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University of Illinois – Urbana-Champaign

1. Seceleanu, Alexandra. The syzygy theorem and the weak Lefschetz Property.

Degree: PhD, 0439, 2011, University of Illinois – Urbana-Champaign

URL: http://hdl.handle.net/2142/26091

This thesis consists of two research topics in commutative algebra.
In the first chapter, a comprehensive analysis is given of the Weak Lefschetz property in the case of ideals generated by powers of linear forms in a standard graded polynomial ring of characteristic zero. The main point to take away from these developments is that, via the inverse system dictionary,
one is able to relate the failure of the Weak Lefschetz property to the geometry of the fat point scheme associated to the powers of linear forms. As a natural outcome of this research we describe conjectures on the asymptotical
behavior of the family of ideals that is being studied.
In the second chapter, we solve some relevant cases of the Evans-Griffith syzygy conjecture in the case of (regular) local rings of unramif ed mixed characteristic p, with the case of syzygies of prime ideals of Cohen-Macaulay
local rings of unramified mixed characteristic being noted. We reduce the remaining considerations to modules annihilated by p^s, s > 0, that have finite projective dimension over a hypersurface ring. Our main results are
obtained as a byproduct of two theorems that establish a weak order ideal property for kth syzygy modules under conditions allowing for comparison ofsyzygies over the original ring versus the hypersurface ring.
*Advisors/Committee Members: Schenck, Henry K. (advisor), Griffith, Phillip A. (Committee Chair), Schenck, Henry K. (committee member), Dutta, Sankar P. (committee member), Evans, Graham (committee member).*

Subjects/Keywords: syzygy; syzygy theorem; weak Lefschetz Property; fat points; homological conjectures

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Seceleanu, A. (2011). The syzygy theorem and the weak Lefschetz Property. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/26091

Chicago Manual of Style (16^{th} Edition):

Seceleanu, Alexandra. “The syzygy theorem and the weak Lefschetz Property.” 2011. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed September 25, 2020. http://hdl.handle.net/2142/26091.

MLA Handbook (7^{th} Edition):

Seceleanu, Alexandra. “The syzygy theorem and the weak Lefschetz Property.” 2011. Web. 25 Sep 2020.

Vancouver:

Seceleanu A. The syzygy theorem and the weak Lefschetz Property. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2011. [cited 2020 Sep 25]. Available from: http://hdl.handle.net/2142/26091.

Council of Science Editors:

Seceleanu A. The syzygy theorem and the weak Lefschetz Property. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2011. Available from: http://hdl.handle.net/2142/26091

2. Beder, Jesse. The Grade Conjecture and asymptotic intersection multiplicity.

Degree: PhD, 0439, 2013, University of Illinois – Urbana-Champaign

URL: http://hdl.handle.net/2142/42274

In this thesis, we study Peskine and Szpiro's Grade Conjecture and its connection with asymptotic intersection multiplicity χ_∞. Given an A-module M of finite projective dimension and a system of parameters x_{1}, …, x_{r} for M, we show, under certain assumptions on M, that χ_∞(M, A/\underline{x}) > 0. We also give a necessary and sufficient condition on M for the existence of a system of parameters \underline{x} with χ_∞(M, A/\underline{x}) > 0.
We then prove that if the Grade Conjecture holds for a given module M, then there is a system of parameters \underline{x} such that χ_∞(M, A/\underline{x}) > 0. We also prove the Grade Conjecture for complete equidimensional local rings in any characteristic.
*Advisors/Committee Members: Dutta, Sankar P. (advisor), Griffith, Phillip A. (Committee Chair), Dutta, Sankar P. (committee member), Schenck, Henry K. (committee member), Haboush, William J. (committee member).*

Subjects/Keywords: commutative algebra; grade conjecture; characteristic p; frobenius; intersection multiplicity

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Beder, J. (2013). The Grade Conjecture and asymptotic intersection multiplicity. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/42274

Chicago Manual of Style (16^{th} Edition):

Beder, Jesse. “The Grade Conjecture and asymptotic intersection multiplicity.” 2013. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed September 25, 2020. http://hdl.handle.net/2142/42274.

MLA Handbook (7^{th} Edition):

Beder, Jesse. “The Grade Conjecture and asymptotic intersection multiplicity.” 2013. Web. 25 Sep 2020.

Vancouver:

Beder J. The Grade Conjecture and asymptotic intersection multiplicity. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2013. [cited 2020 Sep 25]. Available from: http://hdl.handle.net/2142/42274.

Council of Science Editors:

Beder J. The Grade Conjecture and asymptotic intersection multiplicity. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2013. Available from: http://hdl.handle.net/2142/42274