Title: Modules over Gorenstein rings Author: David Pospíšil Department: Department of Algebra Supervisor: Prof. RNDr. Jan Trlifaj, DSc. Supervisor's e-mail address: trlifaj@karlin.mff.cuni.cz Abstract: The dissertation collects my actual contributions to the clas- sification of (co)tilting modules and classes over Gorenstein rings. Com- pared with the original intent we get a more general result in classification of (co)tilting classes namely for general commutative noetherian rings (see the third paper in this dissertation). The dissertation consists of an introduction and three papers with coauthors. The first paper (published in Contemp. Math.) contains a classification of all (co)tilting modules and classes over 1- Gorenstein commutative rings. The second paper (published in J. Algebra) contains a classification of all tilting classes over regular rings of Krull dimen- sion 2 and also a classification of all tilting modules in the local case. Finally the third paper (preprint) contains a classification of all (co)tilting classes and also torsion pairs over general commutative noetherian rings. All these classi- fications are in terms of subsets of the spectrum of the ring and by associated prime ideals of modules. Keywords: (co)tilting module, (co)tilting class, torsion pair, Gorenstein ring, regular ring,...

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.