---
_id: '66859'
abstract:
- lang: eng
  text: Let $\mathbb{k}$ be a finite field and $\mathcal{A}, \mathcal{B}$ be $\mathbb{k}$-linear
    $\mathsf{Ext}$-finite hereditary abelian categories. A theorem of Cramer asserts
    that, under suitable assumptions, a derived equivalence $\mathcal{D}^b(\mathcal{A})
    \!\longrightarrow\! \mathcal{D}^b(\mathcal{B})$ between two such categories induces
    an algebra isomorphism of the corresponding double Hall algebras $\mathsf{DH}_\mathcal{A}
    \!\longrightarrow\! \mathsf{DH}_\mathcal{B}$. It turns out that a counting formula
    for certain distinguished triangles in $\mathcal{D}^b(\mathcal{A})$, on which
    Cramer's proof relies, is incorrect in general. We give a corrected proof of Cramer's
    theorem which preserves the overall strategy of his approach.
author:
- first_name: Igor
  full_name: Burban, Igor
  id: '72064'
  last_name: Burban
- first_name: Daniel
  full_name: Perniok, Daniel
  id: '64242'
  last_name: Perniok
citation:
  ama: Burban I, Perniok D. Double Hall algebras and derived equivalences revisited.
    Published online 2026.
  apa: Burban, I., &#38; Perniok, D. (2026). <i>Double Hall algebras and derived equivalences
    revisited</i>.
  bibtex: '@article{Burban_Perniok_2026, title={Double Hall algebras and derived equivalences
    revisited}, author={Burban, Igor and Perniok, Daniel}, year={2026} }'
  chicago: Burban, Igor, and Daniel Perniok. “Double Hall Algebras and Derived Equivalences
    Revisited,” 2026.
  ieee: I. Burban and D. Perniok, “Double Hall algebras and derived equivalences revisited.”
    2026.
  mla: Burban, Igor, and Daniel Perniok. <i>Double Hall Algebras and Derived Equivalences
    Revisited</i>. 2026.
  short: I. Burban, D. Perniok, (2026).
date_created: 2026-08-26T12:08:41Z
date_updated: 2026-08-26T12:15:11Z
external_id:
  arxiv:
  - '2608.24331'
language:
- iso: eng
status: public
title: Double Hall algebras and derived equivalences revisited
type: preprint
user_id: '64242'
year: '2026'
...