{"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","project":[{"grant_number":"26525","_id":"34cd0f74-11ca-11ed-8bc3-bf0492a14a24","name":"Topology of open smooth varieties with a torus action"}],"publisher":"Institute of Science and Technology Austria","publication_status":"published","_id":"17156","type":"dissertation","oa_version":"Published Version","abstract":[{"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.","lang":"eng"}],"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>","short":"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>.","ieee":"K. P. Rychlewicz, “Equivariant cohomology and rings of functions,” Institute of Science and Technology Austria, 2024.","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>.","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>"},"file":[{"content_type":"application/zip","checksum":"1610063569f5452f8a5acef728c2fc26","file_size":2761814,"relation":"source_file","file_id":"17179","file_name":"thesis.zip","date_updated":"2024-06-26T21:00:14Z","date_created":"2024-06-26T20:56:27Z","creator":"krychlew","access_level":"closed"},{"creator":"krychlew","date_created":"2024-06-26T20:58:24Z","date_updated":"2024-06-26T20:58:24Z","access_level":"open_access","relation":"main_file","file_name":"thesis.pdf","file_id":"17180","file_size":3695952,"content_type":"application/pdf","checksum":"7bbadb1fbc9ed2a1ecf54597f88af99c"}],"related_material":{"record":[{"relation":"part_of_dissertation","status":"public","id":"17157"}]},"title":"Equivariant cohomology and rings of functions","date_published":"2024-06-25T00:00:00Z","year":"2024","doi":"10.15479/at:ista:17156","file_date_updated":"2024-06-26T21:00:14Z","month":"06","has_accepted_license":"1","supervisor":[{"first_name":"Tamás","last_name":"Hausel","full_name":"Hausel, Tamás","orcid":"0000-0002-9582-2634","id":"4A0666D8-F248-11E8-B48F-1D18A9856A87"}],"language":[{"iso":"eng"}],"department":[{"_id":"TaHa"},{"_id":"GradSch"}],"day":"25","page":"117","keyword":["equivariant cohomology","zero schemes","algebraic groups","Lie algebras"],"tmp":{"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)","image":"/images/cc_by_nc_sa.png","short":"CC BY-NC-SA (4.0)"},"oa":1,"date_updated":"2024-07-05T12:01:50Z","article_processing_charge":"No","date_created":"2024-06-23T15:07:06Z","author":[{"first_name":"Kamil P","last_name":"Rychlewicz","id":"85A07246-A8BF-11E9-B4FA-D9E3E5697425","full_name":"Rychlewicz, Kamil P"}],"degree_awarded":"PhD","alternative_title":["ISTA Thesis"],"publication_identifier":{"issn":["2663-337X"]},"license":"https://creativecommons.org/licenses/by-nc-sa/4.0/","status":"public","ddc":["516"]}