Decompositions of Betti Diagrams

In this dissertation, we are concerned with decompositions of Betti diagrams over standard graded rings and the information about that ring and its modules that can be recovered from these decompositions. In Chapter 2, we study the structure of modules over short Gorenstein graded rings and determine a necessary condition for a matrix of nonnegative integers to be the Betti diagram of such a module. We also describe the cone of Betti diagrams over the ring k[x, y]/(x 2 , y 2), and we provide an algorithm for decomposing Betti diagrams, even for modules of infinite projective dimension. Chapter 3 represents work done jointly with Christine Berkesch, Jesse Burke, and Daniel Erman. There we give a complete description of the cone of Betti diagrams over a standard graded hypersurface ring of the form k[x, y]/(q), where q is a homogeneous quadric. In this setting we also provide an algorithm for decomposing Betti diagrams. In both Chapters 2 and 3, the coefficients of the decompositions paint a picture of some aspect of the module theory over the ring. iii DEDICATION I dedicate this dissertation to Marilyn Johnson. Botte buona fa buon vino. iv ACKNOWLEDGMENTS I offer this dissertation along with my heartfelt gratitude to my advisors Lucho Avramov and Roger Wiegand for their guidance, patience, and support. In the cone of tremendous and inspiring mathematicians, they each span an extremal ray. Lucho and Roger, I have learned immeasurable amounts from both of you already, but I think it will take my entire career to unpack all the advice you have given me. I look forward to that immensely! I also thank my committee, Sri, for their comments (on the thesis and otherwise) and enormous thanks to Judy for being an incredible role model and mentor. Judy, if I can be one iota as awesome as you are, I will consider my career a success. I have been lucky to have had the best officemates throughout graduate school. The award for tolerating me the longest will have to be shared by Amanda Croll and Nathan Corwin; they put up with me for five years. They never even redeemed their Be-Quiet-Courtney cards! Amanda, it has been a joy to become a commuta-tive algebraist alongside you. I look forward to our future work together, and our lifelong friendship! Nathan, here it is: Highlight of math class:/the sun glancing off Nathan's/breathtaking cheekbones. have also survived …