Abstract:

In this manuscript we explore properties of minimal free resolutions and their relationship to the Toralgebra structure for trivariate monomial ideals. We begin with an indepth analysis of minimal free resolutions of S = R / I, where R = k[x; y; z] is a polynomial ring over a field k, and I is a monomial ideal that is primary to the homogeneous maximal ideal m of R. We will de ne a special form of the minimal free resolution of S, and then determine when we get nonzero elements from I as entries in the matrices of the resolution. We find a complete answer to this question for the second matrix of our special resolution for all trivariate monomial ideals. For the third matrix, we provide a complete answer for generic monomial ideals. We also observe differences for resolutions of generic monomial ideals in comparison to nongeneric monomial ideals. We will find that our results on free resolutions relate to the Toralgebra structure for S. In [4] Avramov describes the Toralgebra structure A = TorR(k; S), for rings of codepth 3. His description of this structure is comprised of 5 categories. We will explore this structure, and will determine which of the 5 categories can be realized by monomial ideals. We will also learn how to describe the Toralgebra structure from the minimal free resolution of S. Finally, we will find classes of monomial ideals with the desired Toralgebra structure, and give a complete classification for generic monomial ideals. 