ALMOST ADJACENT IDEALS IN TWO-DIMENSIONAL MUHLY RATIONAL SINGULARITIES

2013 ◽  
Vol 13 (03) ◽  
pp. 1350115
Author(s):  
V. VAN LIERDE
Keyword(s):  
Residue Field ◽  
Closed Domain ◽  
Rational Singularity ◽  
Two Dimensional ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Rational Singularities ◽  
Integrally Closed

Let (R, m) be a two-dimensional Muhly rational singularity, i.e. the residue field R/m is algebraically closed and the associated graded ring is an integrally closed domain. The goal of this paper is to use immediate quadratic transforms and degree coefficients to investigate complete ideals that are almost adjacent to m, i.e. [Formula: see text].

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

1992 ◽  
Vol 111 (3) ◽  
pp. 423-432 ◽  
Author(s):  
Bernard L. Johnston ◽  
Jugal Verma
Keyword(s):  
Hilbert Series ◽  
Primary Ideal ◽  
Local Rings ◽  
Classical Theorem ◽  
Two Dimensional ◽  
Graded Rings ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Associated Graded Rings ◽  
Image Position

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].

scholarly journals Hilbert-Samuel function and Grothendieck group

2000 ◽  
Vol 43 (1) ◽  
pp. 73-94
Author(s):  
Koji Nishida
Keyword(s):  
Local Ring ◽  
Grothendieck Group ◽  
Residue Field ◽  
Primary Ideal ◽  
Hilbert Polynomial ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Noetherian Local Ring ◽  
Graded Module ◽  
Primary Ideals

AbstractLet (A, m) be a Noetherian local ring such that the residue field A/m is infinite. Let I be arbitrary ideal in A, and M a finitely generated A-module. We denote by ℓ(I, M) the Krull dimension of the graded module ⊕n≥0InM/mInM over the associated graded ring of I. Notice that ℓ(I, A) is just the analytic spread of I. In this paper, we define, for 0 ≤ i ≤ ℓ = ℓ(I, M), certain elements ei(I, M) in the Grothendieck group K0(A/I) that suitably generalize the notion of the coefficients of Hilbert polynomial for m-primary ideals. In particular, we show that the top term eℓ (I, M), which is denoted by eI(M), enjoys the same properties as the ordinary multiplicity of M with respect to an m-primary ideal.

scholarly journals THE STRUCTURE OF THE SALLY MODULE OF INTEGRALLY CLOSED IDEALS

2016 ◽  
Vol 227 ◽  
pp. 49-76 ◽  
Author(s):  
KAZUHO OZEKI ◽  
MARIA EVELINA ROSSI
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Hilbert Coefficients ◽  
Integrally Closed ◽  
Numerical Invariants ◽  
Minimal Reduction ◽  
Integrally Closed Ideals ◽  
Extremal Value

The first two Hilbert coefficients of a primary ideal play an important role in commutative algebra and in algebraic geometry. In this paper we give a complete algebraic structure of the Sally module of integrally closed ideals $I$ in a Cohen–Macaulay local ring $A$ satisfying the equality $\text{e}_{1}(I)=\text{e}_{0}(I)-\ell _{A}(A/I)+\ell _{A}(I^{2}/QI)+1,$ where $Q$ is a minimal reduction of $I$, and $\text{e}_{0}(I)$ and $\text{e}_{1}(I)$ denote the first two Hilbert coefficients of $I,$ respectively, the multiplicity and the Chern number of $I.$ This almost extremal value of $\text{e}_{1}(I)$ with respect to classical inequalities holds a complete description of the homological and the numerical invariants of the associated graded ring. Examples are given.

Non-splitting of prime divisors

1988 ◽  
Vol 103 (2) ◽  
pp. 251-256 ◽  
Author(s):  
Judith D. Sally
Keyword(s):  
Prime Divisor ◽  
Valuation Ring ◽  
Blow Up ◽  
Residue Field ◽  
General Setting ◽  
Two Dimensional ◽  
Regular Local Ring ◽  
Prime Divisors ◽  
Integrally Closed ◽  
Integrally Closed Ideals

In this study of complete, or integrally closed, ideals in a two-dimensional regular local ring (R, m), Zariski established a one-to-one correspondence between prime divisors of R, i.e. rank 1 discrete valuations v birationally dominating R with residue field of transcendence degree 1 over R/m, and m-primary simple complete ideals Iv in R; cf. [17] and [18]. In this correspondence, the blow-up of such an ideal has unique exceptional prime and the localization at this prime is the valuation ring of a prime divisor of R. In this paper, we will study such ideals in a more general setting, so we begin by recalling some definitions and background results.

scholarly journals The Hilbert Coefficients of the Fiber Cone and the a-Invariant of the Associated Graded Ring

2009 ◽  
Vol 61 (4) ◽  
pp. 762-778 ◽  
Author(s):  
Clare D'Cruz ◽  
Tony J. Puthenpurakal
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Noetherian Local Ring ◽  
Hilbert Coefficients ◽  
Fiber Cone

Abstract.Let (A,m) be a Noetherian local ring with infinite residue field and let I be an ideal in A and let be the fiber cone of I. We prove certain relations among the Hilbert coefficients f0(I), f1(I), f2(I) of F(I) when the a-invariant of the associated graded ring G(I) is negative.

scholarly journals $$v_c$$-Noetherian domains and Krull domains

2021 ◽  
Author(s):  
Gyu Whan Chang
Keyword(s):  
Dedekind Domain ◽  
Closed Domain ◽  
One Dimensional ◽  
Star Operation ◽  
Krull Domain ◽  
Noetherian Domain ◽  
Integrally Closed ◽  
Krull Domains

AbstractLet D be an integrally closed domain, $$\{V_{\alpha }\}$$ { V α } be the set of t-linked valuation overrings of D, and $$v_c$$ v c be the star operation on D defined by $$I^{v_c} = \bigcap _{\alpha } IV_{\alpha }$$ I v c = ⋂ α I V α for all nonzero fractional ideals I of D. In this paper, among other things, we prove that D is a $$v_c$$ v c -Noetherian domain if and only if D is a Krull domain, if and only if $$v_c = v$$ v c = v and every prime t-ideal of D is a maximal t-ideal. As a corollary, we have that if D is one-dimensional, then $$v_c = v$$ v c = v if and only if D is a Dedekind domain.

scholarly journals NORMAL DOMAINS WITH MONOMIAL PRESENTATIONS

2009 ◽  
Vol 19 (03) ◽  
pp. 287-303 ◽  
Author(s):  
ISABEL GOFFA ◽  
ERIC JESPERS ◽  
JAN OKNIŃSKI
Keyword(s):  
Commutative Algebra ◽  
Class Group ◽  
Semigroup Algebra ◽  
Closed Domain ◽  
Finitely Generated ◽  
Defining Relations ◽  
Integrally Closed ◽  
Algebra Over A Field

Let A be a finitely generated commutative algebra over a field K with a presentation A = K 〈X1,…, Xn | R〉, where R is a set of monomial relations in the generators X1,…, Xn. So A = K[S], the semigroup algebra of the monoid S = 〈X1,…, Xn | R〉. We characterize, purely in terms of the defining relations, when A is an integrally closed domain, provided R contains at most two relations. Also the class group of such algebras A is calculated.

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