  
  [1X8 [33X[0;0YMaps[133X[101X
  
  [33X[0;0YA  [5Xhomalg[105X  map  is a data structures for maps (module homomorphisms) between
  finitely  generated modules. Each map in [5Xhomalg[105X knows its source (--> [2XSource[102X
  ([14Xhomalg:  Source[114X))  and  its  target  (-->  [2XRange[102X ([14Xhomalg: Range[114X)). A map is
  represented  by a [5Xhomalg[105X matrix relative to the current set of generators of
  the source resp. target [5Xhomalg[105X module. As with modules (--> Chapter [14X7[114X), maps
  in  [5Xhomalg[105X  are realized in an intrinsic manner: If the presentations of the
  source or/and target module are altered after the map was constructed, a new
  adapted  representation matrix of the map is automatically computed whenever
  needed.  For  this the internal transition matrices of the modules are used.
  [5Xhomalg[105X  uses  the so-called [13Xassociative[113X convention for maps. This means that
  maps  of  left  modules  are  applied  from the right, whereas maps of right
  modules from the left.[133X
  
  
  [1X8.1 [33X[0;0YMaps: Categories and Representations[133X[101X
  
  [1X8.1-1 IsHomalgMap[101X
  
  [33X[1;0Y[29X[2XIsHomalgMap[102X( [3Xphi[103X ) [32X Category[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YThe [5XGAP[105X category of [5Xhomalg[105X maps.[133X
  
  [33X[0;0Y(It   is  a  subcategory  of  the  [5XGAP[105X  categories  [10XIsHomalgModuleOrMap[110X  and
  [10XIsHomalgStaticMorphism[110X.)[133X
  
  [4X[32X  Code  [32X[104X
    [4XDeclareCategory( "IsHomalgMap",[104X
    [4X        IsHomalgModuleOrMap and[104X
    [4X        IsHomalgStaticMorphism );[104X
  [4X[32X[104X
  
  [1X8.1-2 IsHomalgSelfMap[101X
  
  [33X[1;0Y[29X[2XIsHomalgSelfMap[102X( [3Xphi[103X ) [32X Category[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YThe [5XGAP[105X category of [5Xhomalg[105X self-maps.[133X
  
  [33X[0;0Y(It   is   a   subcategory   of   the   [5XGAP[105X   categories   [10XIsHomalgMap[110X   and
  [10XIsHomalgEndomorphism[110X.)[133X
  
  [4X[32X  Code  [32X[104X
    [4XDeclareCategory( "IsHomalgSelfMap",[104X
    [4X        IsHomalgMap and[104X
    [4X        IsHomalgEndomorphism );[104X
  [4X[32X[104X
  
  [1X8.1-3 IsMapOfFinitelyGeneratedModulesRep[101X
  
  [33X[1;0Y[29X[2XIsMapOfFinitelyGeneratedModulesRep[102X( [3Xphi[103X ) [32X Representation[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YThe [5XGAP[105X representation of maps between finitley generated [5Xhomalg[105X modules.[133X
  
  [33X[0;0Y(It  is  a representation of the [5XGAP[105X category [2XIsHomalgChainMorphism[102X ([14Xhomalg:
  IsHomalgChainMorphism[114X),   which   is   a   subrepresentation   of   the  [5XGAP[105X
  representation [10XIsStaticMorphismOfFinitelyGeneratedObjectsRep[110X.)[133X
  
  
  [1X8.2 [33X[0;0YMaps: Constructors[133X[101X
  
  [1X8.2-1 HomalgMap[101X
  
  [33X[1;0Y[29X[2XHomalgMap[102X( [3Xmat[103X, [3XM[103X, [3XN[103X ) [32X function[133X
  [33X[1;0Y[29X[2XHomalgMap[102X( [3Xmat[103X[, [3Xstring[103X] ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X map[133X
  
