Formal Fibers Research Articles

Controlling Formal Fibers of Countably Many Principal Prime Ideals

Controlling Formal Fibers of Countably Many Principal Prime Ideals

On Some Permanence Properties of (Derived) Splinters

We show that noetherian splinters ascend under essentially étale homomorphisms. Along the way, we also prove that the henselization of a noetherian local splinter is always a splinter and that the completion of a local splinter with geometrically regular formal fibers is a splinter. Finally, we give an example of a (nonexcellent) Gorenstein local splinter with mild singularities whose completion is not a splinter. Our results provide evidence for a strengthening of the direct summand theorem, namely that regular maps preserve the splinter property.

A uniform treatment of Grothendieck's localization problem

Let $f\colon Y \to X$ be a proper flat morphism of locally noetherian schemes. Then the locus in $X$ over which $f$ is smooth is stable under generization. We prove that, under suitable assumptions on the formal fibers of $X$, the same property holds for other local properties of morphisms, even if $f$ is only closed and flat. Our proof of this statement reduces to a purely local question known as Grothendieck's localization problem. To answer Grothendieck's problem, we provide a general framework that gives a uniform treatment of previously known cases of this problem, and also solves this problem in new cases, namely for weak normality, seminormality, $F$-rationality, and the ‘Cohen–Macaulay and $F$-injective’ property. For the weak normality statement, we prove that weak normality always lifts from Cartier divisors. We also solve Grothendieck's localization problem for terminal, canonical, and rational singularities in equal characteristic zero.

Structure of spectra of precompletions

Let T be a complete local (Noetherian) ring, and let A be a local subring of T such that the completion of A with respect to its maximal ideal is T. We investigate the possible structures of the partially ordered set Spec(A). Specifically, we explore the minimal prime ideals of A and their formal fibers, the maximal chains of prime ideals in A, and the number of prime ideals in A containing combinations of minimal prime ideals of A.

On étale fundamental groups of formal fibres of $p$-adic curves

We investigate a certain class of (geometric) finite (Galois) coverings of formal fibres of $p$-adic curves and the corresponding quotient of the (geometric) etale fundamental group. A key result in our investigation is that these (Galois) coverings can be compactified to finite (Galois) coverings of proper $p$-adic curves. We also prove that the maximal prime-to-$p$ quotient of the geometric etale fundamental group of a (geometrically connected) formal fibre of a $p$-adic curve is (pro-)prime-to-$p$ free of finite computable rank.

Almost excellent unique factorization domains

Let ( T , 𝔪 ) be a complete local (Noetherian) domain such that depth T > 1 . In addition, suppose T contains the rationals, | T | = | T ∕ 𝔪 | , and the set of all principal height-1 prime ideals of T has the same cardinality as T . We construct a universally catenary local unique factorization domain A such that the completion of A is T and such that there exist uncountably many height-1 prime ideals 𝔮 of A such that ( T ∕ ( 𝔮 ∩ A ) T ) 𝔮 is a field. Furthermore, in the case where T is a normal domain, we can make A “close” to excellent in the following sense: the formal fiber at every prime ideal of A of height not equal to 1 is geometrically regular, and uncountably many height-1 prime ideals of A have geometrically regular formal fibers.

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 $|T| = |T/\mathfrak m|$, where $\mathfrak m$ 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.

The maximal dimension of formal fibers of local rings of an algebraic scheme of finite type

For a scheme [Formula: see text] of finite type over a Noetherian local ring [Formula: see text] with a closed point [Formula: see text] of the special fiber, we show that the maximal dimension of the formal fibers of the local algebra [Formula: see text] equals to [Formula: see text] provided that either [Formula: see text] is complete of dimension one or the dimensions of the formal fibers of [Formula: see text] are less than [Formula: see text]. This extends Matsumura’s theorem for algebraic varieties.

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 $\mathfrak{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 $|T|=|T/\mathfrak{m} |$. Let $d$ and $t$ be integers such that $1\leq d \leq \dim T-1$, $0 \leq t \leq \dim T - 1$ and $d - 1 \leq t$. Assume that, for every $\mathfrak{p} \in Ass T$, $ht \mathfrak{p} \leq d-1$ and that if $z$ is a regular element of $T$ and $Q \in Ass (T/zT)$, 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 $t$.

Some results on the cofiniteness and annihilators of local cohomology modules

ABSTRACTLet (R,𝔪) be a Noetherian local ring. In this paper, we give a new characterization for the universal catenaricity of R and the Cohen–Macaulayness of all its formal fibers, and we study the cofiniteness and annihilators of local cohomology modules.

One-dimensional stable rings

A commutative ring R is stable provided every ideal of R containing a nonzerodivisor is projective as a module over its ring of endomorphisms. The class of stable rings includes the one-dimensional local Cohen–Macaulay rings of multiplicity at most 2, as well as certain rings of higher multiplicity, necessarily analytically ramified. The former are important in the study of modules over Gorenstein rings, while the latter arise in a natural way from generic formal fibers and derivations.We characterize one-dimensional stable local rings in several ways. The characterizations involve the integral closure R‾ of R and the completion of R in a relevant ideal-adic topology. For example, we show: If R is a reduced stable ring, then there are exactly two possibilities for R: (1) R is a Bass ring, that is, R is a reduced Noetherian local ring such that R‾ is finitely generated over R and every ideal of R is generated by two elements; or (2) R is a bad stable domain, that is, R is a one-dimensional stable local domain such that R‾ is not a finitely generated R-module.

