Regular Local Ring Of Dimension Research Articles

On associated graded modules of maximal Cohen–Macaulay modules over hypersurface rings

Let [Formula: see text] where [Formula: see text] be a complete regular local ring of dimension [Formula: see text], [Formula: see text] for some [Formula: see text] and [Formula: see text] an MCM [Formula: see text]-module with [Formula: see text] then we prove that [Formula: see text]. If [Formula: see text] is a complete hypersurface ring of dimension [Formula: see text] with infinite residue field and [Formula: see text], let [Formula: see text] be an MCM [Formula: see text]-module with [Formula: see text] or [Formula: see text] then we prove that [Formula: see text]. Our paper is the first systematic study of depth of associated graded modules of MCM modules over hypersurface rings.

Analytic spread of filtrations and symbolic algebras

In this paper we define and explore the analytic spread ℓ ( I ) $\ell (\mathcal {I})$ of a filtration in a local ring. We show that, especially for divisorial and symbolic filtrations, some basic properties of the analytic spread of an ideal extend to filtrations, even when the filtration is non-Noetherian. We also illustrate some significant differences between the analytic spread of a filtration and the analytic spread of an ideal with examples. In the case of an ideal I $I$ , we have the classical bounds ht ( I ) ⩽ ℓ ( I ) ⩽ dim R $\mbox{ht}(I)\leqslant \ell (I)\leqslant \dim R$ . The upper bound ℓ ( I ) ⩽ dim R $\ell (\mathcal {I})\leqslant \dim R$ is true for filtrations I $\mathcal {I}$ , but the lower bound is not true for all filtrations. We show that for the filtration I $\mathcal {I}$ of symbolic powers of a height two prime ideal p $\mathfrak {p}$ in a regular local ring of dimension three (a space curve singularity), so that ht ( I ) = 2 $\mbox{ht}(\mathcal {I}) =2$ and dim R = 3 $\dim R=3$ , we have that 0 ⩽ ℓ ( I ) ⩽ 2 $0\leqslant \ell (\mathcal {I})\leqslant 2$ and all values of 0, 1 and 2 can occur. In the cases of analytic spread 0 and 1 the symbolic algebra is necessarily non-Noetherian. The symbolic algebra is non-Noetherian if and only if ℓ ( p ( n ) ) = 3 $\ell (\mathfrak {p}^{(n)})=3$ for all symbolic powers of p $\mathfrak {p}$ and if and only if ℓ ( I a ) = 3 $\ell (\mathcal {I}_a)=3$ for all truncations I a $\mathcal {I}_a$ of I $\mathcal {I}$ .

Regular local rings of dimension four and Gorenstein syzygetic prime ideals

Let R be a Noetherian local ring. We prove that R is regular of dimension at most 4 if, and only if, every prime ideal, defining a Gorenstein quotient ring, is syzygetic. We deduce a characterization of these rings in terms of the André-Quillen homology.

Remark on Rees valuations of finitely supported complete monomial ideals in a regular local ring

Let be an equicharacteristic regular local ring of dimension and and consider the sequence of d-dimensional local monomial quadratic transformations where is a local monomial quadratic transformation of Ri for We prove that where and is the special -simple complete ideal associated with Let I and J be finitely supported complete -primary monomial ideals in R. We show that if the set of base points of the ideal IJ is linearly ordered, then

Motivic and ell -adic realizations of the category of singularities of the zero locus of a global section of a vector bundle

We study the motivic and ell -adic realizations of the dg category of singularities of the zero locus of a global section of a line bundle over a regular scheme. We will then use the formula obtained in this way together with a theorem due to D. Orlov and J. Burke–M. Walker to give a formula for the ell -adic realization of the dg category of singularities of the zero locus of a global section of a vector bundle. In particular, we obtain a formula for the ell -adic realization of the dg category of singularities of the special fiber of a scheme over a regular local ring of dimension n.

Completion of skew completable unimodular rows

In this paper, we prove that if R is a local ring of dimension \(d\ge 3\), d odd and \(\frac{1}{(d-1)!}\in R\) then any skew completable unimodular row \(v\in Um_{d}(R[X])\) is completable. It is also proved that skew completable unimodular rows of size \(d\ge 3\) over a regular local ring of dimension d are first row of a 2- stably elementary matrix.

