In der algebraischen Geometrie und kommutativen Algebra sind die lokalen Kohomologiemoduln seit ihrer Einführung vor gut 50 Jahren von großem Interesse. Dabei handelt es sich um eine mathematische Konstruktion, die Anfang der 60er Jahre von Grothendieck in [Gro67] gemacht wurde, um geometrische Fragen zu beantworten. Mittlerweile ist die Theorie der lokalen Kohomologie ein fester Bestandteil für die Untersuchung von kommutativen noetherschen Ringen. Betrachtet man Ringe als Funktionen auf Räumen, so lassen sich auch geometrische und topologische Inhalte untersuchen.

Diese Arbeit entwickelt aufbauend auf bekannten Resultaten die Modultheorie über der äußeren Algebra in Teilen weiter, insbesondere werden die Tiefe eines Moduls und Moduln mit linearer injektiver Auflösung untersucht. Angewendet werden die Resultate auf die Orlik-Solomon Algebra eines Matroids.

We prove that solid closure and graded dagger closure agree for homogeneous ideals in two dimensional $\mathbb{N}$-graded domains of finite type over a field. We also prove that dagger closure is trivial for ideals in regular rings containing a field and that graded dagger closure is trivial for $\mathbb{N}$-graded regular rings of finite type over a field. Finally, we prove an inclusion result for graded dagger closure for homogeneous primary ideals in certain section rings of abelian varieties.

Bekannte Theoreme von Carlson und Griffiths gestatten es, die Variation von Hodgestrukturen assoziiert zu einer Familie von glatten Hyperflächen sowie das Cupprodukt auf der mittleren Kohomologie explizit zu beschreiben. Wir benutzen M. Saitos Theorie der gemischten Hodgemoduln, um diesen Kalkül auf die Variation der Hodgestruktur der Schnittkohomologie von Familien nodaler Hyperflächen zu verallgemeinern. / Well known theorems of Carlson and Griffiths provide an explicit description of the variation of Hodge structures associated to a family of smooth hypersurfaces together with the cupproduct pairing on the middle cohomology. We give a generalization to families of nodal hypersurfaces using M. Saitos theory of mixed Hodge modules.

The two main parts of this doctoral thesis are a theorem that tight closure is contained in continuous closure via axes closure on the one hand and an algorithm to decide semistability of sheaves (or geometric vector bundles) via reduction to a linear algebra problem on the other hand. The sheaf stability algorithm was explicitly implemented by the author.

Für eine Primzahl p, sei L eine endliche Erweiterung von $QQ_p$ mit Ganzheitsring $O_L$ und Restklassenk\"{o}rper $kk_L$. Sei ferner n eine positive ganze Zahl. In dieser Arbeit beschreiben wir die Kategorie der endlich erzeugten stetigen Darstellungen der n-ten direkten Potenz der absoluten Galoisgruppe $G_L$ von L mit Koeffizienten in $O_L$, unter Verwendung einer verallgemeinerten Version der $(phi, Gamma)$-Moduln von Fontaine. In Kapitel 4 beweisen wir, dass die Kategorie der stetigen Darstellungen der n-ten direkten Potenz von $G_L$ auf endlichen dimensionalen $kk_L$-Vektorräumen und die Kategorie étaler $(phi, Gamma)$-Moduln über einem n-variablen Laurentreihenring über $kk_L$ äquivalent sind. In Kapitel 5 erweitern wir diese Äquivalenz, um zu beweisen, dass die Kategorie der stetigen Darstellungen der n-ten direkten Potenz von $G_L$ auf endlich erzeugten $O_L$-Moduln und die Kategorie étaler $(phi, Gamma)$-Moduln über einem n-variablen Laurentreihenring über $O_L$ äquivalent sind. Einerseits erhalten wir, wenn wir n=1 und L willkürlich lassen, die Verfeinerung von Fontaine ursprünglicher Konstruktion gemäß Kisin, Rin und Schneider, die Lubin-Tate Theorie verwenden. Wenn wir andererseits n willkürlich lassen und $L=QQ_p$, erhalten wir die Theorie von Zábrádi von multivariablen zyklotomischen $(phi, Gamma)$-Moduln, die Fontaines Verwendung einer einzelnen freien Variablen verallgemeinert. Daher bietet unsere Arbeit einen gemeinsamen Rahmen für diese beiden Verallgemeinerungen. / For a prime number p, let L be a finite extension of $QQ_p$ with ring of integers $O_L$ and residue field $kk_L$. We also let n be a positive integer. In this thesis we describe the category of finitely generated continuous representations of the n-th direct power of the absolute Galois group $G_L$ of L with coefficients in $O_L$ using a generalized version of Fontaine's $(phi, Gamma)$-modules. In Chapter 4 we prove that the category of continuous representations of the n-th direct power of $G_L$ on finite dimensional $kk_L$-vector spaces is equivalent to the category of étale $(phi, Gamma)$-modules over a n-variable Laurent series ring over $kk_L$. In Chapter 5 we extend this equivalence to prove that the category of continuous representations of the n-th direct power of $G_L$ on finitely generated $O_L$-modules is equivalent to the category of étale $(phi, Gamma)$-modules over a n-variable Laurent series ring over $O_L$. On the one hand, if we let n=1 and $L$ be arbitrary, we obtain the refinement of Fontaine's original construction due to Kisin, Rin and Schneider, which uses Lubin-Tate theory. On the other hand, if we let n be arbitrary and $L=QQ_p$, we recover Zábrádi's theory of multivariable cyclotomic $(phi,Gamma)$-modules that generalizes Fontaine's use of a single free variable. Therefore, our thesis provides a common framework for both of these generalizations.

