The syzygy theorem and the Weak Lefschetz property.

摘要

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 unramified 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 ps, 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 k th syzygy modules under conditions allowing for comparison of syzygies over the original ring versus the hypersurface ring.