The tree of quadratic transforms of a regular local ring of dimension two

Let D be a 2-dimensional regular local ring and let Q(D) denote the quadratic tree of 2-dimensional regular local overrings of D. We explore the topology of the tree Q(D) and the family R(D) of rings obtained as intersections of rings in Q(D). If A is a finite intersection of rings in Q(D), then A is Noetherian and the structure of A is well understood. However, other rings in R(D) need not be Noetherian. The two main goals of this paper are to examine topological properties of the quadratic tree Q(D), and to examine the structure of rings in the set R(D).

Noetherian intersections of regular local rings of dimension two

Let D be a 2-dimensional regular local ring and let Q(D) denote the quadratic tree of 2-dimensional regular local overrings of D. We describe the desingularization of projective models over D both algebraically in terms of the saturation of complete ideals and order-theoretically in terms of the quadratic tree Q(D). We prove that this describes all the Noetherian rings that are intersections of rings in Q(D).

On the almost Gorenstein property in the Rees algebras of contracted ideals

We explore the question of when the Rees algebra R(I)=⊕n≥0In of I is an almost Gorenstein graded ring, where R is a 2-dimensional regular local ring and I is a contracted ideal of R. Recently, we showed that R(I) is an almost Gorenstein graded ring for every integrally closed ideal I of R. The main results of the present article show that if I is a contracted ideal with o(I)≤2, then R(I) is an almost Gorenstein graded ring, while if o(I)≥3, then R(I) is not necessarily an almost Gorenstein graded ring, even though I is a contracted stable ideal. Thus both affirmative and negative answers are given.

Trimming a Gorenstein ideal

Let $Q$ be a regular local ring of dimension $3$. We show how to trim a Gorenstein ideal in $Q$ to obtain an ideal that defines a quotient ring that is close to Gorenstein in the sense that its Koszul homology algebra is a Poincare duality algebra $P$ padded with a nonzero graded vector space on which $P_{\ge 1}$ acts trivially. We explicitly construct an infinite family of such rings.

Eakin-Sathaye-Type Theorems for Joint Reductions and Good Filtrations of Ideals

Analogues of Eakin-Sathaye theorem for reductions of ideals are proved for $\mathbb N^{s}$ -graded good filtrations. These analogues yield bounds on joint reduction vectors for a family of ideals and reduction numbers for $\mathbb N$ -graded filtrations. Several examples related to lex-segment ideals, contracted ideals in 2-dimensional regular local rings and the filtration of integral and tight closures of powers of ideals in hypersurface rings are constructed to show effectiveness of these bounds.

Cofiniteness over Noetherian complete local rings

In this article, we prove the following generalization of a result of Hartshorne: Let be a regular local ring of dimension 4. Assume that is a regular system of parameters for S and . Then for each finitely generated S-module N with the socle of is infinite dimensional. Also, using this result, for any commutative Noetherian complete local ring , we characterize the class of all ideals I of R with the property that, for every finitely generated R-module M, the local cohomology modules are I-cofinite for all .

Directed unions of local monoidal transforms and GCD domains

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

Almost Gorenstein Rees Algebras of pg-Ideals, Good Ideals, and Powers of the Maximal Ideals

Let (A,m) be a Cohen–Macaulay local ring, and let I be an ideal of A. We prove that the Rees algebra R(I) is an almost Gorenstein ring in the following cases: (1) (A,m) is a two-dimensional excellent Gorenstein normal domain over an algebraically closed field K≅A/m, and I is a pg-ideal; (2) (A,m) is a two-dimensional almost Gorenstein local ring having minimal multiplicity, and I=mℓ for all ℓ≥1; (3) (A,m) is a regular local ring of dimension d≥2, and I=md−1. Conversely, if R(mℓ) is an almost Gorenstein graded ring for some ℓ≥2 and d≥3, then ℓ=d−1.

English

Let (R,m) be a commutative Noetherian regular local ring of dimension d and I be a proper ideal of R such that mAssR(R/I) = AsshR(I). It is shown that the R- module Hht(I)I (R) is I-cofinite if and only if cd(I,R) = ht(I). Also we present a sufficient condition under which this condition the R-module HiI (R) is finitely generated if and only if it vanishes.

