Characterization of completions of unique factorization domains

Abstract

Abstract. It is shown that a complete local ring is the completion of a unique factorization domain if and only if it is a field, a discrete valuation ring, or it has depth at least two and no element of its prime ring is a zerodivisor. It is also shown that the Normal Chain Conjecture is false and that there exist local noncatenary UFDs.

Semilocal generic formal fibers

“…Our construction is based on the techniques used in [3] and is inspired by the construction of Heitmann in [1]. We start with the prime subring of T , localized at the appropriate prime ideal.…”

Unique factorization in invariant power series rings

“…Examples of unique factorization domains whose completions are not unique factorization domains were known previously (see Fossum [5,Example 19.9] and Halanay [8]), but our examples seem particularly natural. In fact, Heitmann [9,Theorem 8] has shown that every complete local domain of depth at least two is the completion of a subring which is a unique factorization domain; this gives a wealth of complicated examples.…”

Completions of UFDs with Semi-Local Formal Fibers

“…Suppose that T is the completion of a UFD A with semilocal generic formal fiber G. From Heitmann (1993), we know that T must be a field, a DVR, or have depth at least two and no integer of T is a zerodivisor. Now, let P ∈ Ass T with ht P > 0, and let Q be a prime ideal containing P such that ht Q = ht P + 1.…”

“…From Proposition 1 in Heitmann (1993), in order to make A complete to T , we must have IT ∩ A = I for every finitely generated ideal I of A. The following lemma helps us get this condition.…”

“…Similarly, in Heitmann (1993), the author provides necessary and sufficient conditions for a ring to be the completion of a unique factorization domain (UFD).…”