Two-dimensional Regular Local Ring Research Articles

Indecomposable integrally closed modules of rank 3 over two-dimensional regular local rings

We characterise ideals in two-dimensional regular local rings that arise as ideals of maximal minors of indecomposable integrally closed modules of rank three.

Indecomposable integrally closed modules of arbitrary rank over a two-dimensional regular local ring

In this paper, we construct indecomposable integrally closed modules of arbitrary rank over a two-dimensional regular local ring. The modules are quite explicitly constructed from a given complete monomial ideal. We also give structural and numerical results on integrally closed modules. These are used in the proof of indecomposability of the modules. As a consequence, we have a large class of indecomposable integrally closed modules of arbitrary rank whose ideal is not necessarily simple. This extends the original result on the existence of indecomposable integrally closed modules and strengthens the non-triviality of the theory developed by Kodiyalam.

Torsors on semistable curves and degenerations

In this paper, we answer two long-standing questions on the classification of G-torsors on curves for an almost simple, simply connected algebraic group G over the field of complex numbers. The first is the construction of a flat degeneration of the moduli of G-torsors on smooth projective curves when the smooth curve degenerates to an irreducible nodal curve and the second one is to give an intrinsic definition of (semi)stability for a G-torsor on an irreducible projective nodal curve. A generalization of the classical Bruhat–Tits group schemes to two-dimensional regular local rings and an application of the geometric formulation of the McKay correspondence provide the key tools.

Iterated blowups of two-dimensional regular local rings

We explore sequences of iterated blowups of two-dimensional regular local rings. Classical results of Zariski and Abhyankar show that the directed union of blowups of this type is a valuation ring.We show that the value group of such a valuation ring is determined by the irrational number $\gamma$ that is the value of an infinite continued fraction associated to the sequence of blowups.

On finite generation of Noetherian algebras over two-dimensional regular local rings

Let R be a complete regular local ring with an algebraically closed residue field and let A be a Noetherian R-subalgebra of the polynomial ring R[X]. It has been shown in [4] that if dim⁡R=1, then A is necessarily finitely generated over R. In this paper, we give necessary and sufficient conditions for A to be finitely generated over R when dim⁡R=2 and present an example of a Noetherian normal non-finitely generated R-subalgebra of R[X] over R=C[[u,v]].

Constructing indecomposable integrally closed modules over a two-dimensional regular local ring

In this article, we construct integrally closed modules of rank two over a two-dimensional regular local ring. The modules are explicitly constructed from a given complete monomial ideal with respect to a regular system of parameters. Then we investigate their indecomposability. As a consequence, we have a large class of indecomposable integrally closed modules whose Fitting ideal is not simple. This gives an answer to Kodiyalam's question.

A note on Gersten’s conjecture for étale cohomology over two-dimensional henselian regular local rings

We show the Gersten's conjecture for \'etale cohomology over two dimensional henselian regular local rings without assuming equi-characteristic. As application, we obtain the local-global principle for Galois cohomology over mixed characteristic two-dimensional henselian local rings.

A formula for jumping numbers in a two-dimensional regular local ring

In this article we give an explicit formula for the jumping numbers of an ideal of finite colength in a two-dimensional regular local ring with an algebraically closed residue field. For this purpose, we associate a certain numerical semigroup to each vertex of the dual graph of a log-resolution of the ideal.

Generators of reductions of ideals in a local Noetherian ring with finite residue field

Let ( R , m ) (R,\mathfrak {m}) be a local Noetherian ring with residue field k k . While much is known about the generating sets of reductions of ideals of R R if k k is infinite, the case in which k k is finite is less well understood. We investigate the existence (or lack thereof) of proper reductions of an ideal of R R and the number of generators needed for a reduction in the case k k is a finite field. When R R is one-dimensional, we give a formula for the smallest integer n n for which every ideal has an n n -generated reduction. It follows that in a one-dimensional local Noetherian ring every ideal has a principal reduction if and only if the number of maximal ideals in the normalization of the reduced quotient of R R is at most | k | |k| . In higher dimensions, we show that for any positive integer, there exists an ideal of R R that does not have an n n -generated reduction and that if n ≥ dim ⁡ R n \geq \dim R this ideal can be chosen to be m \mathfrak {m} -primary. In the case where R R is a two-dimensional regular local ring, we construct an example of an integrally closed m \mathfrak {m} -primary ideal that does not have a 2 2 -generated reduction and thus answer in the negative a question raised by Heinzer and Shannon.

