Eigenschaften von Determinanten auf freien Moduln von endlichem Rang
This text discusses the properties of determinants on free modules of finite rank.
This text explains (and proves) the Banach–Tarski paradox and similar, easier paradoxes like the Sierpiński–Mazurkiewicz paradox and the Hausdorff paradox.
This text covers Čech cohomology of abelian sheaves on topological spaces with respect to a given open covering. The main result is the fact that the Čech cohomology and the regular cohomology of quasi-coherent sheaves on noetherian separated schemes coincide.
This text contains an introduction to adjunctions in terms of category theory. Three different definitions for adjunctions are given—their equivalence is shown. There is also a small excursus on reflective subcategories.
Ein Verschwindungssatz für die Garbenkohomologie noetherscher topologischer Räume
This text is actually my bachelor’s thesis. It begins with a brief introduction to some topological concepts which are important in algebraic geometry. This is followed up by some fundamental properties of sheaves of abelian groups and an introduction to sheaf cohomology. Finally, a proof for Alexander Grothendieck’s famous “vanishing theorem for sheaf cohomology on noetherian topological spaces” is given.
This text introduces simplicial approximation. Later, barycentric subdivision and generalized barycentric subdivision are introduced in order to prove the general simplicial approximation theorem.
The Dual Isogeny
This text proves the existence of the “dual isogeny" to an isogeny of two elliptic curves. Later some simple properties of this construction are given.
Grothedieck Spectral Sequences
This text introduces the notion of Grothendieck Spectral Sequences (a certain form of spectral sequences) and proves their existance. After that, two examples (base-change for Tor and Ext) are given.
Serres Tor-Formel und Reduktion auf die Diagonale
This text is my master’s thesis. It introduces the required language for Serre's Tor-formula and proves the main theorem for this formula.