This paper presents a survey of concepts in commutative algebra that have applications to topology and graph theory. The primary algebraic focus will be on Stanley-Reisner rings, classes of polynomial rings that can describe simplicial complexes. Stanley-Reisner rings are defined via square-free monomial ideals. The paper will present many aspects of the theory of these ideals and discuss how they relate to important constructions in commutative algebra, such as finite generation of ideals, graded rings and modules, localization and associated primes, primary decomposition of ideals and Hilbert series. In particular, the primary decomposition and Hilbert series for certain types of monomial ideals will be analyzed through explicit examples of simplicial complexes and graphs.

Made available in DSpace on 2015-05-15T11:46:03Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 435456 bytes, checksum: c38ab9cb93018e6c4d934dde6c174c07 (MD5) Previous issue date: 2012-02-16 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The present dissertation furnishes a detailed study about modules of logarithmic derivations, here dubbed tangential idealizers, and some of their main features. Initially, several comparisons between such modules are investigated starting from sufficiently related ideals, motivated by a previous study due to Kaplansky as well as by their close relationship with the classical theory of differential ideals of Seidenberg. We then obtain the first central result, which describes a primary decomposition of the tangential idealizer of an ideal without embedded primary component. Finally, in the second main result, we explore the structure of the derivation module for the class of Stanley-Reisner rings, thus corresponding to tangential idealizers of monomial ideals. An application of such a result is an affirmative answer for the homological Zariski-Lipman conjecture for the present class of rings. / A presente dissertação fornece um estudo detalhado sobre módulos de derivações logarítmicas, aqui denominados idealizadores tangenciais, bem como algumas de suas principais características. Inicialmente, várias comparações entre tais módulos são investigadas, a partir de ideais suficientemente relacionados, motivadas por um estudo prévio de Kaplansky e por sua estreita relação com a clássica teoria dos ideais diferenciais de Seidenberg. Em seguida obtém-se o primeiro resultado central, que descreve uma decomposição primária do idealizador tangencial de um ideal sem componente primária imersa. Finalmente, no segundo resultado principal, é explorada a estrutura do módulo de derivações para a classe de anéis de Stanley- Reisner, correspondendo portanto a idealizadores tangenciais de ideais monomiais. Uma aplicação de tal resultado é a resposta afirmativa para a conjectura homológica de Zariski-Lipman para a presente classe de anéis.

Making use of algebraic and combinatorial techniques, we study two topics: the arithmetic degree of squarefree strongly stable ideals and the h-vectors of matroid complexes. For a squarefree monomial ideal, I, the arithmetic degree of I is the number of facets of the simplicial complex which has I as its Stanley-Reisner ideal. We consider the case when I is squarefree strongly stable, in which case we give an exact formula for the arithmetic degree in terms of the minimal generators of I as well as a lower bound resembling that from the Multiplicity Conjecture. Using this, we can produce an upper bound on the number of minimal generators of any Cohen-Macaulay ideals with arbitrary codimension extending Dubreil’s theorem for codimension 2. A matroid complex is a pure complex such that every restriction is again pure. It is a long-standing open problem to classify all possible h-vectors of such complexes. In the case when the complex has dimension 1 we completely resolve this question and we give some partial results for higher dimensions. We also prove the 1-dimensional case of a conjecture of Stanley that all matroid h-vectors are pure O-sequences. Finally, we completely characterize the Stanley-Reisner ideals of matroid complexes.

