Structure Theory for a Class of Grade Four Gorenstein Ideals

An ideal $I$ in a commutative noetherian ring $R$ is a Gorenstein ideal of $\operatorname {grade} g$ if ${\operatorname {pd} _R}(R / I) = \operatorname {grade} I = g$ and the canonical module $\operatorname {Ext} _R^g(R / I, R)$ is cyclic. Serre showed that if $g = 2$ then $I$ is a complete intersection, and Buchsbaum and Eisenbud proved a structure theorem for the case $g = 3$. We present generic resolutions for a class of Gorenstein ideals of $\operatorname {grade} 4$, and we illustrate the structure of the resolution with various specializations. Among these examples there are Gorenstein ideals of $\operatorname {grade} 4$ in $k[[x, y, z, v]]$ that are $n$-generated for any odd integer $n \geqslant 7$. We construct other examples from almost complete intersections of $\operatorname {grade} 3$ and their canonical modules. In the generic case the ideals are shown to be normal primes. Finally, we conclude by giving an explicit associative algebra structure for the resolutions. It is this algebra structure that we use to classify the different Gorenstein ideals of $\operatorname {grade} 4$, and which may be the key to a complete structure theorem.