Controlling the Dimensions of Formal Fibers of a Unique Factorization Domain at the Height One Prime Ideals

TL;DR

This paper constructs a local UFD with prescribed dimensions of formal fibers at height one primes and the zero ideal, based on a complete local ring with specific properties.

Contribution

It introduces a method to control the dimensions of formal fibers of a UFD at height one primes and the zero ideal, extending prior understanding of formal fiber structures.

Findings

Constructed a local UFD with specified formal fiber dimensions.

Demonstrated the existence of rings with controlled formal fiber dimensions.

Extended the theory of formal fibers in Noetherian local rings.

Abstract

Let T be a complete local (Noetherian) equidimensional ring with maximal ideal m such that the Krull dimension of T is at least two and the depth of T is at least two. Suppose that no integer of T is a zerodivisor and that |T|=|T/m|. Let d and t be integers such that 1 $\leq$ d $\leq$ dimT-1, 0 $\leq$ t $\leq$ dimT - 1, and d - 1 $\leq$ t. Assume that, for every p in AssT, ht(p) $\leq$ d-1 and that if z is a regular element of T and Q is in Ass(T/zT), then ht(Q) $\leq$ d. We construct a local unique factorization domain A such that the completion of A is T and such that the dimension of the formal fiber ring at every height one prime ideal of A is d - 1 and the dimension of the formal fiber ring of A at (0) is t.