GAP (LocalizeRingForHomalg) - Chapter 1: Introduction

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1 Introduction
1.1 What is the Role of the LocalizeRingForHomalg Package in the homalg Project?
1.2 Functionality
1.3 The Math Behind This Package
1.4 Which Ring to Use?

### 1 Introduction

#### 1.1 What is the Role of the LocalizeRingForHomalg Package in the homalg Project?

The homalg project [tpa22] aims at providing a general and abstract framework for homological computations. The package LocalizeRingForHomalg enables the homalg project to construct localizations from commutative rings in homalg at their maximal ideals.

#### 1.2 Functionality

The package LocalizeRingForHomalg on the one hand builds on the package MatricesForHomalg and on the other hands adds functionality to MatricesForHomalg. It uses the computability (i.e. capability to solve linear systems) of a commutative ring R declared in MatricesForHomalg to construct the localization R_m of R at a maximal ideal m (given by a finite set of generators). This localized ring R_m is again computable and can thus be used by MatricesForHomalg.

Furthermore, via the package RingsForHomalg, an interface to Singular is used to compute in localized polynomial rings with the help of Mora's algorithm.

#### 1.3 The Math Behind This Package

The math behind this package is a simple trick in allowing global computation to be done instead of local computations. This works on any commutative computable ring (in the sense of homalg [BLH20]) without need of implementing new low level algorithms. Details can be found in the paper [BLH11]. This ring can be constructed by `LocalizeAt` (4.3-14) and `LocalizeAtZero` (4.3-15).

Furthermore we use the package RingsForHomalg to communicate with Singular and use the implementation of Mora's algorithm there. This is restricted to polynomial rings and needs the package RingsForHomalg. This ring can be constructed by `LocalizePolynomialRingAtZeroWithMora` (4.3-16).

#### 1.4 Which Ring to Use?

Since there are two kinds of rings included in this package, we want to offer a short comparison of these.

As usually one important part of such a comparison is the computation time. In our experience the general localization is much faster than Mora's algorithm for large examples.

The main advantage of using local bases with Mora's algorithm is the possibility of computing Hilbert polynomials and other combinatorical invariants. This is not possible with our localization algorithm. But it is possible to do a large computation without Mora's algorithm, which perhaps would not terminate in acceptable time, and afterwards compute a local standard basis of the - in comparison to intermediate computations usually much smaller - result to get the combinatorical information and invariants.

Furthermore we remark, that our localization algorithm works on any maximal ideal in any computable commutative ring, whereas Mora's algorithm only works for polynomial rings at the maximal ideal generated by the indeterminates. Of course by affine transformation Mora's algorithm will work on any maximal ideal in a polynomial ring where the residue class field is isomorphic to the ground field.

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