Modules with Pure Resolutions

We show that the property of a standard graded algebra R being Cohen-Macaulay is characterized by the existence of a pure Cohen-Macaulay R-module corresponding to any degree sequence of length at most depth(R). We also give a relation in terms of graded Betti numbers, called the Herzog-Kuhl equations, for a pure R-module M to satisfy the condition dim(R) - depth(R) = dim(M) - depth(M). When R is Cohen-Macaulay, we prove an analogous result characterizing all graded Cohen-Macaulay R-modules.Comment: 9 page

Betti cones over fibre products

Let $R$ be a fibre product of standard graded algebras over a field. We study the structure of syzygies of finitely generated graded $R$-modules. As an application of this, we show that the existence of an $R$-module of finite regularity and infinite projective dimension forces $R$ to be Koszul. We also look at the extremal rays of the Betti cone of finitely generated graded $R$-modules, and show that when $\operatorname{depth}(R)=1$, they are spanned by the Betti tables of pure $R$-modules if and only if $R$ is Cohen-Macaulay with minimal multiplicity.Comment: Comments welcome

Approximating Artinian Rings by Gorenstein Rings and 3-Standardness of the Maximal Ideal

We study two different problems in this dissertation. In the first part, we wish to understand how one can approximate an Artinian local ring by a Gorenstein Artin local ring. We make this notion precise in Chapter 2, by introducing a number associated to an Artin local ring, called its Gorenstein colength . We study the basic properties and give bounds on this number in this chapter. We extend results due to W. Teter, C.Huneke and A. Vraciu by studying the relation of Gorenstein colength with self-dual ideals. In particular, we also answer the question as to when the Gorenstein colength is at most two. In Chapter 3, we show that there is a natural upper bound for Gorenstein colength of some special rings. We compute the Gorenstein colengths of these rings by constructing some Gorenstein Artin rings. We further show that the Gorenstein colength of Artinian quotients of two-dimensional regular local rings are also bounded above by the same upper bound by using a formula due to Hoskin and Deligne. Given two Gorenstein Artin local rings, L. Avramov and W. F. Moore construct another Gorenstein Artin local ring called a connected sum. We use this to improve a result of C. Huneke and A. Vraciu in Chapter 4. We also define the notion of a connected sum more generally and apply it to give bounds on the Gorenstein colengths of fibre products of Artinian local rings. In the second part of the thesis, we study a notion called n-standardness of ideals primary to the maximal ideal in a Cohen-Macaulay local ring. We first prove the equivalence of n-standardness to the vanishing of a certain Koszul homology module up to a certain degree. We go over the properties of Koszul complexes and homology needed for this purpose in Chapter 5. In Chapter 6, we study conditions under which the maximal ideal is 3-standard. We first prove results when the residue field is of prime characteristic and use the method of reduction to prime characteristic to extend the results to the characteristic zero case. As an application, we see that this helps us extend a result due to T. Puthenpurakal in which he shows that a certain length associated to a minimal reduction of the maximal ideal does not depend on the minimal reduction chosen

Linear quotients of connected ideals of graphs

As a higher analogue of the edge ideal of a graph, we study the $t$-connected ideal $\operatorname{J}_{t}$. This is the monomial ideal generated by the connected subsets of size $t$. For trees, we show that $\operatorname{J}_{t}$ has a linear resolution iff the tree is $t$-gap-free, and that this is equivalent to having linear quotients. We then show that if $G$ is any gap-free and $t$-claw-free graph, then $\operatorname{J}_{t}(G)$ has linear quotients and hence, linear resolution.Comment: Comments welcome

Associated graded rings and connected sums

summary:In 2012, Ananthnarayan, Avramov and Moore gave a new construction of Gorenstein rings from two Gorenstein local rings, called their connected sum. In this article, we investigate conditions on the associated graded ring of a Gorenstein Artin local ring $Q$, which force it to be a connected sum over its residue field. In particular, we recover some results regarding short, and stretched, Gorenstein Artin rings. Finally, using these decompositions, we obtain results about the rationality of the Poincaré series of $Q$

Monitoring Radiographic Brain Tumor Progression

Determining radiographic progression in primary malignant brain tumors has posed a significant challenge to the neuroncology community. Glioblastoma multiforme (GBM, WHO Grade IV) through its inherent heterogeneous enhancement, growth patterns, and irregular nature has been difficult to assess for progression. Our ability to detect tumor progression radiographically remains inadequate. Despite the advanced imaging techniques, detecting tumor progression continues to be a clinical challenge. Here we review the different criteria used to detect tumor progression, and highlight the inherent challenges with detection of progression