Related papers: Completions of Countable Excellent Domains and Cou…

We find necessary and sufficient conditions for a complete local ring to be the completion of a noncatenary local (Noetherian) domain, as well as necessary and sufficient conditions for it to be the completion of a noncatenary local…

We find necessary and sufficient conditions for a complete local (Noetherian) ring to be the completion of an uncountable local (Noetherian) domain with a countable spectrum. Our results suggest that uncountable local domains with countable…

Let $T$ be a complete local (Noetherian) ring of characteristic zero. We find necessary and sufficient conditions for $T$ to be the completion of a quasi-excellent local domain. In the case that $T$ contains the rationals, we provide…

We characterize which complete local (Noetherian) rings T containing the rationals are the completion of a countable excellent local ring S. We also discuss the possibilities for the map from the minimal prime ideals of T to the minimal…

Let $T$ be a complete local ring. We present necessary and sufficient conditions for $T$ to be the completion of a local (Noetherian) unique factorization domain $A$ such that there exist height one prime ideals $\{J_k\}_{k = 1}^{\infty}$…

Let $T$ be a complete local (Noetherian) ring. For each $i \in \mathbb{N}$, let $C_i$ be a nonempty countable set of nonmaximal pairwise incomparable prime ideals of $T$, and suppose that if $i \neq j$, then either $C_i = C_j$ or no element…

Given a complete local (Noetherian) ring $T$, we find necessary and sufficient conditions on $T$ such that there exists a local domain $A$ with $|A| < |T|$ and $\widehat{A} = T$, where $\widehat{A}$ denotes the completion of $A$ with…

Let $T$ be a local (Noetherian) ring and let $Q_1$ and $Q_2$ be prime ideals of $T$. We find sufficient conditions for there to exist a quasi-excellent local subring $B$ of $T$ satisfying the following conditions: (1) the completion of $B$…

We demonstrate a class of local (Noetherian) unique factorization domains (UFDs) that are noncatenary at infinitely many places. In particular, if $A$ is in our class of UFDs, then the prime spectrum of $A$ contains infinitely many disjoint…

We investigate the rings in which the set of nonzero elements is positive-existential (i.e. a finite union of projections of "algebraic" sets). In the case of Noetherian domains, we prove in particular that this condition is satisfied…

Let (T,M) be a complete local domain containing the integers. Let p1 \subseteq p2 \subseteq ... \subseteq pn be a chain of nonmaximal prime ideals T such that T_pn is a regular local ring. We construct a chain of excellent local domains An…

We prove that, for any $n\geq 0$, there exists an uncountable, $n$-dimensional, excellent, regular local ring with countable spectrum.

Let (T,m) be a complete local (Notherian) ring, C a finite set of pairwise incomparable nonmaximal prime ideals of T, and p a nonzero element. We provide necessary and sufficient conditions for T to be the completion of an integral domain A…

A ring has bounded factorizations if every cancellative nonunit $a \in R$ can be written as a product of atoms and there is a bound $\lambda(a)$ on the lengths of such factorizations. The bounded factorization property is one of the most…

Let R* be an ideal-adic completion of a Noetherian integral domain R and let L be a subfield of the total quotient ring of R* such that L contains R. Let A denote the intersection of L with R*. The integral domain A sometimes inherits nice…

We consider properties of extensions of Krull domains such as flatness that involve behavior of extensions and contractions of prime ideals. Let (R,m) be an excellent normal local domain with field of fractions K, let y be a nonzero element…

We show that there exists a complete local Noetherian normal domain of prime characteristic whose perfection is a non-coherent GCD domain, answering a question of Patankar in the negative concerning characterizations of $F$-coherent rings.…

Using an old example of Nagata, we construct a Noetherian ring of prime characteristic p, whose Frobenius morphism is locally finite, but not finite.

We prove that two arbitrary ideals $I \subset J$ in an equidimensional and universally catenary Noetherian local ring have the same integral closure if and only if they have the same multiplicity sequence. We also obtain a Principle of…

We show that for a Noetherian ring $A$ that is $I$-adically complete for an ideal $I$, if $A/I$ admits a dualizing complex, so does $A$. This gives an alternative proof of the fact that a Noetherian complete local ring admits a dualizing…