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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 (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 24, 2020. http://hdl.handle.net/2142/26091.

MLA Handbook (7^{th} Edition):

Seceleanu, Alexandra. “The syzygy theorem and the weak Lefschetz Property.” 2011. Web. 24 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 24]. 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