#### Identifier

etd-04102014-151839

#### Degree

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Document Type

Dissertation

#### Abstract

In recent years, the combinatorial properties of monomials ideals and binomial ideals have been widely studied. In particular, combinatorial interpretations of minimal free resolutions have been given in both cases. In this present work, we will generalize existing techniques to obtain two new results. If Lambda is an integer lattice in the n-dimensional integers satisfying some mild conditions, S is the polynomial ring with n variables and R is the group algebra of S[Lambda], then the first result is resolutions of Lambda-invariant submodules of the Laurent polynomial ring in n variables as R-modules. A consequence will be the ability to resolve submodules of the polynomial ring with variables that a Lambda-cosets of the n-dimensional integers modulo Lambda. In particular ideals J of S modulo the lattice ideal associated to Lambda. Second, we will provide a detailed account in three dimensions on how to lift the aforementioned resolutions to resolutions of ideals with monomial and binomial generators in the 3-dimensional polynomial ring .

#### Date

2014

#### Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

#### Recommended Citation

McGuire, Trevor, "Combinatorial minimal free resolutions of ideals with monomial and binomial generators" (2014). *LSU Doctoral Dissertations*. 3146.

http://digitalcommons.lsu.edu/gradschool_dissertations/3146

#### Committee Chair

Madden, James