# Local cohomology and regularity of powers of monomial ideals

## Description

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.