Constructing Local Generic Formal Fibers

Semilocal generic formal fibers

“…However, in [4], it was shown that such rings do exist and perhaps even more surprisingly, in [5], it was shown that these integral domains can be constructed to be excellent. In this paper, we show that these domains are more plentiful than one might suspect (both in the nonexcellent and excellent case).…”

Completions of UFDs with Semi-Local Formal Fibers

“…Some progress toward this goal has already been made. Loepp (1997) provides the following result. Theorem 1.3 (Loepp, 1997).…”

“…Loepp (1997) provides the following result. Theorem 1.3 (Loepp, 1997). Let T M be a complete local domain of dimension at least two, satisfying Serre's (S2) condition and T/M = T .…”

“…To show that out conditions are necessary is not difficult. The sufficiency of our conditions can be proven by construction, following the construction in Loepp (1997). In fact, the proof follows, once it is shown that we can weaken the conditions on T in Lemma 12 of that article.…”

Generic fiber rings of mixed power series/polynomial rings

“…Charters and Loepp in [1] (see also [6,Theorem 17]) determine necessary and sufficient conditions for T to be the completion of a Noetherian local domain T such that the generic formal fiber of T has as maximal elements precisely the prime ideals in C. If T is of characteristic zero, Charters and Loepp give necessary and sufficient conditions to obtain such a domain T that is excellent. The finite set C may be chosen to contain prime ideals of different heights.…”