Some Algebraic Invariants of Squarefree Monomial Ideals

Abstract

Hilbert gave the idea of associating free resolution with finitely generated module to describe the structure of a module. Since then, there has been a lot of progress on the structure and properties of finite free resolutions. The two algebraic invariants namely Castelnuovo-Mumford regularity (or regularity) and projective dimension are associated with minimal graded free resolution of a finitely generated graded module. Regularity measures the complexity of module and projective dimension measures how far a module is from being projective. Projective dimension has a relation with the depth of a module by Auslander–Buchsbaum formula. The depth of a module has been the subject of several studies during the last decades. Let A be a finitely generated multigraded module. In 1982, Stanley conjectured that depth of A is a lower bound for the Stanley depth of A. This conjecture was later disproved by Dual et al. in 2015. However, there still looks to be profound and attractive relationship between the two invariants, which is yet to be understood. Squarefree monomial ideals has been a fascinating area of study in commutative algebra and has a strong connection to combinatorics, which continues to inspire muchofcurrent research. The goal of this thesis is to study some algebraic invariants of quotient rings of some squarefree monomial ideals. These algebraic invariants include depth, Stanley depth, regularity, projective dimension, and Krull dimension. We find the precise values of aforementioned invariants of residue class rings of edge ideals of perfect semiregular trees. We f ind depth, projective dimension and lower bounds for Stanley depth of the quotient rings of edge ideals associated with all cubic circulant graphs. We discuss the said invariants for the quotient rings of the edge ideals associated with some classes of four and five regular circulant graphs.