Combinatorial and Homological Aspects of Monomial Algebras and Numerical Semigroups

Abstract

Our aim throughout this thesis is to illuminate combinatorial and homological properties of algebraic structures arising in combinatorial commutative algebra, combinatorics, and additive number theory. We devote specific attention to Noetherian (standard graded) local rings (with infinite residue fields) that admit desirable properties, e.g., analytically unramified one-dimensional Cohen-Macaulay local rings and monomial algebras such as (i.) numerical semigroup rings, (ii.) edge rings of finite simple graphs, and (iii.) generalized two-dimensional Veronese subrings. We introduce two new classes of non-Gorenstein Cohen-Macaulay local rings --- namely the Gorenstein canonical blow-up (GCB) rings and divisive numerical semigroup rings --- in Chapter 3. We demonstrate that Arf rings, far-flung Gorenstein rings, nearly Gorenstein rings of minimal multiplicity, numerical semigroup rings of multiplicity at most three, and divisive numerical semigroup rings are GCB. We define two new invariants of Noetherian (standard graded) local rings in Chapter 4. We illustrate that these invariants refine the notion of embedding dimension and relate to reductions of the maximal ideal of reduction number one. We provide general bounds for these invariants and compute them explicitly in some cases. We offer a treatise on the invariants for standard graded algebras over fields and edge rings of finite simple graphs, and we demonstrate that these invariants give rise to subtle algebraic invariants of finite simple graphs. Last, in Chapter 5, we introduce a generalization of two-dimensional Veronese subrings --- called pseudo-Veronese subrings --- and we prove that their homological properties are determined by the underlying monomial generators.