Johannes Loher

Eigenschaften von Determinanten auf freien Moduln von endlichem Rang

This text discusses the properties of determinants on free modules of finite rank.

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Das Banach-Tarski-Paradoxon

This text explains (and proofs) the Banach–Tarski paradox and similar, easier paradoxes, like the Sierpiński–Mazurkiewicz paradox and the Hausdorff paradox.

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Čech-Kohomologie

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.

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Adjunktionen 1

This text contains an introduction to adjunctions in terms of category theory. Three different definitions for adjunctions are given and the equivalence of them is shown. There is also a small excursus on reflective subcategories.

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Ein Verschwindungssatz für die Garbenkohomologie noetherscher topologischer Räume

This text is actually my bachelor 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.

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Simplicial Approximation

This text introduces simplicial approximation. Later, barycentric subdivision and generalized barycentric subdivision are introduced in order to to prove the general simplicial approximation theorem.

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The Dual Isogeny

This text proves the existance of the “dual isogeny" to an isogeny of two elliptic curves. Later some simple properties of this construction are given.

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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.

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