Dokument: Operations on Milnor-Witt K-theory
| Titel: | Operations on Milnor-Witt K-theory | |||||||
| URL für Lesezeichen: | https://docserv.uni-duesseldorf.de/servlets/DocumentServlet?id=70462 | |||||||
| URN (NBN): | urn:nbn:de:hbz:061-20250819-110250-8 | |||||||
| Kollektion: | Dissertationen | |||||||
| Sprache: | Englisch | |||||||
| Dokumententyp: | Wissenschaftliche Abschlussarbeiten » Dissertation | |||||||
| Medientyp: | Text | |||||||
| Autor: | Wittich, Thor Harald [Autor] | |||||||
| Dateien: |
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| Beitragende: | Prof. Dr. Halupczok, Immanuel [Betreuer/Doktorvater] Prof. Dr. Wendt, Matthias [Gutachter] | |||||||
| Stichwörter: | Motivic Homotopy Theory, Milnor-Witt K-theory, Cohomology Operations | |||||||
| Dewey Dezimal-Klassifikation: | 500 Naturwissenschaften und Mathematik » 510 Mathematik | |||||||
| Beschreibung: | The aim of this thesis is to study operations on Milnor-Witt K-theory. This invariant of smooth schemes arises naturally in motivic homotopy theory as the motivic 0-stem of the motivic sphere spectrum, and many other invariants are modules over it.
The starting point for studying operations on Milnor-Witt K-theory is a paper of Vial, where the module of all uniformly bounded operations on Milnor K-theory is computed. It turns out that this module of operations is generated by certain divided power operations. By a result of Morel, Milnor-Witt K-theory can be seen as a quadratic refinement of Milnor K-theory. Therefore this thesis deals with a generalization of Vial's aforementioned result to Milnor-Witt K-theory. We first compute all additive and all stable operations on Milnor-Witt K-theory. After this we construct divided power operations for any homotopy module. This is our first main result. Next we study operations on the canonical generators of Milnor-Witt K-theory. Our second main result is a fulldescription of themodule of all operations on these generators. Following a strategy of Garrell from the theory of quadratic forms, we study how a general operation changes when adding/subtracting a generator to the argument. We refer to these changes as shifts. Using those shifts we compute the module of all operationson Milnor-Witt K-theory, which turns out to be essentially generated by the our divided power operations. This is our next main result. Finally, we retrieve and generalize both Vial's and Garrell's computations of operations on Milnor K-theory and on powers of the fundamental ideal of the Witt ring, respectively. This also leads our last main result, which is a description of operations between Milnor, Witt and Milnor-Witt K-theory in mixed degrees. | |||||||
| Lizenz: |  Dieses Werk ist lizenziert unter einer Creative Commons Namensnennung 4.0 International Lizenz | |||||||
| Fachbereich / Einrichtung: | Mathematisch- Naturwissenschaftliche Fakultät | |||||||
| Dokument erstellt am: | 19.08.2025 | |||||||
| Dateien geändert am: | 19.08.2025 | |||||||
| Promotionsantrag am: | 05.06.2024 | |||||||
| Datum der Promotion: | 07.10.2024 |
