Multigraded and Symmetric Syzygies

This dissertation studies the resolutions of and over quotients of polynomial rings in both multigraded and symmetric settings. The first main results deal with multiplicative structures on the multigraded Koszul homology a monomial ring, with applications to resolutions over those rings in the general and symmetric setting. We then describe the syzygies of the ideal of permanents of a 2 by n generic matrix in terms of multigrading and symmetric group representations. In Chapter 2, we study special multigraded cycles of a Koszul complex called monomial cycles. Let S be a polynomial ring over a field, R be a quotient of S by a monomial ideal, and K be the Koszul complex of the variables over S. We determine when a cycle in K tensored with R with a single monomial coefficient corresponds to a non-zero homology class. We prove that such a cycle is a boundary if and only if the coefficient is contained in an ideal we introduce called the boundary ideal. This gives insight as to when the natural multiplication structure on Koszul homology vanishes, which has applications to resolutions over R. Our main application is a classification of Golod monomial ideals in four or fewer variables. In Chapter 3, we determine a class of monomial ideals for which the Koszul homology groups exhibit a basis consisting entirely of monomial cycles. This class includes the well-known case of the stable monomial ideals. In Chapter 4, we apply the results of Chapter 2 to the symmetric monomial ideals. We determine when products of monomial cycles vanish in terms of the generating partitions of the ideal. In Chapter 5, we describe the multigraded symmetric syzygies of the ideal of 2 by 2 permanents of a generic 2 by n matrix.