---
res:
  bibo_abstract:
  - "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.@eng"
  bibo_authorlist:
  - foaf_Person:
      foaf_givenName: Kamil P
      foaf_name: Rychlewicz, Kamil P
      foaf_surname: Rychlewicz
      foaf_workInfoHomepage: http://www.librecat.org/personId=85A07246-A8BF-11E9-B4FA-D9E3E5697425
  bibo_doi: 10.15479/at:ista:17156
  dct_date: 2024^xs_gYear
  dct_isPartOf:
  - http://id.crossref.org/issn/2663-337X
  dct_language: eng
  dct_publisher: Institute of Science and Technology Austria@
  dct_subject:
  - equivariant cohomology
  - zero schemes
  - algebraic groups
  - Lie algebras
  dct_title: Equivariant cohomology and rings of functions@
...