The almost Gorenstein Rees algebras over two-dimensional regular local rings

Let (R,m) be a two-dimensional regular local ring with infinite residue class field. Then the Rees algebra R(I)=⨁n≥0In of I is an almost Gorenstein graded ring in the sense of [6] for every m-primary integrally closed ideal I in R.

The blow-up of a simple ideal

Let R be a two-dimensional regular local ring with maximal ideal m and infinite residue field and let I be a complete simple m-primary residually rational ideal of R; let Σ=BlI(R) be the blow-up of I, and R=:R0⊂R1⊂⋯⊂Rn be the sequence determined by I; Ri is a quadratic transform of Ri−1. Σ is a normal surface; we show that Σ has one resp. two singular points if Rn is free resp. not free. The singular points are rational singularities; we determine their multiplicities.

Lengths and multiplicities of integrally closed modules over a two-dimensional regular local ring

Let (R,m) be a two-dimensional regular local ring with infinite residue field. We prove an analogue of the Hoskin–Deligne length formula for a finitely generated, torsion-free, integrally closed R-module M. As a consequence, we get a formula for the Buchsbaum–Rim multiplicity of F/M, where F=M⁎⁎.

Quasi-one-fibered ideals of order one in dimension two

Simple complete $\M$-primary ideals are fundamental in the Zariski-Lipman theory of complete ideals in a two-dimensional regular local ring $(R,\M)$. Complete $\M$-primary ideals of order one constitute a particular class of such ideals containing, for example, all the first neighborhood ideals of $R$. A number of interesting properties of complete $\M$-primary ideals of order one have been proved by several authors. For example, these ideals have only one Rees valuation (and hence are one-fibered) and they have the very simple form $(x_1^n,x_2)R$, $n\in\n_{+}$, with $x_1,x_2$ a minimal ideal basis of $\M$. In the present paper we investigate how far some results concerning these complete $\M$-primary ideals of order one can be extended to complete quasi-one-fibered $\M$-primary ideals of order one in a natural generalization of $R$: a two-dimensional normal Noetherian local domain with algebraically closed residue field and the associated graded ring an integrally closed domain.

Muhly local domains and Zariski's theory of complete ideals

Let $(R,\M)$ be a two-dimensional Muhly local domain, that is, a two-dimensional integrally closed Noetherian local domain with algebraically closed residue field and with the associated graded ring $\text{gr}_{\M}R$ an integrally closed domain. In this paper we show that a number of fundamental results of Zariski's theory of complete ideals in two-dimensional regular local rings are not necessarily valid in $R$. However, if the associated graded ring $\text{gr}_{\M}R$ satisfies an additional assumption as in work of Muhly and Sakuma, then we are able to show that ``any product of contracted ideals is contracted'' holds in $R$ if and only if $R$ has minimal multiplicity.

Integrally closed rings in birational extensions of two-dimensional regular local rings

AbstractLet D be an integrally closed local Noetherian domain of Krull dimension 2, and let f be a nonzero element of D such that fD has prime radical. We consider when an integrally closed ring H between D and Df is determined locally by finitely many valuation overrings of D. We show such a local determination is equivalent to a statement about the exceptional prime divisors of normalized blow-ups of D and, when D is analytically normal, this property holds for D if and only if it holds for the completion of D. This latter fact, along with MacLane's notion of key polynomials, allows us to prove that in some central cases where D is a regular local ring and f is a regular parameter of D, then H is determined locally by a single valuation. As a consequence, we show that if H is also the integral closure of a finitely generated D-algebra, then the exceptional prime ideals of the extension H/D are comaximal. Geometrically, this translates into a statement about intersections of irreducible components in the closed fiber of the normalization of a proper birational morphism.

