4 Localize Rings
  
  The  package LocalizeRingForHomalg defines the classes of local(ized) rings,
  local  ring  elements and local matrices. These three objects can be used as
  data structures defined in MatricesForHomalg on which the homalg project can
  rely to do homological computations over localized rings.
  
  A  homalg  local ring element contains two homalg ring elements, a numerator
  (-->  Numerator  (4.3-4))  and  a  denominator  (--> Denominator (4.3-6)). A
  homalg  local  matrix  contains  a  global homalg matrix as a numerator (-->
  Numerator  (4.3-5))  and  a  ring  element as a denominator (--> Denominator
  (4.3-7)).   New   constructors   for   ring   elements   and   matrices  are
  HomalgLocalRingElement  (4.3-17)  and HomalgLocalMatrix (4.3-18) in addition
  to  the  standard  contructors  introduced  in  other packages of the homalg
  project.
  
  The local rings most prominently can be used with methods known from general
  homalg  rings.  The  methods for doing the computations are presented in the
  appendix  (A),  since  they are not for external use. The new attributes and
  operations are documented here.
  
  Since   the  objects  inplemented  here  are  representations  from  objects
  elsewhere  in the homalg project (i.e. MatricesForHomalg), we want to stress
  that  there  are  many  other  operations  in  homalg,  which can be used in
  connection  with  the ones presented here. A few of them can be found in the
  examples and the appendix of this documentation.
  
  
  4.1 Category and Representations
  
  4.1-1 IsHomalgLocalRingRep
  
  IsHomalgLocalRingRep( R )  Representation
  Returns:  true or false
  
  The representation of homalg local rings.
  
  (It is a subrepresentation of the GAP representation
  IsHomalgRingOrFinitelyPresentedModuleRep.)
  
    Code  
    DeclareRepresentation( "IsHomalgLocalRingRep",
            IsHomalgRing
            and IsHomalgRingOrFinitelyPresentedModuleRep,
            [ "ring" ] );
  
  
  4.1-2 IsHomalgLocalRingElementRep
  
  IsHomalgLocalRingElementRep( r )  Representation
  Returns:  true or false
  
  The representation of elements of homalg local rings.
  
  (It is a representation of the GAP category IsHomalgRingElement.)
  
    Code  
    DeclareRepresentation( "IsHomalgLocalRingElementRep",
            IsHomalgRingElement,
            [ "pointer" ] );
  
  
  4.1-3 IsHomalgLocalMatrixRep
  
  IsHomalgLocalMatrixRep( A )  Representation
  Returns:  true or false
  
  The representation of homalg matrices with entries in a homalg local ring.
  
  (It is a representation of the GAP category IsHomalgMatrix.)
  
    Code  
    DeclareRepresentation( "IsHomalgLocalMatrixRep",
            IsHomalgMatrix,
            [ ] );
  
  
  
  4.2 Rings: Attributes
  
  4.2-1 GeneratorsOfMaximalLeftIdeal
  
  GeneratorsOfMaximalLeftIdeal( R )  attribute
  Returns:  a homalg matrix
  
  Returns  the  generators  of  the maximal ideal, at which R was created. The
  generators are given as a column over the associated global ring.
  
  4.2-2 GeneratorsOfMaximalRightIdeal
  
  GeneratorsOfMaximalRightIdeal( R )  attribute
  Returns:  a homalg matrix
  
  Returns  the  generators  of  the maximal ideal, at which R was created. The
  generators are given as a row over the associated global ring.
  
  
  4.3 Operations and Functions
  
  4.3-1 AssociatedGlobalRing
  
  AssociatedGlobalRing( R )  operation
  Returns:  a (global) homalg ring
  
  The global homalg ring, from which the local ring R was created.
  
  4.3-2 AssociatedGlobalRing
  
  AssociatedGlobalRing( r )  operation
  Returns:  a (global) homalg ring
  
  The global homalg ring, from which the local ring element r was created.
  
