Narrow Results

Let K be a principal ideal domain, and A_n, with n ≥ 3, be a finitely generated torsion-free abelian group of rank n. Let Ω be a finite subset of KA_n\{0} and U(KA_n) the group of units of KA_n. For a multiplicative monoid P generated by U(KA_n) and Ω, we prove that any generating set for contains infinitely many elements not in . Furthermore, we present a way of constructing elements of not in for n ≥ 3. In the case where K is not a field the aforementioned results hold for n ≥ 2.

This paper discusses a computational treatment of the localization $A_L$ of an affine coordinate ring $A$ at a prime ideal $L$ and its associated graded algebra ${\mathrm{\text{Gr}}}_{𝔞}(A_L)$ with the means of computer algebra. Building on Mora’s paper [T. Mora, La queste del Saint ${\mathrm{\text{Gr}}}_{𝔞}(A_L)$: A computational approach to local algebra, Discrete Appl. Math.33 (1991) 161–190], we present shorter proofs on two of the central statements and expand on the applications touched by Mora: resolutions of ideals, systems of parameters and Hilbert polynomials, as well as dimension and regularity of $A_L$. All algorithms are implemented in the library graal.lib for the computer algebra system Singular.