---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '20323'
abstract:
- lang: eng
  text: We establish several results combining discrete Morse theory and microlocal
    sheaf theory in the setting of finite posets and simplicial complexes. Our primary
    tool is a computationally tractable description of the bounded derived category
    of sheaves on a poset with the Alexandrov topology. We prove that each bounded
    complex of sheaves on a finite poset admits a unique (up to isomorphism of complexes)
    minimal injective resolution, and we provide algorithms for computing minimal
    injective resolution of an injective complex, as well as several useful functors
    between derived categories of sheaves. For the constant sheaf on a simplicial
    complex, we give asymptotically tight bounds on the complexity of computing the
    minimal injective resolution using those algorithms. Our main result is a novel
    definition of the discrete microsupport of a bounded complex of sheaves on a finite
    poset. We detail several foundational properties of the discrete microsupport,
    as well as a microlocal generalization of the discrete homological Morse theorem
    and Morse inequalities.
acknowledgement: This project has received funding from the European Research Council
  (ERC) under the European Union’s Horizon 2020 research and innovation programme,
  grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant
  no. Z342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization
  in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35
article_number: '108068'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Adam
  full_name: Brown, Adam
  id: 70B7FDF6-608D-11E9-9333-8535E6697425
  last_name: Brown
- first_name: Ondrej
  full_name: Draganov, Ondrej
  id: 2B23F01E-F248-11E8-B48F-1D18A9856A87
  last_name: Draganov
  orcid: 0000-0003-0464-3823
citation:
  ama: Brown A, Draganov O. Discrete microlocal Morse theory. <i>Journal of Pure and
    Applied Algebra</i>. 2025;229(10). doi:<a href="https://doi.org/10.1016/j.jpaa.2025.108068">10.1016/j.jpaa.2025.108068</a>
  apa: Brown, A., &#38; Draganov, O. (2025). Discrete microlocal Morse theory. <i>Journal
    of Pure and Applied Algebra</i>. Elsevier. <a href="https://doi.org/10.1016/j.jpaa.2025.108068">https://doi.org/10.1016/j.jpaa.2025.108068</a>
  chicago: Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” <i>Journal
    of Pure and Applied Algebra</i>. Elsevier, 2025. <a href="https://doi.org/10.1016/j.jpaa.2025.108068">https://doi.org/10.1016/j.jpaa.2025.108068</a>.
  ieee: A. Brown and O. Draganov, “Discrete microlocal Morse theory,” <i>Journal of
    Pure and Applied Algebra</i>, vol. 229, no. 10. Elsevier, 2025.
  ista: Brown A, Draganov O. 2025. Discrete microlocal Morse theory. Journal of Pure
    and Applied Algebra. 229(10), 108068.
  mla: Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” <i>Journal
    of Pure and Applied Algebra</i>, vol. 229, no. 10, 108068, Elsevier, 2025, doi:<a
    href="https://doi.org/10.1016/j.jpaa.2025.108068">10.1016/j.jpaa.2025.108068</a>.
  short: A. Brown, O. Draganov, Journal of Pure and Applied Algebra 229 (2025).
corr_author: '1'
date_created: 2025-09-10T05:40:09Z
date_published: 2025-10-01T00:00:00Z
date_updated: 2026-10-02T11:57:00Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1016/j.jpaa.2025.108068
ec_funded: 1
external_id:
  arxiv:
  - '2209.14993'
file:
- access_level: open_access
  checksum: 39bcad462278c9322ef810af7db67f56
  content_type: application/pdf
  creator: dernst
  date_created: 2025-12-30T07:55:08Z
  date_updated: 2025-12-30T07:55:08Z
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file_date_updated: 2025-12-30T07:55:08Z
fulldoi: https://doi.org/10.1016/j.jpaa.2025.108068
has_accepted_license: '1'
intvolume: '       229'
issue: '10'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '10'
oa: 1
oa_version: Published Version
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '788183'
  name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: Z00342
  name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: I02979-N35
  name: Persistence and stability of geometric complexes
publication: Journal of Pure and Applied Algebra
publication_identifier:
  issn:
  - 0022-4049
publication_status: published
publisher: Elsevier
quality_controlled: '1'
related_material:
  record:
  - id: '18981'
    relation: earlier_version
    status: public
scopus_import: '1'
status: public
title: Discrete microlocal Morse theory
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 229
year: '2025'
...