Resolutions And Tor Algebra Structures For Trivariate Monomial Ideals

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Resolutions And Tor Algebra Structures For Trivariate Monomial Ideals

Show simple item record Painter, Jared Lafayette en_US 2012-07-25T19:08:44Z 2012-07-25T19:08:44Z 2012-07-25 January 2012 en_US
dc.identifier.other DISS-11702 en_US
dc.description.abstract 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 [4] 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. en_US
dc.description.sponsorship Jorgensen, David en_US
dc.language.iso en en_US
dc.publisher Mathematics en_US
dc.title Resolutions And Tor Algebra Structures For Trivariate Monomial Ideals en_US
dc.type Ph.D. en_US
dc.contributor.committeeChair Jorgensen, David en_US Mathematics en_US Mathematics en_US University of Texas at Arlington en_US doctoral en_US Ph.D. en_US

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