Regular Local Ring Of Dimension Research Articles

Proximity relations for real rank one valuations dominating a local regular ring

We study 0-dimensional real rank one valuations centered in a regular local ring of dimension $n\\geq 2$ such that the associated valuation ring can be obtained from the regular ring by a sequence of quadratic transforms. We define two classical invariants associated to the valuation (the refined proximity matrix and the multiplicity sequence) and we show that are equivalent data of the valuation.

CHOW GROUPS OF COMPLETE REGULAR LOCAL RINGS III

In this paper we will show that the followings ; (1) Let R be a regular local ring of dimension n. Then <TEX>$A_{n-2}$</TEX>(R) = 0. (2) Let R be a regular local ring of dimension n and I be an ideal in R of height 3 such that R/I is a Gorenstein ring. Then [I] = 0 in <TEX>$A_{n-3}$</TEX>(R). (3) Let R = V[[ <TEX>$X_1$</TEX>, <TEX>$X_2$</TEX>, …, <TEX>$X_{5}$</TEX> ]]/(p+ <TEX>$X_1$</TEX><TEX>$^{t1}$</TEX> + <TEX>$X_2$</TEX><TEX>$^{t2}$</TEX> + <TEX>$X_3$</TEX><TEX>$^{t3}$</TEX> + <TEX>$X_4$</TEX><TEX>$^2$</TEX>+ <TEX>$X_{5}$</TEX> <TEX>$^2$</TEX>/), where p <TEX>$\neq$</TEX>2, <TEX>$t_1$</TEX>, <TEX>$t_2$</TEX>, <TEX>$t_3$</TEX> are arbitrary positive integers and V is a complete discrete valuation ring with (p) = mv. Assume that R/m is algebraically closed. Then all the Chow group for R is 0 except the last Chow group.group.oup.

Noncommutative Rational Double Points

In this paper we study noncommutative analogues of rational double points. The approach is to consider the action of a finite group G on certain noncommutative analogues of k[[x,y]] which were studied by Artin and Stafford (“Regular Local Rings of Dimension 2,” manuscript). An explicit description in terms of generators and relations is given for a large class of such algebras when G is cyclic. Finally, we show that these algebras are AS-Gorenstein of dimension two, have finite representation type and, in many cases, are regular in codimension one.

Betti numbers of modules of essentially monomial type

Let R be a Noetherian local ring. In this paper we supply formulae for computing the ranks of syzygy and Betti numbers of R-modules of essentially monomial type. These modules are defined with respect to various R-regular sequences. For example, finite length modules of monomial type over regular local rings of dimension n are modules of essentially monomial type with respect to R-regular sequences of length n. If a module is of essentially monomial type with respect to an R-regular sequence of length n, then the rank of its i-th syzygy is at least (MDl) and its i-th Betti number is at least (n).

Cycles on arithmetic surfaces

We develop a localized intersection theory for arithmetic schemes on the model of Fulton's intersection theory. We prove a Lefschetz fixed point formula for arithmetic surfaces, and give an application to a conjecture of Serre on the existence of Artin's representations for regular local rings of dimension 2 and unequal characteristic.

Unique factorization of *-products of one-fibered monomial ideals

Let R k[x 1,x 2,x 3] be a 3-dimensional polynomial ring over a field k.We show that no non-trivial relations exist among *-products of complete one-fibered (x 1,x 2,x 3)-primary monomial ideals. This is connected with questions about how Zariski's work on the unique factorization of complete ideals in regular local rings of dimension two might be generalized to higher dimensions.

Koassoziierte Primideale in 2-Dimensionalen Regulären Ringen

Suppose that R is a 2-dimensional regular local ring, pa prime ideal of height 1, and u = r/s an element of the quotient field of R. We give criteria for pbeing coassociated to the R-module B = R[u], i.e., that there is an Artinian factor module B/X such that p= AnnR(B/X). These criteria are formulated using the ideal a= (r, s) and the intersection numbers σ(αi, r), σ(αi, s), where α1, ... αh are the branches of pin the completion R.

Valuation ideals of order one in 2-dimensional regular local rings

Let υ be a prime divisor of a 2-dimensional regular local ring (R m) with algebraically closed residue field k. Zariski showed that a prime divisor υ of R is uniquely associated to a simple m-primary integrally closed ideal I of R, there exist finitely many simple υ-ideals including I, and all the other υ-ideals can be uniquely factored into products of simple υ-ideals. It is known that such an m-primary ideal I of R can be minimally generated by o(I) + 1 elements.Given a simple integrally closed ideal I of order one with arbitrary rank and its associated prime divisor υ, we find minimal generating sets of all the simple υ-ideals and describe factorizations of all the composite υ-ideals in terms of power products of simple υideals as explicitly as possible.

