Computational Aspects of the Short Resolution
Ignacio García-Marco, Philippe Gimenez, Mario González-Sánchez
TL;DR
The paper develops a Schreyer-like Gröbner-basis framework to compute the short (Noether) resolution of $R/I$ as a module over a Noether normalization $A$, providing both a direct $A$-module presentation and a Schreyer resolution that is in general non-minimal. It gives a detailed combinatorial description for dimension $3$ simplicial toric rings, expressing the multigraded resolution, Hilbert series, and regularity in terms of Apery sets and a finite collection of exceptional sets, and derives explicit pruning rules to obtain minimal short resolutions in this setting. A practical pruning algorithm is proposed for 3D simplicial toric rings, with implementations and caveats illustrating the method’s limitations beyond toric/binomial cases. Finally, the work exhibits that the short resolution and Betti numbers can depend on the characteristic of the base field, presenting concrete examples where ${ m pd}_A(R/I)$ and the full resolution differ between characteristic zero and positive characteristic, underscoring subtle arithmetic dependencies in toric and semigroup-algebra contexts.
Abstract
Let $R:= \Bbbk[x_1,\ldots,x_{n}]$ be a polynomial ring over a field $\Bbbk$, $I \subset R$ be a homogeneous ideal with respect to a weight vector $ω= (ω_1,\ldots,ω_n) \in (\mathbb{Z}^+)^n$, 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 $A :=\Bbbk[x_{n-d+1},\ldots,x_n]$ 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 $\Bbbk$.