  
  [1X1 [33X[0;0YIntroduction[133X[101X
  
  
  [1X1.1  [33X[0;0YWhat  is  the  Role  of the [5XLocalizeRingForHomalg[105X[101X[1X Package in the [5Xhomalg[105X[101X[1X[101X
  [1XProject?[133X[101X
  
  [33X[0;0YThe  [5Xhomalg[105X  project  [tpa22]  aims  at  providing  a  general  and abstract
  framework  for  homological  computations. The package [5XLocalizeRingForHomalg[105X
  enables the [5Xhomalg[105X project to construct localizations from commutative rings
  in [5Xhomalg[105X at their maximal ideals.[133X
  
  
  [1X1.2 [33X[0;0YFunctionality[133X[101X
  
  [33X[0;0YThe  package  [5XLocalizeRingForHomalg[105X  on  the  one hand builds on the package
  [5XMatricesForHomalg[105X   and   on   the   other   hands   adds  functionality  to
  [5XMatricesForHomalg[105X.  It  uses  the  computability  (i.e.  capability to solve
  linear  systems)  of  a  commutative ring [22XR[122X declared in [5XMatricesForHomalg[105X to
  construct  the localization [22XR_m[122X of [22XR[122X at a maximal ideal [22Xm[122X (given by a finite
  set of generators). This localized ring [22XR_m[122X is again computable and can thus
  be used by [5XMatricesForHomalg[105X.[133X
  
  [33X[0;0YFurthermore,  via  the  package  [5XRingsForHomalg[105X, an interface to [5XSingular[105X is
  used  to  compute  in  localized  polynomial  rings  with the help of Mora's
  algorithm.[133X
  
  
  [1X1.3 [33X[0;0YThe Math Behind This Package[133X[101X
  
  [33X[0;0YThe  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 [5Xhomalg[105X [BLH20]) without need of
  implementing  new  low  level  algorithms. Details can be found in the paper
  [BLH11].   This   ring   can  be  constructed  by  [2XLocalizeAt[102X  ([14X4.3-14[114X)  and
  [2XLocalizeAtZero[102X ([14X4.3-15[114X).[133X
  
  [33X[0;0YFurthermore  we  use the package [5XRingsForHomalg[105X to communicate with [5XSingular[105X
  and  use the implementation of Mora's algorithm there. This is restricted to
  polynomial  rings  and  needs  the  package [5XRingsForHomalg[105X. This ring can be
  constructed by [2XLocalizePolynomialRingAtZeroWithMora[102X ([14X4.3-16[114X).[133X
  
  
  [1X1.4 [33X[0;0YWhich Ring to Use?[133X[101X
  
  [33X[0;0YSince  there  are  two  kinds  of rings included in this package, we want to
  offer a short comparison of these.[133X
  
  [33X[0;0YAs  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.[133X
  
  [33X[0;0YThe  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.[133X
  
  [33X[0;0YFurthermore  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.[133X
  
