Mora's holy grail: Algorithms for computing in localizations at prime ideals

This article discusses a computational treatment of the localization A_L of an affine coordinate ring A at a prime ideal L and its associated graded ring Gr_a(A_L) with the means of standard basis techniques. Building on Mora's work, we present alternative proofs on two of the central statements and expand on the applications mentioned 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.