Computational Aspects of the Short Resolution

Abstract

Let $R:=\mathbb{k}[x_1,\dots ,x_n]$ be a polynomial ring over a field $\mathbb{k}$, $I\subset R$ be a homogeneous ideal with respect to a weight vector $\ddot{I}‰=(\ddot{I}{‰}_1,\dots ,\ddot{I}{‰}_n)\in ({\mathbb{Z}}^{+}{)}^n$, and denote by $d$ the Krull dimension of $R/I$. In this paper we study graded free resolutions of $R/I$ as $A$-module whenever $A:=\mathbb{k}[x_{n-d+1},\dots ,x_n]$ is a Noether normalization of $R/I$. We exhibit a Schreyer-like method to compute a (non-necessarily minimal) graded free resolution of $R/I$ as $A$-module. When $R/I$ is a $3$-dimensional simplicial toric ring, we describe how to prune the previous resolution to obtain a minimal one. We finally provide an example of a $6$-dimensional simplicial toric ring whose Betti numbers, both as $R$-module and as $A$-module, depend on the characteristic of $\mathbb{k}$.

28 pages, 3 figures, 1 table