Complete ideals and multiplicities in two-dimensional regular local rings

Let I be a complete m-primary ideal of a two-dimensional regular local ring (R,m). The beautiful theory developed by Zariski about complete ideals of R implies that the Rees valuation rings V of I are in a natural one-to-one correspondence with the minimal primes P of the ideal mR[It] in the Rees algebra R[It]. In the previous work of Huneke, Sally and the authors, the structure of R[It]/P is considered in the case where the residue field R/m=k is relatively algebraically closed in the residue field kv of V. In this paper we consider the structure of R[It]/P without the assumption that k is relatively algebraically closed in kv and obtain the following results: we give necessary and sufficient conditions for R[It]/P to be normal; we determine the multiplicity of R[It]/P; we examine the Cohen–Macaulay property of R[It]/P; and we describe implications for affine components of the blowup of I.

Integrally Closed Ideals and Rees Valuation

Let (R, m, k) be a normal Noetherian analytically irreducible universally catenary local domain of dimension d ≥ 1 with infinite residue field k and field of fractions K, and let I be an m-primary ideal. For P ∈ Min(m R[It]) there exists at least one such that P = Q ∩ R[It]. Let v be the Rees valuation of . We obtain necessary and sufficient conditions for the quotient ring R[It]/P to be a polynomial ring in d variables over k in terms of the v-values of a suitably chosen minimal generating set for I. Let J: = (a 1,…, a d )R be a minimal reduction of I. If I is a normal ideal and is a polynomial ring in d variables over k, we show that the residue field k(v) of V, is generated over k by the images of . If (R, m, k) is a two-dimensional regular local ring with algebraically closed residue field k and I is a product of distinct simple integrally closed m-primary ideals, we show for each positive integer n and each Q ∈ Min(m R[I n t]) that the ring is a two-dimensional normal Cohen–Macaulay graded domain with minimal multiplicity at its maximal homogeneous ideal, with this multiplicity being n.

On the degree function coefficients of complete ideals in two-dimensional regular local rings

Let I be a complete M-primary ideal in a two-dimensional regular local ring (R,M) with an algebraically closed residue field. Let v1, …, vn denote the Rees valuations of I. In the theory of degree functions Rees and Sharp have associated with I positive integers d(I,v1), …, d(I,vn) providing useful information about I. We have called these integers the degree function coefficients of I. Using Zariski’s theory of complete ideals in a two-dimensional regular local ring and recent work of Heinzer and Kim on complete ideals, we obtain new interpretations of the degree function coefficients of I. Some applications are given.

Adjacent Ideals to Simple Complete Ideals in Two-Dimensional Regular Local Rings

Let R be a two-dimensional regular local ring with infinite residue field, and ℘ be a simple complete residually rational ideal of R of order r which determines R h . Let 𝒯 be the set of quadratic transforms T of R h with [T: R h ] = 1, and 𝒮 the set of simple complete ideals of R of order r which are adjacent to ℘ from below. If R h is free respectively a satellite, then there exist T* ∈ 𝒯 respectively T*, T** ∈ 𝒯 and a bijective map between the set 𝒮 and the set 𝒯∖{T*} respectively 𝒯∖{T*, T**}.

Quasi-one-fibered ideals in two-dimensional Muhly local domains

In this paper, a two-dimensional integrally closed Noetherian local domain ( R , M ) with algebraically closed residue field is said to be a Muhly local domain, if the associated graded ring is an integrally closed domain. Using the theory of degree functions developed by Rees and Sharp, and the Zariski–Lipman theory of complete ideals in two-dimensional regular local rings, we study M -primary ideals of R having among their Rees valuations exactly one valuation ≠ ord R . We therefore call these ideals quasi-one-fibered. Some consequences are deduced.