On a Class of Monomial Ideals

Transactions of the American Mathematical Society, 1995

The results in this paper expand the fundamental decomposition theory of ideals pioneered by Emmy Noether. Specifically, let I I be an ideal in a local ring ( R , M ) (R,M) that has M M as an embedded prime divisor, and for a prime divisor P P of I I let I C P ( I ) I{C_P}(I) be the set of irreducible components q q of I I that are P P -primary (so there exists a decomposition of I I as an irredundant finite intersection of irreducible ideals that has q q as a factor). Then the main results show: (a) I C M ( I ) = ∪ { I C M ( Q ) ; Q is a MEC ⁡ of I } I{C_M}(I) = \cup \{ I{C_M}(Q);Q\;{\text {is a }}\operatorname {MEC} {\text { of }}I\} ( Q Q is a MEC of I I in case Q Q is maximal in the set of M M -primary components of I I ); (b) if I = ∩ { q i ; i = 1 , … , n } I = \cap \{ {q_i};i = 1, \ldots ,n\} is an irredundant irreducible decomposition of I I such that q i {q_i} is M M -primary if and only if i = 1 , … , k > n i = 1, \ldots ,k > n , then ∩ { q i ; i = 1 , … , k } \ca...

Journal of Pure and Applied Algebra, 1995

Advances in Mathematics, 1994

Let S be a polynomial ring over an infinite field and let I be a homogeneous ideal of S. Let T d be a polynomial ring whose variables correspond to the monomials of degree d in S. We study the initial ideals of the ideals V d (I) ⊂ T d that define the Veronese subrings of S/I. In suitable orders, they are easily deduced from the initial ideal of I. We show that in(V d (I)) is generated in degree ≤ max(⌈reg(I)/d⌉, 2), where reg(I) is the regularity of the ideal I. (In other words, the d th Veronese subring of any commutative graded ring S/I has a Gröbner basis of degree ≤ max(⌈reg(I)/d⌉, 2).) We also give bounds on the regularity of I in terms of the degrees of the generators of in(I) and some combinatorial data. This implies a version of Backelin's Theorem that high Veronese subrings of any ring are homogeneous Koszul algebras in the sense of Priddy [Pr70]. We also give a general obstruction for a homogeneous ideal I ⊂ S to have an initial ideal in(I) that is generated by quadrics, beyond the obvious requirement that I itself should be generated by quadrics, and the stronger statement that S/I is Koszul. We use the obstruction to show that in certain dimensions, a generic complete intersection of quadrics cannot have an initial ideal that is generated by quadrics. For the application to Backelin's Theorem, we require a result of Backelin whose proof has never appeared. We give a simple proof of a sharpened version, bounding the rate of growth of the degrees of generators for syzygies of any multihomogeneous module over a polynomial ring modulo an ideal generated by monomials, following a method of Bruns and Herzog.

2004

Given a squarefree monomial ideal $I \subseteq R =k[x_1,\ldots,x_n]$, we show that $\widehat\alpha(I)$, the Waldschmidt constant of $I$, can be expressed as the optimal solution to a linear program constructed from the primary decomposition of $I$. By applying results from fractional graph theory, we can then express $\widehat\alpha(I)$ in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of $I$. Moreover, expressing $\widehat\alpha(I)$ as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on $\widehat\alpha(I)$, thus verifying a conjecture of Cooper-Embree-H\`a-Hoefel for monomial ideals in the squarefree case. As an application, we compute the Waldschmidt constant and the resurgence for some families of squarefree monomial ideals. For example, we determine both constants for unions of general linear subspaces of $\mathbb{P}^n$ with few components compared to $n$, and we find the Waldschmidt cons...

Communications in Algebra, 2003

Mathematical Proceedings of the Cambridge Philosophical Society, 2010

Let I, I11,. . ., I1q1,. . ., Ip1,. . ., Ipqp be monomial ideals of a polynomial ring R = K[X1,. . ., Xr] and Ln = I+∩jIn1j + ⋅ ⋅ ⋅ + ∩jIpjn. It is shown that the ai-invariant ai(R/Ln) is asymptotically a quasi-linear function of n for all n ≫ 0, and the limit limn→∞ad(R/Ln)/n exists, where d = dim(R/L1). A similar result holds if I11,. . ., I1q1,. . ., Ip1,. . ., Ipqp are replaced by their integral closures. Moreover all limits $\lim_{n\to\infty} a_i(R/(\cap_j \overline{I_{1j}^n} + \cdots + \cap_j \overline{I_{pj}^n}))/n $ also exist.As a consequence, it is shown that there are integers p > 0 and 0 ≤ e ≤ d = dim R/I such that reg(In) = pn + e for all n ≫ 0 and pn ≤ reg(In) ≤ pn + d for all n > 0 and that the asymptotic behavior of the Castelnuovo–Mumford regularity of ordinary symbolic powers of a square-free monomial ideal is very close to a linear function.

2007

ln this paper, we give some characterizations for prime and primary submodules of a finitely generated free modules over PID's and determine the height of prime submodules. We also characterize the minimal primary decompositions and radicals of submodules of any finitely generated free module over a PID.

Factorization in integral domains, 2017

Let x 1 ,. .. , x d be indeterminates over an infinite field F , let R denote the polynomial ring F [x 1 ,. .. , x d ], and let M denote the maximal ideal (x 1 ,. .. , x d)R. If I is an M-primary ideal the Hilbert polynomial P I (n) = e 0 (I) n + d − 1 d − e 1 (I) n + d − 2 d − 1 + • • • + (−1) d e d (I) gives the length of the R-module R/I n for sufficiently large positive integers n. The integral closure I of I is the unique largest ideal of R containing I and having the same coefficient e 0 (i.e., multiplicity) as I, and the Ratliff-Rush ideal I of I is the unique largest ideal containing I and having the same Hilbert polynomial as I. Kishor Shah has shown in [S1] that there exists a unique chain of ideals 1 I ⊆ I = I {d} ⊆ • • • ⊆ I {k} ⊆ • • • ⊆ I {0} = I , where, for 0 ≤ k ≤ d, the ideal I {k} is maximal with the property of having the same coefficients e 0 ,. .. , e k of its Hilbert polynomial as those of I. The ideal I {k} is called the k-th coefficient ideal of I. If I = I {k} , we say I is an e k-ideal. We are particularly interested in the case where R is of dimension two. In this setting, an M-primary ideal I has reduction number at most one (i.e., if J is a minimal reduction of I, then JI = I 2) if and only if the Rees algebra R[It] is Cohen-Macaulay [HM, Prop. 2.6],[JV, Theorem 4.1], or [S2, Corollary 4(f)]. Moreover, the coefficients e 1 (I) and e 2 (I) are nonnegative, and it follows from [Hu, Theorem 2.1] that I has reduction number at most one if and only if λ(R/I) = e 0 (I) − e 1 (I), and if this holds, then e 2 (I) = 0. We say that an ideal with these 1 The existence of this unique chain of ideals is shown in [S1, Theorem 1] for an ideal primary for the maximal ideal of a quasi-unmixed local ring with infinite residue field. Since R M is regular and so, in particular, quasi-unmixed, and since the length of R/I n is equal to the length of R M /I n R M , Shah's result also applies in our setting.