Asymptotic properties of infinite directed unions of local quadratic transforms

Let (R,m) be a regular local ring of dimension at least 2. For each valuation domain birationally dominating R, there is an associated sequence {Rn} of local quadratic transforms of R. We consider the case where this sequence {Rn}n≥0 is infinite and examine properties of the integrally closed local domain S=⋃n≥0Rn in the case where S is not a valuation domain. For this sequence, there is an associated boundary valuation ring V=⋃n≥0⋂i≥nVi, where Vi is the order valuation ring of Ri. There exists a unique minimal proper Noetherian overring T of S. T is the regular Noetherian UFD obtained by localizing outside the maximal ideal of S and S=V∩T. In the present paper, we define functions w and e, where w is the asymptotic limit of the order valuations and e is the limit of the orders of transforms of principal ideals. We describe V explicitly in terms of w and e and prove that V is either rank 1 or rank 2. We define an invariant τ associated to S that is either a positive real number or +∞. If τ is finite, then S is archimedean and T is not local. In this case, the function w defines the rank 1 valuation overring W of V and W dominates S. The rational dependence of τ over w(T×) determines whether S is completely integrally closed and whether V has rank 1. We give examples where S is completely integrally closed. If τ is infinite, then S is non-archimedean and T is local. In this case, the function e defines the rank 1 valuation overring E of V. The valuation ring E is a DVR and E dominates T, and in certain cases we prove that E is the order valuation ring of T.

Ideal theory of infinite directed unions of local quadratic transforms

Let (R,m) be a regular local ring of dimension at least 2. Associated to each valuation domain birationally dominating R, there exists a unique sequence {Rn} of local quadratic transforms of R along this valuation domain. We consider the situation where the sequence {Rn}n≥0 is infinite, and examine ideal-theoretic properties of the integrally closed local domain S=⋃n≥0Rn. Among the set of valuation overrings of R, there exists a unique limit point V for the sequence of order valuation rings of the Rn. We prove the existence of a unique minimal proper Noetherian overring T of S, and establish the decomposition S=T∩V. If S is archimedean, then the complete integral closure S⁎ of S has the form S⁎=W∩T, where W is the rank 1 valuation overring of V.

Constructing examples of semigroups of valuations

We work with rational rank 1 valuations centered in regular local rings of dimension 3. Given a regular local ring R that has an algebraically closed coefficient field k and a zero-dimensional valuation ν, we provide an algorithm for constructing generating sequences for ν in R. We then develop a method for determining a valuation on k(x,y,z) through the sequence of defining values. Using the above results we construct examples of valuations centered in k[x,y,z](x,y,z) and investigate their semigroups of values.

Cofiniteness of local cohomology modules over Noetherian local rings

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring. Let $I$ be an ideal of $R$ and let $M$ be a finitely generated $R$-module of dimension $d\geq 1$. In this paper we consider the $I$-cofiniteness property of the local cohomology module $H^{d-1}_I(M)$. More precisely, we prove that the $R$-module $H^{d-1}_I(M)$ is $I$-cofinite if and only if the $R$-module $\Hom_R(R/I,H^{d-1}_I(M))$ is finitely generated. As an immediate consequence of this result, we prove that if $(R,\operatorname{\frak m})$ is a regular local ring of dimension $d\geq 2$ and $I$ is an ideal of $R$ with $\dim R/I\neq 1$, then $H^{d-1}_I(R)=0$ if and only if the $R$-module $\Hom_R(R/I,H^{d-1}_I(R))$ is finitely generated.

On the relationship between depth and cohomological dimension

Let $(S,\mathfrak{m})$ be an $n$-dimensional regular local ring essentially of finite type over a field and let $\mathfrak{a}$ be an ideal of $S$. We prove that if $\text{depth}\,S/\mathfrak{a}\geqslant 3$, then the cohomological dimension $\text{cd}(S,\mathfrak{a})$ of $\mathfrak{a}$ is less than or equal to $n-3$. This settles a conjecture of Varbaro for such an $S$. We also show, under the assumption that $S$ has an algebraically closed residue field of characteristic zero, that if $\text{depth}\,S/\mathfrak{a}\geqslant 4$, then $\text{cd}(S,\mathfrak{a})\leqslant n-4$ if and only if the local Picard group of the completion $\widehat{S/\mathfrak{a}}$ is torsion. We give a number of applications, including a vanishing result on Lyubeznik’s numbers, and sharp bounds on the cohomological dimension of ideals whose quotients satisfy good depth conditions such as Serre’s conditions $(S_{i})$.