Noetherian Local Domain Research Articles

Partial trace ideals, torsion and canonical module

For any finitely generated module M with non-zero rank over a commutative one dimensional Noetherian local domain, the numerical invariant h(M) was introduced and studied in [25]. We establish a bound on it which helps capture information about the torsion submodule of M when M has rank one and generalizes the discussion in [25]. We further study bounds and properties of h(M) in the case when M is the canonical module ωR. This in turn helps in answering a question of S. Greco and then provides classifications in the Gorenstein, almost Gorenstein and far-flung Gorenstein setups.

Finite generation of nilpotent quotients of fundamental groups of punctured spectra

In SGA 2, Grothendieck conjectures that the étale fundamental group of the punctured spectrum of a complete noetherian local domain of dimension at least two with algebraically closed residue field is topologically finitely generated. In this paper, we prove a weaker statement, namely that the maximal pro-nilpotent quotient of the fundamental group is topologically finitely generated. The proof uses p-adic nearby cycles and negative definiteness of intersection pairings over resolutions of singularities as well as some analysis of Lie algebras of certain algebraic group structures on deformation cohomology.

Quantum tilting modules over local rings

AbstractWe show that tilting modules for quantum groups over local Noetherian domains of quantum characteristic 0 exist and the indecomposable tilting modules are parametrized by their highest weight. For this, we introduce a model category associated with a Noetherian ‐domain and a root system . We show that if is of quantum characteristic , the model category contains all ‐modules that admit a Weyl filtration. If is in addition local, we study torsion phenomena in the model category. This leads to a construction of torsion free, or “maximal” objects in . We show that these correspond to tilting modules for the quantum group associated with and .

Partial trace ideals and Berger's conjecture

For any finitely generated module M with non-zero rank over a commutative one-dimensional Noetherian local domain, we study a numerical invariant h(M) based on a partial trace ideal of M. We study its properties and explore relations between this invariant and the colength of the conductor. Finally we apply this to the universally finite module of differentials ΩR/k, where R is a complete k-algebra with k any perfect field, to study a long-standing conjecture due to R. W. Berger.

Anabelian group-theoretic properties of the absolute Galois groups of discrete valuation fields

Let K be a field. Write GK for the absolute Galois group of K. In the present paper, we discuss the slimness [i.e., the property that every open subgroup is center-free] and the elasticity [i.e., the property that every nontrivial topologically finitely generated normal closed subgroup of an open subgroup is open] of GK. These two group-theoretic properties are closely related to [various versions of] the Grothendieck Conjecture in anabelian geometry. For instance, with regard to the slimness, Mochizuki proved that GK is slim if K is a subfield of a finitely generated extension of the field of fractions of the Witt ring W(F‾p) as a consequence of a [highly nontrivial] Grothendieck Conjecture-type result. In the present paper, we generalize this result to the case where K is a subfield of the field of fractions of an arbitrary mixed characteristic Noetherian local domain. Our proof is based on elementary field theories such as Kummer theory. On the other hand, with regard to the elasticity, Mochizuki proved that GK is elastic if K is a finite extension of the field of p-adic numbers. In the present paper, we generalize this result to the case where K is an arbitrary mixed characteristic Henselian discrete valuation field. As a corollary of this generalization, we prove the semi-absoluteness of isomorphisms between the étale fundamental groups of smooth varieties over mixed characteristic Henselian discrete valuation fields. Moreover, we also prove the weak version of the Grothendieck Conjecture for hyperbolic curves of genus 0 over subfields of finitely generated extensions of mixed characteristic higher local fields.

The genericity theorem for the essential dimension of tame stacks

Let $X$ be a regular tame stack. If $X$ is locally of finite type over a field, we prove that the essential dimension of $X$ is equal to its generic essential dimension, this generalizes a previous result of P. Brosnan, Z. Reichstein and the second author. Now suppose that $X$ is locally of finite type over a $1$-dimensional local noetherian domain $R$ with fraction field $K$ and residue field $k$. We prove that $\operatorname{ed}_{k}X_{k} \le \operatorname{ed}_{K}X_{K}$ if $X\to \operatorname{Spec} R$ is smooth and $\operatorname{ed}_{k}X_{k} \le \operatorname{ed}_{K}X_{K}+1$ in general.

