<p>Consider the standard graded polynomial ring in $n$ variables over a field $k$ and fix the Hilbert function of a homogeneous ideal. In the nineties Bigatti, Hulett, and Pardue showed that the Hilbert scheme consisting of all the homogeneous ideals with such a Hilbert function contains an extremal point which simultaneously maximizes all the graded Betti numbers. Such a point is the unique lexsegment ideal associated to the fixed Hilbert function.</p> <p> For such a scheme, we consider the individual strata defined by all ideals with Castelnuovo-Mumford regularity bounded above by <em>m</em>. In 1997 Mall showed that when <em>k </em>is of characteristic 0 there exists an ideal in each nonempty strata with maximal possible Betti numbers among the ideals of the strata. In chapter 4 of this thesis we provide a new construction of Mall's ideal, extend the result to fields of any characteristic, and show that these ideals have other extremal properties. For example, Mall's ideals satisfy an equation similar to Green's hyperplane section theorem.</p> <p> The key technical component needed to extend the results of Mall is discussed in Chapter 3. This component is the construction of a new invariant called the distraction-generic initial ideal. Given a homogeneous ideal <em>I C S</em> we construct the associated distraction-generic initial ideal, D-gin_< (<em>I</em>), by iteratively computing initial ideals and general distractions. The result is a monomial ideal that is strongly stable in any characteristic and which has many properties analogous to the generic initial ideal of <em>I</em>.</p>

<p dir="ltr">This thesis consists of four parts that study different topics in commutative algebra. The main results of the first part of the dissertation are in Chapter 3, which is based on the author’s paper [1]. Let R be a commutative Noetherian ring graded by a torsionfree abelian group G. We introduce the notion of G-graded irreducibility and prove that G-graded irreducibility is equivalent to irreducibility in the usual sense. This is a generalization of a result by Chen and Kim in the Z-graded case. We also discuss the concept of the index of reducibility and give an inequality for the indices of reducibility between any radical non-graded ideal and its largest graded subideal. The second topic is developed in Chapter 4 which is based on the author’s paper [2]. In this chapter, we prove that if P is a prime ideal of inside a polynomial ring S with dim S/P = r, and adjoining s general linear forms to the prime ideal changes the (r − s)-th Hilbert coefficient of the quotient ring by 1 and doesn’t change the 0th to (r − s − 1)-th Hilbert coefficients where s ≤ r, then the depth of S/P is n − s − 1. This criterion also tells us about possible restrictions on the generic initial ideal of a prime ideal inside a polynomial ring. The third part of the thesis is Chapter 5 which is based on the author’s paper [3]. Let R be a polynomial ring over a field. We introduce the concept of sequentially almost Cohen-Macaulay modules, describe the extremal rays of the cone of local cohomology tables of finitely generated graded R-modules which are sequentially almost Cohen-Macaulay, and also describe some cases when the local cohomology table of a module of dimension 3 has a nontrivial decomposition. The last part is Chapter 6 which is based on the author’s paper [4]. We introduce the notion of strongly Lech-independent ideals as a generalization of Lech-independent ideals defined by Lech and Hanes, and use this notion to derive inequalities on multiplicities of ideals. In particular, we prove a new case of Lech’s conjecture, namely, if (R, m) → (S, n) is a flat local extension of local rings with dim R = dim S, the completion of S is the completion of a standard graded ring over a field k with respect to the homogeneous maximal ideal, and the completion of mS is the completion of a homogeneous ideal, then e(R) ≤ e(S).</p>