Semilocal Generic Formal Fibers

Semilocal formal fibers of principal prime ideals

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 containing the prime ideal pA whose formal fiber is semilocal with maximal ideals the elements of C.

Formal Fibers of Prime Ideals in Polynomial Rings

Let (R,m) be a Noetherian local domain of dimension n that is essentially finitely generated over a field and let R^ denote the m-adic completion of R. Matsumura has shown that n-1 is the maximal height possible for prime ideals of R^ in the generic formal fiber of R. In this article we prove that every prime ideal of R^ that is maximal in the generic formal fiber of R has height n-1. We also present a related result concerning the generic formal fibers of certain extensions of mixed polynomial-power series rings.

Completely Controlling the Dimensions of Formal Fiber Rings at Prime Ideals of Small Height

Let $T$ be a complete equicharacteristic local (Noetherian) UFD of dimension $3$ or greater. Assuming that $|T| = |T/m|$, where $m$ is the maximal ideal of $T$, we construct a local UFD $A$ whose completion is $T$ and whose formal fibers at height one prime ideals have prescribed dimension between zero and the dimension of the generic formal fiber. If, in addition, $T$ is regular and has characteristic zero, we can construct $A$ to be excellent.

Generic formal fibers and analytically ramified stable rings

Let $A$ be a local Noetherian domain of Krull dimension $d$. Heinzer, Rotthaus and Sally have shown that if the generic formal fiber of $A$ has dimension $d-1$, then $A$ is birationally dominated by a one-dimensional analytically ramified local Noetherian ring having residue field finite over the residue field of $A$. We explore further this correspondence between prime ideals in the generic formal fiber and one-dimensional analytically ramified local rings. Our main focus is on the case where the analytically ramified local rings are stable, and we show that in this case the embedding dimensi…

形式纤维的Cohen-Macaulay性质与局部上同调模的维数

Cohen-Macaulayness of formal fibers and dimension of local cohomology modules

本文研究了局部环中形式纤维的Cohen-Macaulay性质与局部上同调模维数的关系，证明了在特定条件下环的维数与模的支撑关系。

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

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 fo…