Resolutions And Tor Algebra Structures For Trivariate Monomial Ideals
Painter, Jared Lafayette
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In this manuscript we explore properties of minimal free resolutions and their relationship to the Tor-algebra structure for trivariate monomial ideals. We begin with an in-depth 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 non-generic monomial ideals. We will find that our results on free resolutions relate to the Tor-algebra structure for S. In  Avramov describes the Tor-algebra 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 Tor-algebra structure from the minimal free resolution of S. Finally, we will find classes of monomial ideals with the desired Tor-algebra structure, and give a complete classification for generic monomial ideals.