The Hilbert function of two ideals

1957 ◽  
Vol 53 (3) ◽  
pp. 568-575 ◽  
Author(s):  
P. B. Bhattacharya
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Hilbert Function ◽  
Projective Dimension ◽  
Primary Ideal ◽  
Homogeneous Ideal ◽  
Artin Ring ◽  
Ring Of Polynomials ◽  
1 Samuel

It is well known that Hubert's function of a homogeneous ideal in the ring of polynomials K[x0, …, xm], where K is a field and x0, …, xm are independent indeterminates over K, is, for large values of r, a polynomial in r of degree equal to the projective dimension of (1). Samuel (4) and Northcott (2) have both shown that if the field K is replaced by an Artin ring A, is still a polynomial in r for large values of r. Applying this generalization Samuel (4) has shown that in a local ring Q the length of an ideal qρ, where q is a primary ideal belonging to the maximal ideal m of Q, is, for sufficiently large values of ρ, a polynomial in ρ whose degree is equal to the dimension of Q.

A note on reductions of ideals with an application to the generalized Hilbert function

1954 ◽  
Vol 50 (3) ◽  
pp. 353-359 ◽  
Author(s):  
D. G. Northcott ◽  
D. Rees
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Hilbert Function ◽  
Residue Field ◽  
Primary Ideal ◽  
System Of Parameters ◽  
A Minor ◽  
The Ideal

1. Throughout this note Q will denote a local ring, m will denote its maximal ideal, q will denote a primary ideal belonging to m and k will denote the residue field Q/m. It will not be assumed that k is infinite, but we shall suppose that Q and k both have the same characteristic. Now let υ1, υ2 …,υd be a system of parameters contained in q, so that d = dim Q; then according to the definition given in (2) the ideal (υl υ2,…, υd) is a reduction of q if (υ1 υ2, …, υd) qm = qm+1 for at least one value of m. The use of the concept lies in the fact that such a reduction is, in a certain sense, a very good approximation to q itself; but the notion does, however, suffer from a minor disadvantage in that, if k is finite, q need not have any reductions. In §3 we shall generalize the notion of a reduction in such a way that we overcome this difficulty, and in such a way that the results concerning reductions obtained in (2) acquire some useful extensions.

On the Multiplicity and the Cohen–Macaulayness of Fiber Cones of Graded Modules

2021 ◽  
Vol 28 (01) ◽  
pp. 13-32
Author(s):  
Nguyen Tien Manh
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Primary Ideal ◽  
Graded Algebra ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Graded Modules ◽  
Fiber Cone

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text], [Formula: see text] an ideal of [Formula: see text], [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text], [Formula: see text] a finitely generated [Formula: see text]-module, [Formula: see text] a finitely generated standard graded algebra over [Formula: see text] and [Formula: see text] a finitely generated graded [Formula: see text]-module. We characterize the multiplicity and the Cohen–Macaulayness of the fiber cone [Formula: see text]. As an application, we obtain some results on the multiplicity and the Cohen–Macaulayness of the fiber cone[Formula: see text].

Hilbert's function in a semi-lattice

1959 ◽  
Vol 55 (3) ◽  
pp. 239-243
Author(s):  
A. Learner
Keyword(s):  
Local Ring ◽  
Polynomial Ring ◽  
Hilbert Function ◽  
Homogeneous Ideal ◽  
Samuel 1

Samuel (1) introduced a generalized Hilbert function, written Xq(r, a) and defined for arbitrary ideals a in a local ring Q with maximal ideai m. where q is m-primary.Northcott(2) proved that for a homogeneous ideal ã in a polynomial ring A[X1, …, Xn], where A = Q/q, this is equal to the ordinary Hilbert function χ(r, ã).

scholarly journals Tight closure of powers of ideals and tight hilbert polynomials

2019 ◽  
Vol 169 (2) ◽  
pp. 335-355
Author(s):  
KRITI GOEL ◽  
J. K. VERMA ◽  
VIVEK MUKUNDAN
Keyword(s):  
Local Ring ◽  
Hilbert Function ◽  
Simplicial Complexes ◽  
Primary Ideal ◽  
Hilbert Polynomial ◽  
Prime Characteristic ◽  
Tight Closure ◽  
Powers Of Ideals ◽  
Hilbert Polynomials