Some variations of projectivity

We prove that a ring [Formula: see text] has a module [Formula: see text] whose domain of projectivity consists of only some injective modules if and only if [Formula: see text] is a right noetherian right [Formula: see text]-ring. Also, we consider modules which are projective relative only to a subclass of max modules. Such modules are called max-poor modules. In a recent paper Holston et al. showed that every ring has a p-poor module (that is a module whose projectivity domain consists precisely of the semisimple modules). So every ring has a max-poor module. The structure of all max-poor abelian groups is completely determined. Examples of rings having a max-poor module which is neither projective nor p-poor are provided. We prove that the class of max-poor [Formula: see text]-modules is closed under direct summands if and only if [Formula: see text] is a right Bass ring. A ring [Formula: see text] is said to have no right max-p-middle class if every right [Formula: see text]-module is either projective or max-poor. It is shown that if a commutative noetherian ring [Formula: see text] has no right max-p-middle class, then [Formula: see text] is the ring direct sum of a semisimple ring [Formula: see text] and a ring [Formula: see text] which is either zero or an artinian ring or a one-dimensional local noetherian integral domain such that the quotient field [Formula: see text] of [Formula: see text] has a proper [Formula: see text]-submodule which is not complete in its [Formula: see text]-topology. Then we show that a commutative noetherian hereditary ring [Formula: see text] has no right max-p-middle class if and only if [Formula: see text] is a semisimple ring.

Asymptotic for the number of star operations on one-dimensional Noetherian domains

We study the set of star operations on local Noetherian domains $D$ of dimension $1$ such that the conductor $(D:T)$ (where $T$ is the integral closure of $D$) is equal to the maximal ideal of $D$. We reduce this problem to the study of a class of closure operations (more precisely, multiplicative operations) in a finite extension $k\subseteq B$, where $k$ is a field, and then we study how the cardinality of this set of closures vary as the size of $k$ varies while the structure of $B$ remains fixed.

Sequences of monoidal transformations of a regular noetherian local domain

Let R R be a regular noetherian local ring of dimension d ≥ 2 d\geq 2 . We characterize the sequences ( R i ) i ≥ 0 (R_i)_{i\geq 0} of successive monoidal transforms of R = R 0 R=R_0 such that S = ⋃ i ≥ 0 R i S =\bigcup _{i\geq 0} R_i is a valuation ring. This characterization involves two well-known conditions in the case of quadratic transforms ( ( R i ) i ≥ 0 (R_i)_{i\geq 0} either switches strongly infinitely often or is height one directed), to which we must add the condition that a family of ideals of S S (finitely supported on the exceptional divisors along the sequence) is linearly ordered by inclusion. Moreover and under the assumption that S S is a valuation ring, we compute the limit points (in the Zariski–Riemann space over the quotient field of R R equipped with the patch topology) of the valuation rings associated with the order valuations defined by the centers of the monoidal transforms as well as the limit points of the valuation rings associated with the order valuations defined by the maximal ideals of the rings R i R_i , i ≥ 0 i\geq 0 .

The ideal theory of intersections of prime divisors dominating a normal Noetherian local domain of dimension two

Let R be a normal Noetherian local domain of Krull dimension two. We examine intersections of rank one discrete valuation rings that birationally dominate R . We restrict to the class of prime divisors that dominate R and show that if a collection of such prime divisors is taken below a certain “level,” then the intersection is an almost Dedekind domain having the property that every nonzero ideal can be represented uniquely as an irredundant intersection of powers of maximal ideals.

A Note on Local Weakly SG-Hereditary Domains

A ring R is called weakly SG-hereditary if every ideal of R is SG-projective. In this note, we prove that a local domain R is a Noetherian Warfield domain if and only if it is weakly SG-hereditary. Furthermore, we prove that any countably generated submodule of any free module over a Noetherian local Warfield domain is SG-projective.

Noetherian domains in which every ideal is isomorphic to a trace ideal

This paper seeks an answer to the following question: Let [Formula: see text] be a Noetherian ring with [Formula: see text]. When is every ideal isomorphic to a trace ideal? We prove that for a local Noetherian domain [Formula: see text] with [Formula: see text], every ideal is isomorphic to a trace ideal if and only if either [Formula: see text] is a DVR or [Formula: see text] is one-dimensional divisorial domain, [Formula: see text] is a principal ideal of [Formula: see text] and [Formula: see text] posses the property that every ideal of [Formula: see text] is isomorphic to a trace ideal of [Formula: see text]. Next, we globalize our result by showing that a Noetherian domain [Formula: see text] with [Formula: see text] has every ideal isomorphic to a trace ideal if and only if either [Formula: see text] is a PID or [Formula: see text] is one-dimensional divisorial domain, every invertible ideal of [Formula: see text] is principal and for every non-invertible maximal ideal [Formula: see text] of [Formula: see text], [Formula: see text] is a principal ideal of [Formula: see text] and every ideal of [Formula: see text] is isomorphic to a trace ideal of [Formula: see text]. We close the paper by examining some classes of non-Noetherian domains with this property to provide a large family of original examples.

