Leslie G. Roberts

Monomial ideals and points in projective space

Let A=k[X0,…Xn]/I be the homogeneous co-ordinate ring of s points in generic position in Pn. The first and third authors have formulated natural conjectures for the number of generators of I, and for the Cohen-Macaulay type of A. In this paper we give a simple new proof of the conjectures for n = 2 (all s), and prove that the conjectures hold for ‘most’ s if n≥3.

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.

K2 of n lines in the plane

We calculate SK1(A) where A is the coordinate ring of the reduced affine variety consisting of n straight lines in the plane.

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.

On hilbert functions of reduced and of integral algebras

Abstract We prove that for any finite field k , there exist differentiable O -sequences which are not Hilbert functions of reduced graded k -algebras. We discuss when generic Hilbert functions and the Hilbert function of a complete intersection can be Hilbert functions of reduced or of integral graded algebras.

Kahler differentials and excision for curves

In this paper we continue the study of excision for K1 of algebraic curves begun in [4]. If A c B are commutative rings and I is a B-ideal contained in A, then excision holds if the natural map K1(A, I) + K1(B, I) is an isomorphism. By “curve” we mean an algebra A of finite type over a field k, such that if P is a minimal prime ideal of A, then A/P is of Krull dimension 1. This is an abuse of notation, but is shorter than “co-ordinate ring of an affine curve.” All the curves that we consider in the paper are reduced and irreducible, i.e. A is already a domain. We have not thought about non-reduced curves. For reduced curves the additional assumption of irreducibility involves no loss of generality, as can be seen from the proof of Theorem 9 in [4]. The methods we use in the curve case sometimes work for subrings of R[t], R a commutative ring. This paper is a revision and extension of the unpublished manuscript [S]. In order to prove that excision holds we use the exact sequence of Swan (see Section 1) or its improvement due to Vorst (see Section 2). Usually we find explicit generators of R *,,., &I/I2 and then show that these generators map to 1 in K1(A, I). However in Section 4 we use a trick of Swan [15, p. 2381. Using these methods we show that excision holds if enough integers are invertible (Theorem 3.1, Theorem 3.3) or if the ground field is perfect (Theorem 4.2). We also give counterexamples to excision in all characteristics. In order to prove that elements in the kernel of the excision homomorphism are non-zero we use in this paper only one method, the “12-trick” (see proof of Theorem 6.3). However we have

Hilbert functions of monomial curves

We study the Hilbert function of certain projective monomial curves. We determine which of our curves are Cohen–Macaulay, and find the Cohen–Macaulay type of those that are Cohen–Macaulay.

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.

Bases and Ideal Generators for Projective Monomial Curves

In this article we study bases for projective monomial curves and the relationship between the basis and the set of generators for the defining ideal of the curve. We understand this relationship best for curves in ℙ3 and for curves defined by an arithmetic progression. We are able to prove that the latter are set theoretic complete intersections.

Certain projective curves with unusual Hilbert function

Abstract This paper describes a method of producing projective curves with easily computed Hilbert function.