Periodic modules over Gorenstein local rings

It is proved that the minimal free resolution of a module M over a Gorenstein local ring R is eventually periodic if, and only if, the class of M is torsion in a certain Z[t ±1 ]-module associated to R. This module, denoted J(R), is the free Z[t ±1 ]-module on the isomorphism classes of finitely generated R-modules modulo relations reminiscent of those defining the Grothendieck group of R. The main result is a structure theorem for J(R) when R is a complete Gorenstein local ring; the link between peri-odicity and torsion stated above is a corollary. iii DEDICATION To Nicholas, for everything iv ACKNOWLEDGMENTS I owe my advisor, Srikanth Iyengar, more than words can express for all that he has done for me over the last four years. Thank you, Sri, for your patience, for sharing with me your enthusiasm for mathematics, and for helping me grow immensely as a mathematician. Your enthusiasm is contagious even when mathematics is frustrating, and a conversation with you always gives me the motivation I need to continue. Sri, I believe you are exactly the advisor I needed to succeed in graduate school. Thank you for the advice and support you have provided for me, both academically and personally. I would also like to thank my committee: My experience as a graduate student at UNL has been amazing. Not only do I have an exceptional advisor, but the entire Mathematics Department has been supportive and encouraging. The faculty and staff truly care about graduate students and their academic success as well as their emotional stability. Thank you each for your support and encouragement, and a special thank you to Marilyn for many conversations that kept me sane. The department also provides many opportunities for the development of graduate students as teachers, as academics, and as future faculty members. Thank you to the faculty who provide these opportunities for graduate students. I also owe many thanks to my fellow graduate students for their friendship and support during my time at UNL. For the last five years, I have had the privilege of sharing an office with Courtney Gibbons and Nathan Corwin. Courtney, it has been wonderful to grow mathematically with someone who is also a close friend. I will forever have memories of working through Matsumura and Bruns & Herzog v with you and attending countless conferences together. I also owe Courtney a very …