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One of the common invariants of a graded module over a graded commutative ring is the Betti number. For any graded minimal free resolution F. of a graded R-module, we have corresponding Betti numbers that record information about the grading of F.. Using a speciﬁc index, we can construct a Betti diagram with Betti numbers as entries. Inspired by a set of conjectures of M. Boij and J. Söderberg, an algorithm was given by D. Eisenbud and F. Schreyer allowing the decomposition of Betti diagrams into pure diagrams. In this thesis, we explore the basic concepts of Boij-Söderberg theory, including the construction of minimal free resolutions of graded R-modules, Betti diagrams, and Betti decomposition. We investigate the relationship between the Betti decompositions of graded R-modules that form a short exact sequence and ﬁnd that there is a class of short exact sequences of modules such that the Betti decomposition of the middle module is equivalent to the sum of the Betti decompositions of the outer two modules. We also examine the decompositions of Betti diagrams over a special kind of ring called a complete intersection, which furthers the results of C. Gibbons, J. Jeﬀries, S. Mayes, C. Raicu, B. Stone, B. White (2012) to codimension 4.
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Wyrick-Flax, Fanya R., "Algebraic relations and Boij-Söderberg theory" (2013). Senior Projects Spring 2013. 363.