Sequences of quadratic transforms of a noetherian local domain

Let R be a noetherian local domain. The aim of this paper is to characterize the sequences (Ri)i≥0 of successive quadratic transforms of R=R0 such that S=⋃i≥0Ri is a valuation ring. These characterizations involve ideals of S supported on the exceptional divisors along the sequence. Furthermore, the characterizations are improved remarkably when the Hilbert functions of the rings (Ri)i≥0 stabilize along the sequence. We also give some examples to put our results in perspective.

On the Ideal Case of a Conjecture of Huneke and Wiegand

AbstractA conjecture of Huneke and Wiegand claims that, over one-dimensional commutative Noetherian local domains, the tensor product of a finitely generated, non-free, torsion-free module with its algebraic dual always has torsion. Building on a beautiful result of Corso, Huneke, Katz and Vasconcelos, we prove that the conjecture is affirmative for a large class of ideals over arbitrary one-dimensional local domains. Furthermore, we study a higher-dimensional analogue of the conjecture for integrally closed ideals over Noetherian rings that are not necessarily local. We also consider a related question on the conjecture and give an affirmative answer for first syzygies of maximal Cohen–Macaulay modules.

Convergence of Rees valuations of sequences of one-fibered domains

We show that for an infinite sequence {(Rn,mn)}n≥0 of local normalized quadratic transforms of analytically unramified one-fibered Noetherian local domains, the corresponding sequence of Rees valuations converges. This extends a result of [4], where the authors show that for an infinite sequence of local quadratic transforms of regular local rings, the order valuations converge.

Integral perfectoid big Cohen–Macaulay algebras via André’s theorem

The main result of this article is to prove that any Noetherian local domain of mixed characteristic maps to an integral perfectoid big Cohen–Macaulay algebra. The proof of this result is based on the construction of almost Cohen–Macaulay algebras in mixed characteristic due to Yves Andre. Moreover, we prove that the absolute integral closure of a complete Noetherian local domain of mixed characteristic maps to an integral perfectoid big Cohen–Macaulay algebra.

Integral closures of Noetherian domains

It has long been known that an integral domain A can be Noetherian while its integral closure A′ is not. However, when Heinzer asked whether A′ can have an infinitely generated maximal ideal, the question went unanswered. In this article, a local Noetherian domain A is constructed with the property that A′ has an infinitely generated maximal ideal. Moreover, the completion of A′ is not Noetherian. It had not previously been shown that this can happen.

Some results on the integer-valued polynomials over matrix rings

ABSTRACTIn this paper, we study the ring of integer-valued polynomials over matrix rings where D is an integral domain with the field of fractions K. We introduce equalizing ideal of Mn(𝔞) in Int(Mn(D)) where 𝔞 is an ideal of D. We show that, if is finite then the set of distinct ideals of the form of Int(Mn(D)) is finite. Also, we prove that . Using this inclusion we obtain that, if D is a Noetherian local one-dimensional domain with finite residue field, which is not unibranched, then the set of ideals of the form is finite, where 𝔪 is the maximal ideal of D. We present some properties of maximal ideals of Int(Mn(D)). Also, we generalize the Skolem closure of an ideal of Int(Mn(D)). Among the other results, we present some relations between prime ideals of Mn(D) and prime ideals of Int(Mn(D)). Finally, we state a lower bound of Krull dimension of integer-valued polynomials ring Int(Mn(D)).

The Ideal Completion of a Noetherian Local Domain

The ideal topology on a integral domain R is the linear topology which has as a fundamental system of neighborhoods of 0 the nonzero ideals of R. We investigate the properties of the ideal topology on a Noetherian local domain (R, 𝔪), and we establish connections between the 𝔪-adic completion and the ideal completion. We give conditions under which the completion in the ideal topology is Noetherian, and we show that, unlike the 𝔪-adic completion, the completion in the ideal topology is not always Noetherian.

Quasi-one-fibered and projectively full ideals in 2-dimensional Muhly rational singularities

Let (R, m) be a 2-dimensional rational singularity. Assume that R is a Muhly domain: (R, m) is an integrally closed Noetherian local domain with algebraically closed residue field and the associated graded ring $$\mathrm{gr}_mR$$ is an integrally closed domain. Let I be a complete m-primary ideal of R and let $$v_1, \ldots , v_n$$ be the Rees valuations of I. For every $$v_i \ne \mathrm{ord}_R$$ , there is a complete m-primary ideal $$I_i$$ of R such that $$I_i$$ is quasi-one-fibered (i.e. $$I_i$$ has exactly one Rees valuation different from $$\mathrm{ord}_R$$ ) and the degree coefficient of $$I_i$$ with respect to $$v_i$$ is $$d(I_i,v_i)=1$$ . We show the following formula: for some $$s, u_1, \ldots , u_n \geqslant 0$$ . We then prove that if $$\mathrm{ord}_R$$ is not a Rees valuation of I and if , then I is projectively full.