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Classification of plethories in characteristic zero
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-0588-9369
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We classify plethories over fields of characteristic zero, thus answering a question of Borger-Wieland and Bergman. All plethories over characteristic zero fields are linear, in the sense that they are free plethories on a bialgebra. For the proof we need some facts from the theory of ring schemes where we extend previously known results. We also classify plethories with trivial Verschiebung over a perfect field of non-zero characteristic and indicate future work.

Identifiers
URN: urn:nbn:se:kth:diva-177184OAI: oai:DiVA.org:kth-177184DiVA, id: diva2:871805
Note

QS 2015

Available from: 2015-11-17 Created: 2015-11-17 Last updated: 2018-05-18Bibliographically approved
In thesis
1. Classification of plethories in characteristic zero
Open this publication in new window or tab >>Classification of plethories in characteristic zero
2015 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

We classify plethories over fields of characteristic zero, thus answering a question of Borger-Wieland and Bergman-Hausknecht. All plethories over characteristic zero fields are linear, in the sense that they are free plethories on a bialgebra. For the proof we need some facts from the theory of ring schemes where we extend previously known results. We also classify plethories with trivial Verschiebung over a perfect field of non-zero characteristic and indicate future work. 

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2015. p. vii, 11
Series
TRITA-MAT-A ; 2015:13
Keyword
Plethories, Witt vectors, ring schemes
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-177021 (URN)978-91-7595-775-3 (ISBN)
Presentation
2015-12-07, 3418, Lindstedtsvägen 25, KTH, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

QC 20151117

Available from: 2015-11-17 Created: 2015-11-13 Last updated: 2015-11-17Bibliographically approved
2. Abelian affine group schemes, plethories, and arithmetic topology
Open this publication in new window or tab >>Abelian affine group schemes, plethories, and arithmetic topology
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In Paper A we classify plethories over a field of characteristic zero. All plethories over characteristic zero fields are ``linear", in the sense that they are free plethories on a bialgebra. For the proof of this classification we need some facts from the theory of ring schemes where we extend previously known results. We also give a classification of plethories with trivial Verschiebung over a perfect field k of characteristic p>0.

In Paper B we study tensor products of abelian affine group schemes over a perfect field k. We first prove that the tensor product G_1 \otimes G_2 of two abelian affine group schemes G_1,G_2 over a perfect field k exists. We then describe the multiplicative and unipotent part of the group scheme G_1 \otimes G_2. The multiplicative part is described in terms of Galois modules over the absolute Galois group of k. In characteristic zero the unipotent part of G_1 \otimes G_2 is the group scheme whose primitive elements are P(G_1) \otimes P(G_2). In positive characteristic, we give a formula for the tensor product in terms of Dieudonné theory. 

In Paper C we use ideas from homotopy theory to define new obstructions to solutions of embedding problems and compute the étale cohomology ring of the ring of integers of a totally imaginary number field with coefficients in Z/2Z. As an application of the obstruction-theoretical machinery, we give an infinite family of totally imaginary quadratic number fields such that Aut(PSL(2,q^2)), for q an odd prime power, cannot be realized as an unramified Galois group over K, but its maximal solvable quotient can. 

In Paper D we compute the étale cohomology ring of an arbitrary number field with coefficients in Z/nZ for n an arbitrary positive integer. This generalizes the computation in Paper C. As an application, we give a formula for an invariant defined by Minhyong Kim.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2018. p. 43
Series
TRITA-MAT-A ; 2018:28
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-228197 (URN)978-91-7729-831-1 (ISBN)
Public defence
2018-06-12, E3, Lindstedtsvägen 3, Stockholm, 13:00
Opponent
Supervisors
Note

QC 20180518

Available from: 2018-05-18 Created: 2018-05-18 Last updated: 2018-05-18Bibliographically approved

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