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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]/(x2,y2), 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.
Advisors: Professors Luchezar Avramov and Roger Wiegand