Hoa Lê - Academia.edu (original) (raw)

Papers by Hoa Lê

Nagoya mathematical journal

Let N denote the set of non-negative integers. An affine semigroup is a finitely generated submon... more Let N denote the set of non-negative integers. An affine semigroup is a finitely generated submonoid S of the additive monoid N m for some positive integer m. Let k[S] denote the semigroup ring of S over a field k. Then one can identify k[S] with the subring of a polynomial ring Wu , t m ] generated by the monomials t x = % 1-t%?, x = (x»-,x m)eS. Let Q denote the field of rational numbers. Let σ: Q m-* Q be a linear functional such that σ(S)^N and σ(x) = 0, x e S, implies x == 0. Then one can define an JV-grading on k[S] by setting deg t x = σ(x) for all x e S. Such a procedure is called specializing to an N-grading [13, p. 190]. If T^N n is another affine semigroup and k[T] is specialized to an iV-grading by a linear functional τ: Q n-> Q, then one can define a new affine semigroup W^N m X N n by setting W: = (Sχ T)ΠF, where F denotes the set of all elements (x, y)eQ m X Q n with σ(x) = τ(y). We call k[W] the Segre product of the ΛΓ-graded rings k[S] and k[T] with respect to σ and r (cf. [9, p. 125]). The class of rings of the form k[W] includes, for example, the usual Segre product of polynomial rings, the Segre-Veronese graded algebra and the Rees algebras of certain rings generated by monomials. Several authors have been dealt with the Cohen-Macaulayness and the Gorensteiness of Segre products of special classes of affine semigroup rings [1], [2], [3], [4], [16]. The main result of this paper is a combinatorial criterion for k[W] to be a Cohen-Macaulay (res. Gorenstein) in terms of S and T (Theorem 2.1). It is based on a combinatorial criterion of [16] for an affine semigroup ring to be Cohen-Macaulay (res. Gorenstein) which uses certain simplicial complexes associated with the affine semigroup (see Section 1), We shall see that the associated simplicial complexes of W are the joins Keceived August 26, 1986.

Nagoya mathematical journal

Let N denote the set of non-negative integers. An affine semigroup is a finitely generated submon... more Let N denote the set of non-negative integers. An affine semigroup is a finitely generated submonoid S of the additive monoid N m for some positive integer m. Let k[S] denote the semigroup ring of S over a field k. Then one can identify k[S] with the subring of a polynomial ring Wu , t m ] generated by the monomials t x = % 1-t%?, x = (x»-,x m)eS. Let Q denote the field of rational numbers. Let σ: Q m-* Q be a linear functional such that σ(S)^N and σ(x) = 0, x e S, implies x == 0. Then one can define an JV-grading on k[S] by setting deg t x = σ(x) for all x e S. Such a procedure is called specializing to an N-grading [13, p. 190]. If T^N n is another affine semigroup and k[T] is specialized to an iV-grading by a linear functional τ: Q n-> Q, then one can define a new affine semigroup W^N m X N n by setting W: = (Sχ T)ΠF, where F denotes the set of all elements (x, y)eQ m X Q n with σ(x) = τ(y). We call k[W] the Segre product of the ΛΓ-graded rings k[S] and k[T] with respect to σ and r (cf. [9, p. 125]). The class of rings of the form k[W] includes, for example, the usual Segre product of polynomial rings, the Segre-Veronese graded algebra and the Rees algebras of certain rings generated by monomials. Several authors have been dealt with the Cohen-Macaulayness and the Gorensteiness of Segre products of special classes of affine semigroup rings [1], [2], [3], [4], [16]. The main result of this paper is a combinatorial criterion for k[W] to be a Cohen-Macaulay (res. Gorenstein) in terms of S and T (Theorem 2.1). It is based on a combinatorial criterion of [16] for an affine semigroup ring to be Cohen-Macaulay (res. Gorenstein) which uses certain simplicial complexes associated with the affine semigroup (see Section 1), We shall see that the associated simplicial complexes of W are the joins Keceived August 26, 1986.