chap1.txt

  
  1 Introduction
  
  
  1.1  What  is  the  Role  of the LocalizeRingForHomalg Package in the homalg
  Project?
  
  The  homalg  project  [hpa10]  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 [Bar10]) 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.