AbstractLet (R, ) be an analytically unramified local ring of positive prime characteristic p. For an ideal I, let I* denote its tight closure. We introduce the tight Hilbert function $$H_I^*\left( n \right) = \Im \left( {R/\left( {{I^n}} \right)*} \right)$$ and the corresponding tight Hilbert polynomial $$P_I^*\left( n \right)$$, where I is an m-primary ideal. It is proved that F-rationality can be detected by the vanishing of the first coefficient of $$P_I^*\left( n \right)$$. We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes.

scholarly journals Notes on irreducible ideals

1985 ◽  
Vol 31 (3) ◽  
pp. 321-324
Author(s):  
David J. Smith
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Noetherian Ring ◽  
Primary Ideal ◽  
Regular Local Ring ◽  
Finite Intersection ◽  
Irreducible Components ◽  
Primary Ideals

Every ideal of a Noetherian ring may be represented as a finite intersection of primary ideals. Each primary ideal may be decomposed as an irredundant intersection of irreducible ideals. It is shown that in the case that Q is an M-primary ideal of a local ring (R, M) satisfying the condition that Q: M = Q + Ms−1 where s is the index of Q, then all irreducible components of Q have index s. (Q is “index-unmixed”.) This condition is shown to hold in the case that Q is a power of the maximal ideal of a regular local ring, and also in other cases as illustrated by examples.

scholarly journals The structure of sally modules and Buchsbaumness of associated graded rings

2013 ◽  
Vol 212 ◽  
pp. 97-138 ◽  
Author(s):  
Kazuho Ozeki
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Primary Ideal ◽  
Graded Rings ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Noetherian Local Ring ◽  
Hilbert Coefficients ◽  
Associated Graded Rings

AbstractLet A be a Noetherian local ring with the maximal ideal m, and let I be an m-primary ideal in A. This paper examines the equality on Hilbert coefficients of I first presented by Elias and Valla, but without assuming that A is a Cohen–Macaulay local ring. That equality is related to the Buchsbaumness of the associated graded ring of I.

scholarly journals The structure of sally modules and Buchsbaumness of associated graded rings

2013 ◽  
Vol 212 ◽  
pp. 97-138 ◽  
Author(s):  
Kazuho Ozeki
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Primary Ideal ◽  
Graded Rings ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Noetherian Local Ring ◽  
Hilbert Coefficients ◽  
Associated Graded Rings

AbstractLetAbe a Noetherian local ring with the maximal ideal m, and letIbe an m-primary ideal inA. This paper examines the equality on Hilbert coefficients ofIfirst presented by Elias and Valla, but without assuming thatAis a Cohen–Macaulay local ring. That equality is related to the Buchsbaumness of the associated graded ring ofI.

scholarly journals Upper bounds of Hilbert coefficients and Hilbert functions

2008 ◽  
Vol 145 (1) ◽  
pp. 87-94 ◽  
Author(s):  
JUAN ELIAS
Keyword(s):  
Finite Number ◽  
Local Ring ◽  
Maximal Ideal ◽  
Upper Bound ◽  
Upper Bounds ◽  
Primary Ideal ◽  
Hilbert Functions ◽  
Hilbert Coefficients ◽  
Hilbert Coefficient

AbstractLet (R, m) be a d-dimensional Cohen–Macaulay local ring. In this paper we prove, in a very elementary way, an upper bound of the first normalized Hilbert coefficient of a m-primary ideal I ⊂ R that improves all known upper bounds unless for a finite number of cases, see Remark 2.3. We also provide new upper bounds of the Hilbert functions of I extending the known bounds for the maximal ideal.

scholarly journals Local rings with quasi-decomposable maximal ideal

2018 ◽  
Vol 168 (2) ◽  
pp. 305-322 ◽  
Author(s):  
SAEED NASSEH ◽  
RYO TAKAHASHI
Keyword(s):  
Direct Summand ◽  
Local Ring ◽  
Maximal Ideal ◽  
Projective Dimension ◽  
Complete Classification ◽  
Local Rings ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Direct Sum

AbstractLet (R, 𝔪) be a commutative noetherian local ring. In this paper, we prove that if 𝔪 is decomposable, then for any finitely generated R-module M of infinite projective dimension 𝔪 is a direct summand of (a direct sum of) syzygies of M. Applying this result to the case where 𝔪 is quasi-decomposable, we obtain several classifications of subcategories, including a complete classification of the thick subcategories of the singularity category of R.

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

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.

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