From: BrendaSueCook (brenda@math.missouri.edu)
Date: Wed Jul 18 2007 - 13:32:20 CDT
Ph.D. DEFENSE
Department of Mathematics
Sabine El-Khoury
University of Missouri-Columbia
A Class of Artinian Gorenstein Algebras of Embedding Dimension Four
Abstract: An ideal $I$ in a commutative noetherian ring $R$ is a
Gorenstein ideal of height $g$ if height $I = \mbox{pd}_R$ $R/I = g$ and
the canonical module $\mbox{Ext}^g_R(R/I,R)$ is a cyclic $R/I$-module.
Serre showed that if $g = 2$ then $I$ is a complete intersection, and
Buchsbaum and Eisenbud proved a structure theorem for the case $g = 3$.
When $g = 4$, Kustin and Miller proved a structure theorem for ideals
generated by $(f,g,h,wJ)$ where $(f,g,h)$ is a regular sequence and $J$
is height three Gorenstein. In our case we let $I$ be a homogeneous
height $4$ Gorenstein ideal in $R = K[x,y,z,w]$ generated in degree $2$
or higher. Let $I_t$ be the ideal generated by the elements of degree
$t$ in $I$. Suppose that the height of $I_2 = 1$ then $I_2 = (wx, wy,
wz)$ or $I_2 = (wx,wy,w^2)$ after a linear change of variables.
Iaorrobino and Srinivasan proved a structure theorem for ideals $I$
where $I_2 = (wx, wy, wz)$. This then we construct the free resolution
and the structure for a class of ideals $I$ with $I_2 = (wx,wy,w^2)$.
Dissertation Advisor: Hema Srinivasan
3:30 p.m.
Monday, July 23
312 Math Sciences
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