Primary decomposition of certain determinantal ideals

Rocky Mountain Journal of Mathematics, 2009

Given the monomial ideal I = (x α 1 1 ,. .. , x αn n) ⊂ K[x1,. .. , xn] where αi are positive integers and K a field and let J be the integral closure of I. It is a challenging problem to translate the question of the normality of J into a question about the exponent set Γ(J) and the Newton polyhedron N P (J). A relaxed version of this problem is to give necessary or sufficient conditions on α1,. .. , αn for the normality of J. We show that if αi ∈ {s, l} with s and l arbitrary positive integers, then J is normal.

Communications in Algebra, 2003

In this article we investigate when a homogeneous ideal in a graded ring is normal, that is, when all positive powers of the ideal are integrally closed. We are particularly interested in homogeneous ideals in an ℕ-graded ring Aof the form A≥m ≔ ⊕ℓ≥m Aℓand monomial ideals in a polynomial ring over a field. For ideals of the form A≥mwe generalize a recent

2016

We compute the primary decomposition of certain ideals generated by subsets of minors in a generic matrix or in a generic symmetric matrix, or subsets of Pfaffians in a generic skew-symmetric matrix. Specifically, the ideals we consider are generated by minors that have at least some given number of rows and columns in certain submatrices.

Journal of Algebra, 1997

Emmy Noether showed that every ideal in a Noetherian ring admits a decomposition into irreducible ideals. In this paper we explicitly calculate this decomposition in a fundamental case. Specifically, let R be a commutative ring with identity, let x 1 , . . . , x d (d > 1) be an R -sequence, let X = (x 1 , . . . , x d )R, and let I be a monomial ideal (that is, a proper ideal generated by monomials x e 1 1 · · · x e d d ) such that Rad(I) = Rad(X). Then the main result gives a canonical and unique decomposition of I as an irredundant finite intersection of ideals of the form (x

Mediterranean Journal of Mathematics, 2019

If l = lcm(a1,. .. , an) and the integral closure of x a 1 1 ,. .. , x an n , x l n+1 ⊂ R[xn+1] is not normal, then we show that the integral closure of x a 1 1 ,. .. , x an n , x s n+1 is not normal for any s > l. Also, we give a shorter proof of a main result of Coughlin (Classes of Normal Monomial Ideals. Ph.D. thesis, 2004).

Formalized Mathematics, 2021

Summary. We formalize in the Mizar System [3], [4], definitions and basic propositions about primary ideals of a commutative ring along with Chapter 4 of [1] and Chapter III of [8]. Additionally other necessary basic ideal operations such as compatibilities taking radical and intersection of finite number of ideals are formalized as well in order to prove theorems relating primary ideals. These basic operations are mainly quoted from Chapter 1 of [1] and compiled as preliminaries in the first half of the article.

Contemporary Mathematics, 2003

: 13C99, 13H99, 14D25 Let R be a Noetherian ring and I an ideal in R. Then there exists an integer k such that for all n ≥ 1 there exists a primary decomposition Also, for each homogeneous ideal I in a polynomial ring over a field there exists an integer k such that the Castelnuovo-Mumford regularity of I n is bounded above by kn. The regularity part follows from the primary decompositions part, so the heart of this paper is the analysis of the primary decompositions. In [S], this was proved for the primary components of height at most one over the ideal. This paper proves the existence of such a k but does not provide a formula for it. In the paper [SS], Karen E. Smith and myself find explicit k for ordinary and Frobenius powers of monomial ideals in polynomial rings over fields modulo a monomial ideal and also for Frobenius powers of a special ideal first studied by Katzman. Explicit k for the Castelnuovo-Mumford regularity for special ideals is given in the papers by Chandler [C] and Geramita, Gimigliano and Pitteloud [GGP]. Another method for proving the existence of k for primary decompositions of powers of an ideal in Noetherian rings which are locally formally equidimensional and analytically unramified is given in the paper by Heinzer and Swanson [HS]. The primary decomposition result is not valid for all primary decompositions. Here is an example: let I be the ideal (X 2 , XY ) in the polynomial ring k[X, Y ] in two variables X and Y over a field k. For each positive integer m, I = (X) ∩ (X 2 , XY, Y m ) is an irredundant primary decomposition of I. However, for each integer k there exists an integer m, say m = k + 1, such that (X, Y ) k ⊆ (X 2 , XY, Y m ). Hence the result can only

Journal of Pure and Applied Algebra, 1995

We compute the primary decomposition of certain ideals generated by subsets of minors in a generic matrix or in a generic symmetric matrix, or subsets of Pfaffians in a generic skew-symmetric matrix. Specifically, the ideals we consider are generated by minors that have at least some given number of rows and columns in certain submatrices.

2019

Let $ X $ be an $ m \times n $ matrix of distinct indeterminates over a field $ K $, where $ m \le n $. Set the polynomial ring $K[X] := K[X_{ij} : 1 \le i \le m, 1 \le j \le n] $. Let $ 1 \le k < l \le n $ be such that $ l - k + 1 \ge m $. Consider the submatrix $ Y_{kl} $ of consecutive columns of $ X $ from $ k $th column to $ l $th column. Let $ J_{kl} $ be the ideal generated by `diagonal monomials' of all $ m \times m $ submatrices of $ Y_{kl} $, where diagonal monomial of a square matrix means product of its main diagonal entries. We show that $ J_{k_1 l_1} J_{k_2 l_2} \cdots J_{k_s l_s} $ has a linear free resolution, where $ k_1 \le k_2 \le \cdots \le k_s $ and $ l_1 \le l_2 \le \cdots \le l_s $. This result is a variation of a theorem due to Bruns and Conca. Moreover, our proof is self-contained, elementary and combinatorial.

Mathematics of Computation, 2002

We explicitly calculate the normal cones of all monomial primes which de ne the curves of the form (t L ; t L+1 ; : : : ; t L+n ), where n 4. All of these normal cones are reduced and Cohen-Macaulay, and their reduction numbers are independent of the reduction. These monomial primes are new examples of integrally closed ideals for which the product with the maximal homogeneous ideal is also integrally closed.

Communications in Algebra, 1993

Journal of Symbolic Computation, 1988

We present an algorithm to compute the primary decomposition of any ideal in a polynomial ring over a factorially closed algorithmic principal ideal domain R. This means that the ring R is a constructive PID and that we are given an algorithm to factor polynomials over fields which are finitely generated over R or residue fields of R. We show how basic ideal theoretic operations can be performed using Gr6bner bases and we exploit these constructions to inductively reduce the problem to zero dimensional ideals. Here we again exploit the structure of Gr6bner bases to directly compute the primary decomposition using polynomial factorization. We also show how the reduction process can be applied to computing radicals and testing ideals for primality.

2010

In this paper, we use the tools of Gr\"{o}bner bases and combinatorial secant varieties to study the determinantal ideals $I_t$ of the extended Hankel matrices. Denote by $c$-chain a sequence $a_1,\...,a_k$ with $a_i+c<a_{i+1}$ for all $i=1,\...,k-1$. Using the results of $c$-chain, we solve the membership problem for the symbolic powers $I_t^{(s)}$ and we compute the primary decomposition of the product $I_{t_1}\... I_{t_k}$ of the determinantal ideals. Passing through the initial ideals and algebras we prove that the product $I_{t_1}\... I_{t_k}$ has a linear resolution and the multi-homogeneous Rees algebra $\Rees(I_{t_1},\...,I_{t_k})$ is defined by a Gr\"obner basis of quadrics.