Vanishing of Ext and Tor over complete intersections

Let (R,m) be a local complete intersection, that is, a local ring whose m-adic completion is the quotient of a complete regular local ring by a regular sequence. Let M and N be finitely generated R-modules. This dissertation concerns the vanishing of Tor(M, N) and Ext(M, N). In this context, M satisfies Serre's condition (S_{n}) if and only if M is an nth syzygy. The complexity of M is the least nonnegative integer r such that the nth Betti number of M is bounded by a polynomial of degree r-1 for all sufficiently large n. We use this notion of Serre's condition and complexity to study the vanishing of Tor_{i}(M, N). In particular, building on results of C. Huneke, D. Jorgensen and R. Wiegand [32], and H. Dao [21], we obtain new results showing that good depth properties on the R-modules M, N and MtensorN force the vanishing of Tor_{i}(M, N) for all i>0. We give examples showing that our results are sharp. We also show that if R is a one-dimensional domain and M and MtensorHom(M,R) are torsion-free, then M is free if and only if M has complexity at most one. If R is a hypersurface and Ext^{i}(M, N) has finite length for all i>>0, then the Herbrand difference [18] is defined as length(Ext^{2n}(M, N))-(Ext^{2n-1}(M, N)) for some (equivalently, every) sufficiently large integer n. In joint work with Hailong Dao, we generalize and study the Herbrand difference. Using the Grothendieck group of finitely generated R-modules, we also examined the number of consecutive vanishing of Ext^{i}(M, N) needed to ensure that Ext^{i}(M, N) = 0 for all i>>0. Our results recover and improve on most of the known bounds in the literature, especially when R has dimension two.

Contesting Sphere Boundaries Online: Private/Technical/Public Discourses in Polycystic Ovarian Syndrome Discussion Groups

The internet is fast becoming a means for people to obtain information, creating a unique forum for the intersection of the public, technical, and private spheres. To ground my research theoretically, I used Jürgen Habermas’s sphere theory. Habermas (1987) explains that the technical sphere colonizes the private sphere, which decreases democratic potential. In particular, the internet is a place for altering technical colonization of the private and public spheres. My research focuses on women’s health because it is a particularly useful case study for examining sphere tensions. Historically, the biomedical health establishment has been a powerful agent of colonization, resulting in detrimental effects for women and their health. The purpose of this study is to examine how the internet encourages expert and female patient deliberation, which empowers women to challenge the experts and, thus, make conversations between the private/technical spheres more democratic. I used PCOS (Polycystic Ovarian Syndrome) as a case to observe the changing sphere boundaries by studying the discourse that took place on multiple patient and doctor websites over a four-year period. Through my research, I found that the PCOS women challenge the biomedical model by appropriating medical language. By understanding the medical talk, the women are able to feel confident when discussing their health conditions with the doctor and with each other. The PCOS women also become lay-experts who have personal and medical experience with PCOS, reducing private sphere colonization. This case study exemplifies how female empowerment can influence expert culture, challenging our conventional understanding of democracy.

Class Notes for Math 905: Commutative Algebra, Instructor Sylvia Wiegand

Topics include: Rings, ideals, algebraic sets and affine varieties, modules, localizations, tensor products, intersection multiplicities, primary decomposition, the Nullstellensatz

Class Notes for Math 918: Homological Conjectures, Instructor Tom Marley

This course was an overview of what are known as the “Homological Conjectures,” in particular, the Zero Divisor Conjecture, the Rigidity Conjecture, the Intersection Conjectures, Bass’ Conjecture, the Superheight Conjecture, the Direct Summand Conjecture, the Monomial Conjecture, the Syzygy Conjecture, and the big and small Cohen Macaulay Conjectures. Many of these are shown to imply others. This document contains notes for a course taught by Tom Marley during the 2009 spring semester at the University of Nebraska-Lincoln. The notes loosely follow the treatment given in Chapters 8 and 9 of Cohen-Macaulay Rings, by W. Bruns and J. Herzog, although many other sources, including articles and monographs by Peskine, Szpiro, Hochster, Huneke, Grith, Evans, Lyubeznik, and Roberts (to name a few), were used. Special thanks to Laura Lynch for putting these notes into LaTeX.