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arxiv: 2606.26255 · v1 · pith:CMTOTBHXnew · submitted 2026-06-24 · 🧮 math.RT

Hochschild (co)homology and cyclic homology via a graded Euler characteristic with applications to higher preprojective algebras

Jon Wallem Anundsen , Mads Hustad Sand{\o}y This is my paper

Pith reviewed 2026-06-26 00:58 UTC · model grok-4.3

classification 🧮 math.RT
keywords Hochschild homologycyclic homologyhigher preprojective algebrasgraded Euler characteristicCartan matrixrepresentation finite algebrastype A algebras
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The pith

For higher preprojective algebras, the graded vector space structures of all Hochschild (co)homology and cyclic homology groups follow from those of the center and the zeroth Hochschild homology via a graded Euler characteristic.

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

The paper generalizes a method originally used for ordinary preprojective algebras of ADE type to their higher versions. By exploiting a graded Euler characteristic derived from the graded Cartan matrix, the structures in higher degrees can be deduced once the center and degree-zero Hochschild homology are known. This applies specifically to higher preprojective algebras arising from tensor products of representation-finite hereditary algebras of type A. A reader would care because direct computation of these homologies is typically difficult, and the method reduces the problem to simpler data.

Core claim

The authors show that for higher preprojective algebras, it suffices to know the graded vector space structure of the center and the zeroth Hochschild homology in order to determine the graded vector space structures of the Hochschild cohomology, Hochschild homology, and cyclic homology in all degrees, by generalizing the graded Euler characteristic approach of Etingof and Eu.

What carries the argument

the graded Euler characteristic computed via the graded Cartan matrix, which encodes the necessary relations to determine higher homology groups from the center and HH_0

Load-bearing premise

The structural features allowing the graded Euler characteristic to determine all homology groups from the center and zeroth homology extend from ordinary preprojective algebras to the higher ones considered here.

What would settle it

An explicit computation for a specific higher preprojective algebra where the higher-degree Hochschild homology structures deviate from what the graded Euler characteristic predicts based on the center and HH_0.

read the original abstract

Computing the structure of the Hochschild (co)homology and the cyclic homology of an algebra can be hard work, but Etingof and Eu showed that it can be done surprisingly easily for preprojective algebras of ADE Dynkin type, at least if one only wants to know the graded vector space structure of each Hochschild cohomology group. Their method is based on exploiting strong structural features of such a preprojective algebra via a graded Euler characteristic that can computed using the algebra's graded Cartan matrix. In this paper, we present a generalization of the method used by Etingof and Eu to higher preprojective algebras. We also apply our generalization to the higher preprojective algebras of the 2-representation finite algebras that arise as tensor products of representation finite hereditary algebras of type A. For this, it turns out to be enough to know the graded vector space structure of the center and the zeroth Hochschild homology to be able to deduce the graded vector space structure of the Hochschild (co)homology and the cyclic homology in all other degrees.

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

0 major / 3 minor

Summary. The manuscript generalizes the Etingof-Eu method based on the graded Euler characteristic (computed from the graded Cartan matrix) to higher preprojective algebras arising from 2-representation-finite tensor products of type-A hereditary algebras. It proves that the graded vector-space structures of the center and HH_0 suffice to determine the graded structures of all other HH^i, HH_i and cyclic homology groups, by verifying graded Calabi-Yau duality and vanishing of higher Ext groups directly for these algebras rather than assuming them from the classical ADE case.

Significance. If the results hold, the work supplies an explicit, direct-verification-based extension of the Etingof-Eu technique to a new class of algebras, yielding a parameter-free computational route from center and HH_0 data alone. This strengthens the toolkit for homological invariants in higher representation theory and Calabi-Yau structures.

minor comments (3)
  1. [Abstract] Abstract: the sentence 'it turns out to be enough to know the graded vector space structure of the center and the zeroth Hochschild homology' would be clearer if it briefly named the two structural properties (graded Calabi-Yau duality and Ext vanishing) that make the deduction possible.
  2. [§2] §2 (or wherever the higher preprojective algebra is defined): include an explicit reference or short recap of the tensor-product construction from type-A hereditary algebras to ensure the class under consideration is unambiguously delimited.
  3. [Main theorem] Notation for graded vector spaces: ensure consistent use of parentheses or subscripts when denoting the graded pieces of HH^i and HH_i across all statements of the main theorem.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The referee's description accurately reflects the paper's scope and results.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via direct verification

full rationale

The manuscript generalizes the Etingof-Eu graded Euler characteristic (computed from the graded Cartan matrix) to higher preprojective algebras arising from 2-representation-finite tensor products of type-A hereditary algebras. It establishes that graded center and HH_0 data suffice to determine all other graded HH^i, HH_i and cyclic homology groups by verifying the required formal properties (graded Calabi-Yau duality and vanishing of higher Ext groups) directly on the algebras in question, rather than importing them via self-citation or ansatz. No step reduces a claimed prediction to a fitted parameter, self-definition, or load-bearing self-citation chain; the central result follows from the explicit structural checks and the prior external method without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review provides insufficient detail to identify concrete free parameters, axioms, or invented entities beyond implicit reliance on standard properties of graded algebras and preprojective constructions from prior literature.

axioms (1)
  • domain assumption Graded Cartan matrix encodes the information needed for the Euler characteristic computation in preprojective algebras
    Invoked as the basis for the original and generalized method.

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