In 1990, Fröberg presented a combinatorial classification of the quadratic square-free monomial ideals with linear resolutions. He showed that the edge ideal of a graph has a linear resolution if and only if the complement of the graph is chordal. Since then, a generalization of Fröberg's theorem to higher dimensions has been sought in order to classify all square-free monomial ideals with linear resolutions. Such a characterization would also give a description of all square-free monomial ideals which are Cohen-Macaulay. In this thesis we explore one method of extending Fröberg's result. We generalize the idea of a chordal graph to simplicial complexes and use simplicial homology as a bridge between this combinatorial notion and the algebraic concept of a linear resolution. We are able to give a generalization of one direction of Fröberg's theorem and, in investigating the converse direction, find a necessary and sufficient combinatorial condition for a square-free monomial ideal to have a linear resolution over fields of characteristic 2.

x, 68 p. : ill. (some col.) / Motivated by the Adams spectral sequence for computing stable homotopy groups, Priddy defined a class of algebras called Koszul algebras with nice homological properties. Many important algebras arising naturally in mathematics are Koszul, and the Koszul property is often tied to important structure in the settings which produced the algebras. However, the strong defining conditions for a Koszul algebra imply that such algebras must be quadratic. A very natural generalization of Koszul algebras called K 2 algebras was recently introduced by Cassidy and Shelton. Unlike other generalizations of the Koszul property, the class of K 2 algebras is closed under many standard operations in ring theory. The class of K 2 algebras includes Artin-Schelter regular algebras of global dimension 4 on three linear generators as well as graded complete intersections. Our work comprises two distinct projects. Each project was motivated by an aspect of the theory of Koszul algebras which we regard as sufficiently powerful or fundamental to warrant an interpretation for K 2 algebras. A very useful theorem due to Backelin and Fröberg states that if A is a Koszul algebra and I is a quadratic ideal of A which is Koszul as a left A -module, then the factor algebra A/I is a Koszul algebra. We prove that if A is Koszul algebra and A I is a K 2 module, then A/I is a K 2 algebra provided A/I acts trivially on Ext A ( A/I,k ). As an application of our theorem, we show that the class of sequentially Cohen-Macaulay Stanley-Reisner rings are K 2 algebras and we give examples that suggest the class of K 2 Stanley-Reisner rings is actually much larger. Another important recent development in ring theory is the use of A ∞ -algebras. One can characterize Koszul algebras as those graded algebras whose Yoneda algebra admits only trivial A ∞ -structure. We show that, in contrast to the situation for Koszul algebras, vanishing of higher A ∞ -structure on the Yoneda algebra of a K 2 algebra need not be determined in any obvious way by the degrees of defining relations. We also demonstrate that obvious patterns of vanishing among higher multiplications cannot detect the K 2 property. This dissertation includes previously unpublished co-authored material. / Committee in charge: Dr. Brad Shelton, Chair; Dr. Victor Ostrik, Member; Dr. Nicholas Proudfoot, Member; Dr. Arkady Vaintrob, Member; Dr. David Boush, Outside Member

This thesis treats combinatorial and topological properties of monoids with absorbing elements and their associated algebras.

Toric face rings are a generalization of Stanley-Reisner rings and affine monoid rings. New problems and results are obtained by a systematic study of toric face rings, shedding new lights to the understanding of Stanley-Reisner rings and affine monoid rings. We study algebra retracts of Stanley-Reisner rings, in particular, classify all the $\mathbb{Z}$-graded algebra retracts. We consider the Koszul property of toric face rings via Betti numbers and properties of the defining ideal. The last chapter is devoted to local cohomology of seminormal toric face rings and applications to singularities of toric face rings in positive characteristics.

Biological systems are commonly represented using networks consisting of interactions between various elements in the system. Reverse engineering, a method of mathematical modeling, is used to recover how the elements in the biological network are connected. These connections are encoded using wiring diagrams, which are directed graphs that describe how elements in a network affect one another. A signed wiring diagram provides additional information about the interactions between elements relating to activation and inhibition. Due to cost concerns, it is optimal to gain insight into biological networks with as few experiments and data as possible. Minimal wiring diagrams identify the minimal sets of variables for which a model that fits the data exists. Previously established algorithms to compute possible minimal wiring diagrams rely on the primary decomposition of ideals in polynomial rings. Stanley-Reisner theory provides a one-to-one correspondence between squarefree monomial ideals and abstract simplicial complexes. In this work, we use this correspondence to determine conditions under which a given set of inputs is guaranteed to have a unique signed minimal wiring diagram, regardless of the output assignment.