Plücker formula and the equisingularity set of a linear system

We study the process of resolution of linear systems on 2-dimensional regular local rings by quadratic transformations and we give an effective characterization of the Zariski open set of equisingular elements in the linear system. We give a description of the behaviour at the first infinitesimal neighborhood by means of the Plücker Formulas.

Note on Birational Extensions in D-Dimension

Let $(R,m)$ be a $d$-dimensional regular local ring with quotient field $K$ and $(S,n)$ be a $d$-dimensional normal local domain birationally dominating $R$ with $l(mS) = d$. In this paper, it is shown that the following three properties hold. $S$ is dominated by the $m$-adic prime divisor of $R$;$n^i \cap R = m^i$, for all $i \ge 1$;$R/m = S/n$.

Excellent Rings with Local Generic Formal Fibers

Let (T,M) be a complete regular local ring of dimension at least 2, containing the rationals, such that the cardinality of the residue fieldT/Mis at least the cardinality of the real numbers. Supposep≠Mis a prime ideal ofT. We construct an excellent regular local ringAsuch that the completion ofAisTand the generic formal fiber ofAis local (this means that the ringT⊗AKis a local ring whereKis the quotient field ofA) withp⊗AKits maximal ideal. In addition, ifqis a prime ideal ofTwithq∩A=P≠(0), thenq=PT.

Betti Numbers of Modules of Exponent Two over Regular Local Rings

Let (R,m,K) be a regular local ring of dimensionnand letMbe a finite length module overR. This paper gives an affirmative answer to Horrocks' questions whenm2M=0, that is, in this case the rank of theith syzygy ofMis at least[formula]and theith Betti number ofMis at least[formula].

Product of distinct simple integrally closed ideals in 2-dimensional regular local rings

Let (R, m) be a two-dimensional regular local ring and I an mprimary integrally closed ideal in R. In this paper, we give equivalent conditions for I to be a product of distinct simple m-primary integrally closed ideals (i.e., I = II ... Il, where II,, I are distinct simple m-primary integrally closed ideals of R) in terms of the regularity of R[It]/p for all p E Min(mR[It]) and in terms of how to choose a minimal generating set for I over its minimal reductions.

Cores of ideals in $2$-dimensional regular local rings.

Cores of ideals in $2$-dimensional regular local rings.

Sequence of Valuation Ideals of Prime Divisors of the Second Kind in 2-Dimensional Regular Local Rings

Sequence of Valuation Ideals of Prime Divisors of the Second Kind in 2-Dimensional Regular Local Rings

The Value Semigroups of Prime Divisors of the Second Kind in 2-Dimensional Regular Local Rings

The Value Semigroups of Prime Divisors of the Second Kind in 2-Dimensional Regular Local Rings

Adjacent integrally closed ideals in dimension two

Let I be an m-primary integrally closed ideal in a 2-dimensional regular local ring R. Zariski proved that I can be uniquely factored into a product of simple integrally closed ideals, and Lipman later proved that the Hilbert function H I( n) = λ( R/ I n ) of I is a polynomial for all n ≥ 1. By using these results with many others, we study various properties of adjacent integrally closed ideals in 2-dimensional regular local rings. In particular, multiplicities, factorizations, minimal reductions, and Rees valuations of adjacent integrally closed ideals are studied.

The value semigroups of prime divisors of the second kind in 2-dimensional regular local rings

In this paper, it is shown that the value semigroup of a prime divisor of the second kind on a 2 2 -dimensional regular local ring is symmetric. Further, a necessary and sufficient condition for two prime divisors of the second kind on a 2 2 -dimensional regular local ring to have the same value semigroup is obtained.

On the length formula of Hoskin and Deligne and associated graded rings of two-dimensional regular local rings

Let (R, m) be a 2-dimensional regular local ring and I an m-primary ideal. The aim of this paper is to find conditions on I so that the associated graded ring of I,and the Rees ring of I,where t is an indeterminate, are Cohen–Macaulay (resp. Gorenstein). To this end, we use the results and techniques from Zariski's theory of complete ideals ([14], appendix 5) and its later generalizations and refinements due to Huneke [7] and Lipman[8]. The main result is an application of three deep theorems: (i) a generalization of Macaulay's classical theorem on Hilbert series of Gorenstein graded rings [13], (ii) a generalization of the Briançon–Skoda theorem due to Lipman and Sathaye [9], and (iii) a formula for the length of R/I, where I is a complete m-primary ideal, due to Hoskin[4] and Deligne[1].

Joint reductions of complete ideals

The aim of this paper is to extend and unify several results concerning complete ideals in 2-dimensional regular local rings by using the theory of joint reductions and mixed multiplicities. The theory of complete ideals in a 2-dimensional regular local ring was developed by Zariski in his 1938 paper [Z]. This theory is presented in a simpler and general form in [ZS, Appendix 5] and [H2].