Noether resolutions in dimension $2$

Let $R:= K[x_1,\ldots,x_{n}]$ be a polynomial ring over an infinite field $K$, and let $I \subset R$ be a homogeneous ideal with respect to a weight vector $\omega = (\omega_1,\ldots,\omega_n) \in (\mathbb{Z}^+)^n$ such that $\dim(R/I) = d$. In this paper we study the minimal graded free resolution of $R/I$ as $A$-module, that we call the Noether resolution of $R/I$, whenever $A :=K[x_{n-d+1},\ldots,x_n]$ 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 Gr\"obner 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 $R/I$. Moreover, in the more general setting that $R/I$ is a simplicial semigroup ring of any dimension, we provide its Macaulayfication. 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 and the Macaulayfication of either the coordinate ring of a projective monomial curve $\mathcal{C} \subseteq \mathbb{P}_K^{n}$ associated to an arithmetic sequence or the coordinate ring of any canonical projection $\pi_{r}(\mathcal{C})$ of $\mathcal{C}$ to $\mathbb{P}_K^{n-1}$.