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Thesis (Ph. D.)--University of Wisconsin--Madison, 1964. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaf 39).

archives@tulane.edu / The primary objects studied in this dissertation are ordinary and symbolic powers of monomial ideals in a polynomial ring over a field. In particular, we are interested in studying their local cohomology and Castlenuovo-Mumford regularity. In Chapter 3, we restrict our study to edge ideals of unicyclic graphs, that is, squarefree monomial ideals generated in degree $2$ corresponding to a graph that has a single cycle. When the cycle is even, the symbolic power was known to coincide with the ordinary power. When the cycle is odd, we are able to describe the symbolic powers explicitly, which allows us to compute invariants of the ideal explicitly. Furthermore, in certain cases, we can calculate the Castelnuovo-Mumford regularity. In Chapter 4, we study ideals that can be written as the sum of monomial ideals in different polynomial rings. In order to study the graded local cohomology of these ideals, we use a formula of Takayama which allows us to translate this problem of computing homology of certain simplicial complexes called \textit{degree complexes}. We build up the construction of the degree complexes of ordinary and symbolic powers of sums, and then we use this to discuss their graded local cohomologies. In Chapter 5, we study ideals that can be written as the fiber product of squarefree monomial ideals in different polynomial rings. Building on the construction from Chapter 4, we are able to determine that the nonempty faces in the degree complex of ordinary and symbolic powers of fiber products come from the faces of the degree complexes of powers of the component ideals. This allows us to compute the homology of these degree complexes explicitly. Furthermore, this allows us to compute the regularity of symbolic powers of fiber products of squarefree monomial ideals in terms of the regularities of the component ideals. / 1 / Jonathan O'Rourke

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This thesis is a study of some properties of prime ideals in commutative rings with unity.

Neste trabalho, estudamos a multiplicidade de Hilbert-Samuel, e suas possíveis generalizações, tais como números de Segre e a sequência de multiplicidades de Achilles e Manaresi / In this work is studied the multiplicity of Hilbert-Samuel and its possible generalizations, such as Segre numbers and sequence of multiplicities of Achilles and Manaresi

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acase@tulane.edu / Let $G$ be a finite simple graph and let $I = I(G)$ be its edge ideal. Main goal in this thesis is to relate algebraic invariants of powers of edge ideals and combinatorial data of graphs. In particular, we focus on the Castelnuovo-Mumford regularity of an edge ideal and its powers. The first part of this thesis focuses on regularity of edge ideals. In that regard, we give new bounds on the regularity of $I$ when $G$ contains a Hamiltonian path and when $G$ is a Hamiltonian graph. Moreover, we explicitly compute the regularity of unicyclic graphs and characterize the unicyclic graphs with regularity $\nu(G)+1$ and $\nu(G)+2$ where $\nu(G)$ denotes the induced matching number of $G.$ The second problem is on the regularity of powers of edge ideals. Let $R=k[x_1, \ldots, x_n]$ be a polynomial ring and let $I \subset R$ be a homogeneous ideal. It is a celebrated result of Cutkosky, Herzog,Trung \cite{CHT}, Kodiyalam \cite{Kodi}, Trung and Wang \cite{TW} that regularity of $I^s$ is asymptotically a linear function for $s \gg 0,$ i.e., $as+b$ for integers $a,b$ and $s_0$ when $s \geq s_0.$ It is known that $a$ is equal to 2 when $I=I(G)$ is the edge ideal of a graph. We then turn on our focus on identifying $b$ and $s_0$ via combinatorial data of the graph $G.$ We explicitly compute the regularity of $I^s$ for all $s\geq 1$ when $G$ is a forest, a cycle and a unicyclic graph. We also present a lower bound on the regularity of powers of edge ideals in terms of the induced matching number of a graph. / 1 / Selvi Beyarslan

The class of squarefree monomial ideals is a classical object in commutative algebra, which has a strong connection to combinatorics. Our main goal throughout this dissertation is to study the algebraic properties of squarefree monomial ideals using combinatorial structures and invariants of hypergraphs. We focus on the following algebraic properties and invariants: the persistence property, non-increasing depth property, Castelnuovo-Mumford regularity and projective dimension. It has been believed for a long time that squarefree monomial ideals satisfy the persistence property and non-increasing depth property. In a recent work, Kaiser, Stehlik and Skrekovski provided a family of graphs and showed that the cover ideal of the smallest member of this family gives a counterexample to the persistence and non-increasing depth properties. We show that the cover ideals of all members of their family of graphs indeed fail to have the persistence and non-increasing depth properties. Castelnuovo-Mumford regularity and projective dimension are both important invariants in commutative algebra and algebraic geometry that govern the computational complexity of ideals and modules. Our focus is on finding bounds for the regularity in terms of combinatorial data from associated hypergraphs. We provide two upper bounds for the edge ideal of any vertex decomposable graph in terms of induced matching number and the number of cycles. We then give an upper bound for the edge ideal of a special class of vertex decomposable hypergraphs. Moreover, we generalize a domination parameter from graphs to hypergraphs and use it to give an upper bound for the projective dimension of the edge ideal of any hypergraph. / Mengyao Sun