computation for a concrete cyclic associative algebra A and bimodule M where the second cohomology group does not parametrize the extensions, or where one of the claimed inclusions fails to hold.

read the original abstract

We introduce a cohomology theory for cyclic associative algebras, a subclass of shift associative algebras defined by the identity $(xy)z = x(yz) = y(zx)$. This cohomology, denoted $H^\bullet_{\mathrm{cyc}}(A, M)$, is a subtheory of Hochschild cohomology obtained by restricting to cochains that satisfy a cyclic compatibility condition derived from the defining identity. We prove that $H^2_{\mathrm{cyc}}(A, M)$ classifies cyclic associative extensions of $A$ by a cyclic bimodule $M$. The universal derivation and the module of differential forms $\Omega^\bullet_{\mathbb{F}}(A)$ are constructed, and $(\Omega^\bullet_{\mathbb{F}}(A), d)$ is shown to be the universal cyclic differential graded algebra over $A$. For trivial coefficients, we establish natural inclusions $HC^n(A) \hookrightarrow H^n_{\mathrm{cyc}}(A, \mathbb{F}) \hookrightarrow HH^n(A, \mathbb{F})$, placing our theory intermediate between Connes' cyclic cohomology and Hochschild cohomology.

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 / 2 minor

Summary. The manuscript introduces a cohomology theory H^•_cyc(A, M) for cyclic associative algebras, defined as a restriction of the Hochschild cohomology to cochains satisfying a cyclic compatibility condition derived from the identity (xy)z = x(yz) = y(zx). It asserts that H²_cyc classifies cyclic associative extensions of A by a cyclic bimodule M, that the constructed (Ω^•_F(A), d) is the universal cyclic differential graded algebra over A, and that there are natural inclusions HC^n(A) ↪ H^n_cyc(A, F) ↪ HH^n(A, F) for trivial coefficients F.

Significance. If the central construction yields a genuine subcomplex whose cohomology satisfies the stated properties, the result would establish an intermediate cohomology theory between cyclic and Hochschild cohomology for this algebra class, with the extension classification and universal DGA property providing concrete applications. The placement between HC and HH is a potentially useful observation.

major comments (1)
  1. [Definition of H^•_cyc and the cyclic cochain condition] The verification that the cyclic compatibility condition is preserved by the Hochschild coboundary operator is load-bearing for all main theorems (classification of extensions, universal property of differential forms, and the two inclusions). The abstract indicates the condition is 'derived from the defining identity', but an explicit computation showing that if a cochain satisfies the condition then so does its coboundary is required to confirm that the restricted cochains form a subcomplex. Without this step (likely in the section defining H^•_cyc), the cohomology groups are not defined and the claims cannot hold.
minor comments (2)
  1. [Introduction] The relation between 'cyclic associative algebras' and the broader class of 'shift associative algebras' mentioned in the abstract should be clarified with a brief comparison or reference.
  2. [Construction of differential forms] Notation for the base field or ring in Ω^•_F(A) and the trivial coefficients F should be made consistent throughout.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and for identifying a critical gap in the presentation of our cohomology theory. We address the major comment below and confirm that the requested verification will be added to the revised manuscript.

read point-by-point responses

  1. Referee: [Definition of H^•_cyc and the cyclic cochain condition] The verification that the cyclic compatibility condition is preserved by the Hochschild coboundary operator is load-bearing for all main theorems (classification of extensions, universal property of differential forms, and the two inclusions). The abstract indicates the condition is 'derived from the defining identity', but an explicit computation showing that if a cochain satisfies the condition then so does its coboundary is required to confirm that the restricted cochains form a subcomplex. Without this step (likely in the section defining H^•_cyc), the cohomology groups are not defined and the claims cannot hold.

    Authors: We agree that an explicit verification is required. The original manuscript states that the cyclic condition is derived from the identity (xy)z = x(yz) = y(zx) but does not supply the direct computation confirming that the Hochschild coboundary maps cyclic cochains to cyclic cochains. In the revision we will insert a dedicated lemma immediately after the definition of the cyclic cochain space, computing δφ explicitly on a general cyclic cochain φ and showing, term by term, that the resulting cochain satisfies the same cyclic relations by repeated application of the defining identity of cyclic associative algebras. This will establish that the restricted cochains form a subcomplex and thereby justify all subsequent claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a direct restriction of Hochschild cohomology with independent verification steps

full rationale

The paper defines cyclic cochains via the algebra identity (xy)z = x(yz) = y(zx) and restricts the Hochschild complex, then proves the subcomplex property, extension classification, universal DGA property, and inclusions as separate results. No step reduces by construction to a fitted parameter, self-citation chain, or self-definition of the target claims. The central results require explicit checks (e.g., that the coboundary preserves cyclic cochains) that are not tautological with the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces a new cohomology theory resting on the standard Hochschild cochain complex together with one additional domain assumption about cyclic compatibility; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption The cyclic compatibility condition derived from (xy)z = x(yz) = y(zx) defines a subcomplex of the Hochschild cochain complex.
    Invoked when restricting cochains to obtain H^•_cyc; stated in the abstract definition paragraph.

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discussion (0)

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Reference graph

Works this paper leans on

10 extracted references

  1. [1]

    Abdelwahab, I

    H. Abdelwahab, I. Kaygorodov, B. Sartayev,Shift associative algebras, Rend. Circ. Mat. Palermo, II. Ser.74, 145 (2025)

    2025

  2. [2]

    Dotsenko, N

    V. Dotsenko, N. Ismailov, U. Umirbaev,Polynomial identities in Novikov algebras, Mathematische Zeitschrift 303(60), 3 (2023)

    2023

  3. [3]

    Sartayev, P

    B. Sartayev, P. Kolesnikov,Noncommutative Novikov algebras, Eur. J. Math. 9(2), 35 (2023). 17

    2023

  4. [4]

    Drensky, N

    V. Drensky, N. Ismailov, M. Mustafa, B. Zhakhayev,Free bicommutative super- algebras, J. Algebra 652, 158-187 (2024)

    2024

  5. [5]

    Ismailov, F.A

    N.A. Ismailov, F.A. Mashurov, B.K. Sartayev,On algebras embeddable into bicommutative algebras, Comm. Algebra 52(11), 4778-4785 (2024)

    2024

  6. [6]

    Kaygorodov, F

    I. Kaygorodov, F. Mashurov,Mutations of perm algebras, Revista de la Real Academia de Ciencias Exactas, F´ ısicas y Naturales. Serie A. Matem´ aticas 118, 166 (2024)

    2024

  7. [7]

    Mashurov, B

    F. Mashurov, B. Sartayev,Metabelian Lie and perm algebras, J. Algebra Appl. 23(4), 2450065 (2024)

    2024

  8. [8]

    Hochschild,On the cohomology groups of an associative algebra, Ann

    G. Hochschild,On the cohomology groups of an associative algebra, Ann. Math. 46(1), 58-67 (1945)

    1945

  9. [9]

    Connes,Noncommutative differential geometry, Publ

    A. Connes,Noncommutative differential geometry, Publ. Math. Inst. Hautes ´Etudes Sci. 62, 41-144 (1985)

    1985

  10. [10]

    Cuntz and D

    J. Cuntz and D. Quillen,Algebra extensions and nonsingularity, J. Amer. Math. Soc.8(1995), no. 2, 251–289. 18

    1995