Controlling the generic formal fiber of local domains and their polynomial rings

Let $T$ be a complete local ring with maximal ideal $M$, $C$ a countable set of incomparable prime ideals of $T$, and $B_1$ and $B_2$ sets of prime ideals of $T[[x_1,\ldots,x_n]]$ with cardinality less than that of $T$. We present necessary and sufficient conditions for the existence of a local domain $A$ with completion $T$, such that the generic formal fiber of $A$ has maximal elements equal to the ideals in $C$ and the generic formal fiber of $A[x_1,\ldots,x_n]_{(M\cap A,x_1,\ldots,x_n)}$ contains every element of $B_1$ but no element of $B_2$. If $T$ has characteristic $0$, we present necessary and sufficient conditions for the existence of an excellent local domain $A$ with the above properties.

Formal fibers with countably many maximal elements

Let $T$ be a complete local (Noetherian) ring. Let $C$ be a countable set of pairwise incomparable nonmaximal prime ideals of $T$. We find necessary and sufficient conditions for $T$ to be the completion of a local integral domain whose generic formal fiber has maximal elements precisely the elements of $C$. Furthermore, if the characteristic of $T$ is zero, we provide necessary and sufficient conditions for $T$ to be the completion of an \textit{excellent} local integral domain whose generic formal fiber has maximal elements precisely the elements of $C$. In addition, for a positive integer $k$, we construct local integral domains that contain a prime ideal of height~$k$ whose formal fiber has countably many maximal elements.

Noetherian Dimension and Co-localization of Artinian Modules over Local Rings

Let (R, 𝔪) be a Noetherian local ring. Denote by N-dim RA the Noetherian dimension of an Artinian R-module A. In this paper, we give some characterizations for the ring R to satisfy N-dim RA = dim (R/ Ann RA) for certain Artinian R-modules A. Then the existence of a co-localization compatible with Artinian R-modules is studied and it is shown that if it is compatible with local cohomologies of finitely generated modules, then the base ring is universally catenary and all of its formal fibers are Cohen-Macaulay.

Attached primes of local cohomology modules under localization and completion

Let (R,m) be a Noetherian local ring and M a finitely generated R-module. Following I.G. Macdonald [8], the set of all attached primes of the Artinian local cohomology module Hmi(M) is denoted by AttR(Hmi(M)). In [13, Theorem 3.7], R.Y. Sharp proved that if R is a quotient of a Gorenstein local ring then the shifted localization principle holds true for any local cohomology modules Hmi(M), i.e.(1)AttRp(HpRpi−dim⁡(R/p)(Mp))={qRp|q∈AttRHmi(M),q⊆p} for any p∈Spec(R). In this paper, we improve Sharp's result as follows: the shifted localization principle holds true if and only if R is universally catenary and all its formal fibers are Cohen–Macaulay, if and only if the shifted completion principle(2)AttRˆ(Hmi(M))=⋃p∈AttR(Hmi(M))AssRˆ(Rˆ/pRˆ) holds true for any local cohomology module Hmi(M). This also improves the main result of the paper by T.D.M. Chau and the first author [2].

Non-Cohen-Macaulay Locus and Non Generalized Cohen-Macaulay Locus

Let (R, 𝔪) be a Noetherian local ring and M a finitely generated R-module. In this paper, we improve and extend some known relations among the non-Cohen–Macaulay locus nCM(M), the polynomial type p(M) and the annihilators of local cohomology modules . A characterization for the base ring R being universally catenary with all Cohen–Macaulay formal fibers is given. The nongeneralized Cohen–Macaulay locus nGCM(M) of M is described.

Attached primes of local cohomology modules and structure of Noetherian local rings

Let (R,m) be a Noetherian local ring and M a finitely generated R-module. It is well known that the local cohomology module Hmi(M) is Artinian for all i⩾0. Following I.G. Macdonald [8], denote by AttRHmi(M) the set of attached primes of Hmi(M). This paper is concerned with clarifying the structure of the base ring R via a relation between AttRHmi(M) and AttRˆHmi(M). Some characterizations for R being universally catenary with all Cohen–Macaulay formal fibers are given.

Local rings with zero-dimensional formal fibers

We study Noetherian local rings whose all formal fibers are of dimension zero. Universal catenarity and going-up property of the canonical map to the completion are considered. We present several characterizations of these rings, including a characterization of Weierstrass preparation type. A characterization of local rings with going up property by a strong form of Lichtenbaum–Hartshorne Theorem is obtained. As an application, we give an upper bound for dimension of formal fibers of a large class of algebras over these rings.

Generic formal fibers and analytically ramified stable rings

AbstractLetAbe a local Noetherian domain of Krull dimensiond. Heinzer, Rotthaus, and Sally have shown that if the generic formal fiber ofAhas dimensiond– 1, thenAis birationally dominated by a 1-dimensional analytically ramified local Noetherian ring having residue field finite over the residue field ofA. We explore further this correspondence between prime ideals in the generic formal fiber and 1-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 dimension of the stable ring reflects the embedding dimension of a prime ideal maximal in the generic formal fiber, thus providing a measure of how far the generic formal fiber deviates from regularity. A number of characterizations of analytically ramified local stable domains are also given.

Generic formal fibers and analytically ramified stable rings

AbstractLet 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 1-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 1-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 dimension of the stable ring reflects the embedding dimension of a prime ideal maximal in the generic formal fiber, thus providing a measure of how far the generic formal fiber deviates from regularity. A number of characterizations of analytically ramified local stable domains are also given.