We compute the Hilbert-Kunz functions of two-dimensional rings of type ADE by using representations of their indecomposable, maximal Cohen-Macaulay modules in terms of matrix factorizations, and as first syzygy modules of homogeneous ideals.

My thesis is mostly concerned with algebraic and combinatorial aspects of the theory of hyperplane arrangements. More specifically, I study the Orlik-Terao algebra of a hyperplane arrangement and the broken circuit complex of a matroid. The Orlik-Terao algebra is a useful tool for studying hyperplane arrangements, especially for characterizing some non-combinatorial properties. The broken circuit complex, on the one hand, is closely related to the Orlik-Terao algebra, and on the other hand, plays a crucial role in the study of many combinatorial problem: the coefficients of the characteristic polynomial of a matroid are encoded in the f-vector of the broken circuit complex of the matroid. Among main results of the thesis are characterizations of the complete intersection and Gorenstein properties of the broken circuit complex and the Orlik-Terao algebra. I also study the h-vector of the broken circuit complex of a series-parallel network and relate certain entries of that vector to ear decompositions of the network. An application of the Orlik-Terao algebra in studying the relation space of a hyperplane arrangement is also included in the thesis.

My PhD-project has two main research directions. The first direction is on partial regularities which we define as refinements of the Castelnuovo-Mumford regularity. Main results are: relationship of partial regularities and related invariants, like the a-invariants or the Castelnuovo-Mumford regularity of the syzygy modules; algebraic properties of partial regularities via a filter-regular sequence or a short exact sequence; generalizing a well-known result for the Castelnuovo-Mumford regularity to the case of partial regularities of stable and squarefree stable monomial ideals; finally extending an upper bound proven by Caviglia-Sbarra to partial regularities. The second direction of my project is to develop a theory on monomial preorders. Many interesting statements from the classical theory of monomial orders generalize to monomial preorders. Main results are: a characterization of monomial preorders by real matrices, which extends a result of Robbiano on monomial orders; secondly, leading term ideals with respect to monomial preorders can be studied via flat deformations of the given ideal; finally, comparing invariants of the given ideal and the leading term ideal with respect to a monomial preorder.

The thesis deals with stochastic and algebraic aspects of the integer convex hull. In the first part, the intrinsic volumes of the randomized integer convex hull are investigated. In particular, we obtained an exact asymptotic order of the expected intrinsic volumes difference in a smooth convex body and a tight inequality for the expected mean width difference. In the algebraic part, an exact formula for the Bhattacharya function of complete primary monomial ideas in two variables is given. As a consequence, we derive an effective characterization for complete monomial ideals in two variables.