# Inequalities related to Lech's conjecture and other problems in local and graded algebra

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).

## History

## Degree Type

- Doctor of Philosophy

## Department

- Mathematics

## Campus location

- West Lafayette