Noether resolutions in dimension 2

Abstract

Let R := K [ x 1 ,..., x n ] be a polynomial ring over an infinite field K , and let I ⊂ R be a homogeneous ideal with respect to a weight vector ω = (ω 1 , ..., ω n ) ∈ (Z + ) n such that dim( R/I ) = d. We consider the minimal graded free resolution of R/I as A -module, that we call the Noether resolution of R/I , whenever A := K [ x n-d +1 ,..., x n ] is a Noether normalization of R/I. When d = 2 and I is saturated, we give an algorithm for obtaining this resolution that involves the computation of a minimal Gröbner basis of I with respect to the weighted degree reverse lexicographic order. In the particular case when R/I is a 2-dimensional semigroup ring, we also describe the multigraded version of this resolution in terms of the underlying semigroup. Whenever we have the Noether resolution of R/I or its multigraded version, we obtain formulas for the corresponding Hilbert series of R/I , and when I is homogeneous, we obtain a formula for the Castelnuovo-Mumford regularity of R/I. As an application of the results for 2-dimensional semigroup rings, we provide a new upper bound for the Castelnuovo-Mumford regularity of the coordinate ring of a projective monomial curve. Finally, we describe the multigraded Noether resolution of either the coordinate ring of a projective monomial curve [EQUATION] associated to an arithmetic sequence or of any of its canonical projections [EQUATION].