Minimal homogeneous resolutions, almost complete intersections and Gorenstein Artin algebras
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[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] This work is devoted to the study of the structures of the graded resolutions of codimension three almost complete intersections and the unimodality and SI-sequence of Hilbert functions of Gorenstein Artin algebras in codimension four. One of the main conjectures in Commutative Algebra is the Multiplicity Conjecture by J. Herzog, C. Huneke and H. Srinivasan. They conjectured bounds for multiplicity in terms of the shifts in the minimal graded resolutions. We established the structure of the graded resolution of codimension three almost complete intersections. Using these structure theorems, we proved the multiplicity conjecture for almost complete intersections of codimension three. We proved that the Hilbert functions of codimension four graded Gorenstein Artin algebras R/I are unimodal provided I has a minimal generator in degree less than fi ve, i.e., all Gorenstein h-vectors (1; h1; h2; h3; h4; ...) are unimodal if h4 ≤ 34. This extends the results of Iarrobino-Srinivsan (2005) and Migliore-Nagel-Zanello (2008). It is still an open question whether all Gorenstein h-vectors in codimension four are SI-sequences. It is not even known if they are all unimodal.
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