---
OA_place: publisher
_id: '17156'
abstract:
- lang: eng
  text: "This dissertation is the summary of the author’s work, concerning the relations
    between\r\ncohomology rings of algebraic varieties and rings of functions on zero
    schemes and fixed\r\npoint schemes. For most of the thesis, the focus is on smooth
    complex varieties with\r\nan action of a principally paired group, e.g. a parabolic
    subgroup of a reductive group.\r\nThe fundamental theorem 5.2.11 from co-authored
    article [66] says that if the principal\r\nnilpotent has a unique zero, then the
    zero scheme over the Kostant section is isomorphic\r\nto the spectrum of the equivariant
    cohomology ring, remembering the grading in terms of\r\na C^* action. A similar
    statement is proved also for the G-invariant functions on the total\r\nzero scheme
    over the whole Lie algebra. Additionally, we are able to prove an analogous\r\nresult
    for the GKM spaces, which poses the question on a joint generalisation.\r\nWe
    also tackle the situation of a singular variety. As long as it is embedded in
    a smooth\r\nvariety with regular action, we are able to study its cohomology as
    well by means of\r\nthe zero scheme. In case of e.g. Schubert varieties this determines
    the cohomology ring\r\ncompletely. In largest generality, this allows us to see
    a significant part of the cohomology\r\nring.\r\nWe also show (Theorem 6.2.1)
    that the cohomology ring of spherical varieties appears as\r\nthe ring of functions
    on the zero scheme. The computational aspect is not easy, but one\r\ncan hope
    that this can bring some concrete information about such cohomology rings.\r\nLastly,
    the K-theory conjecture 6.3.1 is studied, with some results attained for GKM\r\nspaces.\r\nThe
    thesis includes also an introduction to group actions on algebraic varieties.
    In\r\nparticular, the vector fields associated to the actions are extensively
    studied. We also\r\nprovide a version of the Kostant section for arbitrary principally
    paired group, which\r\nparametrises the regular orbits in the Lie algebra of an
    algebraic group. Before proving\r\nthe main theorem, we also include a historical
    overview of the field. In particular we bring\r\ntogether the results of Akyildiz,
    Carrell and Lieberman on non-equivariant cohomology\r\nrings."
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Kamil P
  full_name: Rychlewicz, Kamil P
  id: 85A07246-A8BF-11E9-B4FA-D9E3E5697425
  last_name: Rychlewicz
citation:
  ama: Rychlewicz KP. Equivariant cohomology and rings of functions. 2024. doi:<a
    href="https://doi.org/10.15479/at:ista:17156">10.15479/at:ista:17156</a>
  apa: Rychlewicz, K. P. (2024). <i>Equivariant cohomology and rings of functions</i>.
    Institute of Science and Technology Austria. <a href="https://doi.org/10.15479/at:ista:17156">https://doi.org/10.15479/at:ista:17156</a>
  chicago: Rychlewicz, Kamil P. “Equivariant Cohomology and Rings of Functions.” Institute
    of Science and Technology Austria, 2024. <a href="https://doi.org/10.15479/at:ista:17156">https://doi.org/10.15479/at:ista:17156</a>.
  ieee: K. P. Rychlewicz, “Equivariant cohomology and rings of functions,” Institute
    of Science and Technology Austria, 2024.
  ista: Rychlewicz KP. 2024. Equivariant cohomology and rings of functions. Institute
    of Science and Technology Austria.
  mla: Rychlewicz, Kamil P. <i>Equivariant Cohomology and Rings of Functions</i>.
    Institute of Science and Technology Austria, 2024, doi:<a href="https://doi.org/10.15479/at:ista:17156">10.15479/at:ista:17156</a>.
  short: K.P. Rychlewicz, Equivariant Cohomology and Rings of Functions, Institute
    of Science and Technology Austria, 2024.
corr_author: '1'
date_created: 2024-06-23T15:07:06Z
date_published: 2024-06-25T00:00:00Z
date_updated: 2026-04-07T12:55:46Z
day: '25'
ddc:
- '516'
degree_awarded: PhD
department:
- _id: TaHa
- _id: GradSch
doi: 10.15479/at:ista:17156
file:
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  creator: krychlew
  date_created: 2024-06-26T20:56:27Z
  date_updated: 2024-06-26T21:00:14Z
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  file_name: thesis.zip
  file_size: 2761814
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- access_level: open_access
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  date_created: 2024-06-26T20:58:24Z
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  file_name: thesis.pdf
  file_size: 3695952
  relation: main_file
file_date_updated: 2024-06-26T21:00:14Z
has_accepted_license: '1'
keyword:
- equivariant cohomology
- zero schemes
- algebraic groups
- Lie algebras
language:
- iso: eng
license: https://creativecommons.org/licenses/by-nc-sa/4.0/
month: '06'
oa: 1
oa_version: Published Version
page: '117'
project:
- _id: 34cd0f74-11ca-11ed-8bc3-bf0492a14a24
  grant_number: '26525'
  name: Topology of open smooth varieties with a torus action
publication_identifier:
  issn:
  - 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
  record:
  - id: '17157'
    relation: part_of_dissertation
    status: public
status: public
supervisor:
- first_name: Tamás
  full_name: Hausel, Tamás
  id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
  last_name: Hausel
  orcid: 0000-0002-9582-2634
title: Equivariant cohomology and rings of functions
tmp:
  image: /images/cc_by_nc_sa.png
  legal_code_url: https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode
  name: Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC
    BY-NC-SA 4.0)
  short: CC BY-NC-SA (4.0)
type: dissertation
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
year: '2024'
...