Theses

Master Thesis (supervisors: Ralf Meyer and Frank Gounelas)

Title: Towards Hochschild and Cyclic Homology of Dagger Algebras

Abstract: Over the complex numbers, the Hochschild–Kostant–Rosenberg theorem generalises to crossed product algebras arising from (finite) groups actions on affine complex varieties. More precisely, their Hochschild, cyclic, and periodic homologies are given in terms of algebraic cohomologies of the associated quotient varieties. We show that this extends to, in particular, non-algebraically closed base fields k of characteristic 0.  If, however, k is of positive characteristic, a different approach is needed. We assume that k arises as the residue field of a discrete valuation ring with uniformizer π and assume that V is of characteristic 0. This introduces a non-trivial topological component. To properly utilize this aspect in the context of homological computations, we make use of the framework of bornologies. We define a variation of Hochschild homology for bornologically complete V-algebras and verify that this theory admits an axiomatic description. We then show that this theory is well-behaved under base change to the field of fractions of V, to which we can then apply the results from the characteristic 0 case.

Here is a copy of my Master thesis.

Bachelor Thesis (supervisors: Preda Mihăilescu and Katharina Müller)

Title: Studying the structure of Λ-Modules in Iwasawa Theory using Kummer Theory and defining a Projective Radical

Abstract: Let Λ be the Iwasawa algebra. The first part of this thesis views Λ-modules as abstract algebraic structures. We establish the standard structure theorem for finitely generated Λ-modules, with an emphasis on module-theoretic arguments in place of linear-algebraic ones. This approach highlights the algebraic similarities to the classification theorem for finitely generated modules over principal ideal domains and, in fact, makes use of the latter theorem. The second part of the thesis views Λ-modules as concrete Galois groups of sufficiently well-behaved field extensions. To make use of Kummer theory, we specialise to the cyclotomic Z_p-extension of a number field. We construct a generating system for the given field extension, which is built from the Cogalois radicals of a canonical filtration of subextensions and, furthermore, admits the structure of a projective  Λ-module. Finally, we sketch some applications.

Here is a copy of my Bachelor thesis.