Les Reid

Some Results on Normal Homogeneous Ideals

Abstract 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 A ℓand monomial ideals in a polynomial ring over a field. For ideals of the form A ≥m we generalize a recent result of Faridi. We prove that a monomial ideal in a polynomial ring in nindeterminates over a field is normal if and only if the first n − 1 positive powers of the ideal are integrally closed. We then specialize to the case of ideals of the form I( λ ) ≔ , where J( λ ) = ( ,…, ) ⊆ K[x 1,…, x n ]. To state our main result in this setting, we let ℓ = lcm(λ1,…, ,…λ n ), for 1 ≤ i ≤ n, and set λ ′ = (λ1,…, λ i−1, λ i + ℓ, λ i+1,…, λ n ). We prove that if I( λ ′) is normal then I( λ ) is normal and that the converse holds with a small additional assumption.

On weak subintegrality

Abstract In a previous paper the last two authors introduced a condition which gave an elementwise characterization of subintegrality for an extension A ⊆ B of commutative Q-algebras. In the present paper we show that the same condition gives an elementwise characterization of weak subintegrality for an extension A ⊆ B of arbitrary commutative rings. We also give a new characterization of weakly subintegral elements in which the “coefficients” lie in A rather than B.

Finiteness of Subintegrality

Let A and B be commutative algebras containing the rationals, with A contained in B,and B subintegral over A. In an earlier paper the authors showed that if A is excellent of finite Krull dimension then there is a natural isomorphism from B/A to the group of invertible A-submodules of B. In the present paper we remove the requirement that A be excellent of finite Krull dimension.

The structure of generic subintegrality

AbstractIn order to give an elementwise characterization of a subintegral extension of ℚ-algebras, a family of generic ℚ-algebras was introduced in [3]. This family is parametrized by two integral parameters p ⩾ 0,N ⩾ 1, the member corresponding top, N being the subalgebraR = ℚ [{γn¦n ⩾ N}] of the polynomial algebra ℚ[x1,…,xp, z] inp + 1 variables, where

Eulerian properties of non-commuting and non-cyclic graphs of finite groups

\gamma _n = z^n + \sum\nolimits_{i = 1}^p {(_i^n )} x_i z^{n - i}

ON COMPLETE INTERSECTIONS AND THEIR HILBERT FUNCTIONS

. This is graded by weight (z) = 1, weight (xi) =i, and it is shown in [2] to be finitely generated. So these algebras provide examples of geometric objects. In this paper we study the structure of these algebras. It is shown first that the ideal of relations among all the γn’s is generated by quadratic relations. This is used to determine an explicit monomial basis for each homogeneous component ofR, thereby obtaining an expression for the Poincaré series ofR. It is then proved thatR has Krull dimension p+1 and embedding dimensionN + 2p, and that in a presentation ofR as a graded quotient of the polynomial algebra inN + 2p variables the ideal of relations is generated minimally by

The weak subintegral closure of a monomial ideal

\left( \begin{gathered} N + p \\ 2 \\ \end{gathered} \right)