GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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