  4.3-3 AssociatedGlobalRing
  
  AssociatedGlobalRing( mat )  operation
  Returns:  a (global) homalg ring
  
  The global homalg ring, from which the local matrix mat was created.
  
  4.3-4 Numerator
  
  Numerator( r )  operation
  Returns:  a (global) homalg ring element
  
  The  numerator  from  a local ring element r, which is a homalg ring element
  from the computation ring.
  
  4.3-5 Numerator
  
  Numerator( mat )  operation
  Returns:  a (global) homalg matrix
  
  The  numerator  from  a  local matrix mat, which is a homalg matrix from the
  computation ring.
  
  4.3-6 Denominator
  
  Denominator( r )  operation
  Returns:  a (global) homalg ring element
  
  The  denominator from a local ring element r, which is a homalg ring element
  from the computation ring.
  
  4.3-7 Denominator
  
  Denominator( mat )  operation
  Returns:  a (global) homalg ring element
  
  The  denominator  from a local matrix mat, which is a homalg matrix from the
  computation ring.
  
  4.3-8 Name
  
  Name( r )  operation
  Returns:  a string
  
  The name of the local ring element r.
  
  4.3-9 SetMatElm
  
  SetMatElm( mat, i, j, r, R )  operation
  
  Changes  the  entry  (i,j) of the local matrix mat to the value r. Here R is
  the (local) homalg ring involved in these computations.
  
  4.3-10 AddToMatElm
  
  AddToMatElm( mat, i, j, r, R )  operation
  
  Changes the entry (i,j) of the local matrix mat by adding the value r to it.
  Here R is the (local) homalg ring involved in these computations.
  
  4.3-11 MatElmAsString
  
  MatElmAsString( mat, i, j, R )  operation
  Returns:  a string
  
  Returns  the  entry (i,j) of the local matrix mat as a string. Here R is the
  (local) homalg ring involved in these computations.
  
  4.3-12 MatElm
  
  MatElm( mat, i, j, R )  operation
  Returns:  a local ring element
  
  Returns  the  entry  (i,j)  of  the  local matrix mat. Here R is the (local)
  homalg ring involved in these computations.
  
  4.3-13 Cancel
  
  Cancel( a, b )  operation
  Returns:  two ring elements
  
  For  a=a'*c  and  b=b'*c  return  a'  and b'. The exact form of c depends on
  whether a procedure for gcd computation is included in the ring package.
  
  4.3-14 LocalizeAt
  
  LocalizeAt( R, l )  operation
  Returns:  a local ring
  
  If  l is a list of elements of the global ring R generating a maximal ideal,
  the  method creates the corresponding localization of R at the complement of
  the maximal ideal.
  
  4.3-15 LocalizeAtZero
  
  LocalizeAtZero( R )  operation
  Returns:  a local ring
  
  This method creates the corresponding localization of R at the complement of
  the maximal ideal generated by the indeterminates ("at zero").
  
  4.3-16 LocalizePolynomialRingAtZeroWithMora
  
  LocalizePolynomialRingAtZeroWithMora( R )  operation
  Returns:  a local ring
  
  This  method  localizes  the  ring  R  at  zero  and  this localized ring is
  returned.  The  ring table uses Mora's algorithm as implemented Singular for
  low level computations.
  
  4.3-17 HomalgLocalRingElement
  
  HomalgLocalRingElement( numer, denom, R )  function
  HomalgLocalRingElement( numer, R )  function
  Returns:  a local ring element
  
  Creates the local ring element numer/denom or in the second case numer/1 for
  the  local  ring R. Both numer and denom may either be a string describing a
  valid global ring element or from the global ring or computation ring.
  
  4.3-18 HomalgLocalMatrix
  
  HomalgLocalMatrix( numer, denom, R )  function
  HomalgLocalMatrix( numer, R )  function
  Returns:  a local matrix
  
  Creates  the  local matrix numer/denom or in the second case numer/1 for the
  local ring R. Both numer and denom may either be from the global ring or the
  computation ring.