On complete ideals in regular local rings.(English)Zbl 0693.13011

Algebraic geometry and commutative algebra, Vol. I, 203-231 (1988).
[For the entire collection see Zbl 0655.00011.]
In appendix 5 of volume II of a famous text he coauthored, O. Zariski proved that in a 2-dimensional regular local ring, every complete (i.e., integrally closed) ideal can be uniquely factored into a product of simple complete ideals. He also states that his arguments do not seem to generalize to higher dimension, at least not without substantial revision.
The present author, using a different approach has found a partial generalization of Zariski’s result. He shows that in a regular local ring R of arbitrary degree $$at\quad least\quad 2,$$ there is an interesting set of complete ideals (those with finite support) for which a unique factorization of sorts does hold. When applied to the 2-dimensional case, Zariski’s result is recovered.
Although space prohibits more detail here, this paper is fairly easy to read, and is well worth the effort.