The syzygy theorem and the weak Lefschetz Property

syzygy;syzygy theorem;weak Lefschetz Property;fat points;homological conjectures

Seceleanu, Alexandra

关键词: syzygy;    syzygy theorem;    weak Lefschetz Property;    fat points;    homological conjectures;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/26091/Seceleanu_Alexandra.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

This thesis consists of two research topics in commutative algebra.In thefirst 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 asymptoticalbehavior 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-Macaulaylocal 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 areobtained 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.