Castelnuovo–Mumford regularity of simplicial semigroup rings with isolated singularity

Abstract

Let S = K[x 1 ,...,x n ] be the polynomial ring in n ≥ 2 variables over a field K and m its graded maximal ideal. Let f 1 f m ∈ S be homogeneous polynomials of degree d - 1 > 2 generating an m-primary ideal, and let g 1 ,...,g r E S be arbitrary homogeneous polynomials of degree d. In the present paper it will be proved that the Castelnuovo-Mumford regularity of the standard graded K-algebra A = K[{f i x j } i=1,...,m, g1,...,gr ] j=1,...,n is at most (d - 2)(n - 1). By virtue of this result, it follows that the regularity of a simplicial semigroup ring K[C] with isolated singularity is at most e(K[C]) - codim(K[C]), where e(K[C]) is the multiplicity of K[C] and codim(K[C]) is the codimension of K[C].