  [33X[0;0YThis constructor returns a map (homomorphism) of finitely presented modules.
  It  is  represented  by the [5Xhomalg[105X matrix [3Xmat[103X relative to the current set of
  generators  of  the  source  [5Xhomalg[105X  module [3XM[103X and target module [3XN[103X (--> [14X7.2[114X).
  Unless  the  source module is free [13Xand[113X given on free generators the returned
  map  will cautiously be indicated using parenthesis: [21Xhomomorphism[121X. To verify
  if  the  result  is  indeed  a  well  defined  map  use  [2XIsMorphism[102X ([14Xhomalg:
  IsMorphism[114X).  If  the  presentations  of the source or/and target module are
  altered  after  the map was constructed, a new adapted representation matrix
  of  the map is automatically computed whenever needed. For this the internal
  transition  matrices  of  the  modules  are  used.  If source and target are
  identical  objects,  and  only  then,  the  map  is  created  as  a  selfmap
  (endomorphism).  [5Xhomalg[105X  uses the so-called [13Xassociative[113X convention for maps.
  This  means  that  maps  of left modules are applied from the right, whereas
  maps of right modules from the left.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZZ := HomalgRingOfIntegers( );;[127X[104X
    [4X[25Xgap>[125X [27XM := HomalgMatrix( "[ 2, 3, 4,   5, 6, 7 ]", 2, 3, ZZ );[127X[104X
    [4X[28X<A 2 x 3 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XM := LeftPresentation( M );[127X[104X
    [4X[28X<A non-torsion left module presented by 2 relations for 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XN := HomalgMatrix( "[ 2, 3, 4, 5,   6, 7, 8, 9 ]", 2, 4, ZZ );[127X[104X
    [4X[28X<A 2 x 4 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XN := LeftPresentation( N );[127X[104X
    [4X[28X<A non-torsion left module presented by 2 relations for 4 generators>[128X[104X
    [4X[25Xgap>[125X [27Xmat := HomalgMatrix( "[ \[127X[104X
    [4X[25X>[125X [27X1, 0, -2, -4, \[127X[104X
    [4X[25X>[125X [27X0, 1,  4,  7, \[127X[104X
    [4X[25X>[125X [27X1, 0, -2, -4  \[127X[104X
    [4X[25X>[125X [27X]", 3, 4, ZZ );[127X[104X
    [4X[28X<A 3 x 4 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27Xphi := HomalgMap( mat, M, N );[127X[104X
    [4X[28X<A "homomorphism" of left modules>[128X[104X
    [4X[25Xgap>[125X [27XIsMorphism( phi );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xphi;[127X[104X
    [4X[28X<A homomorphism of left modules>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( phi );[127X[104X
    [4X[28X[ [   1,   0,  -2,  -4 ],[128X[104X
    [4X[28X  [   0,   1,   4,   7 ],[128X[104X
    [4X[28X  [   1,   0,  -2,  -4 ] ][128X[104X
    [4X[28X[128X[104X
    [4X[28Xthe map is currently represented by the above 3 x 4 matrix[128X[104X
    [4X[25Xgap>[125X [27XByASmallerPresentation( M );[127X[104X
    [4X[28X<A rank 1 left module presented by 1 relation for 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( last );[127X[104X
    [4X[28XZ/< 3 > + Z^(1 x 1)[128X[104X
    [4X[25Xgap>[125X [27XDisplay( phi );[127X[104X
    [4X[28X[ [   2,   1,   0,  -1 ],[128X[104X
    [4X[28X  [   1,   0,  -2,  -4 ] ][128X[104X
    [4X[28X[128X[104X
    [4X[28Xthe map is currently represented by the above 2 x 4 matrix[128X[104X
    [4X[25Xgap>[125X [27XByASmallerPresentation( N );[127X[104X
    [4X[28X<A rank 2 left module presented by 1 relation for 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( N );[127X[104X
    [4X[28XZ/< 4 > + Z^(1 x 2)[128X[104X
    [4X[25Xgap>[125X [27XDisplay( phi );[127X[104X
    [4X[28X[ [  -8,   0,   0 ],[128X[104X
    [4X[28X  [  -3,  -1,  -2 ] ][128X[104X
    [4X[28X[128X[104X
    [4X[28Xthe map is currently represented by the above 2 x 3 matrix[128X[104X
    [4X[25Xgap>[125X [27XByASmallerPresentation( phi );[127X[104X
    [4X[28X<A non-zero homomorphism of left modules>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( phi );[127X[104X
    [4X[28X[ [   0,   0,   0 ],[128X[104X
    [4X[28X  [   1,  -1,  -2 ] ][128X[104X
    [4X[28X[128X[104X
    [4X[28Xthe map is currently represented by the above 2 x 3 matrix[128X[104X
  [4X[32X[104X
  
  [33X[0;0YTo construct a map with source being a not yet specified free module[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XN;[127X[104X
    [4X[28X<A rank 2 left module presented by 1 relation for 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XSetPositionOfTheDefaultSetOfGenerators( N, 1 );[127X[104X
    [4X[25Xgap>[125X [27XN;[127X[104X
    [4X[28X<A rank 2 left module presented by 2 relations for 4 generators>[128X[104X
    [4X[25Xgap>[125X [27Xpsi := HomalgMap( mat, "free", N );[127X[104X
    [4X[28X<A homomorphism of left modules>[128X[104X
    [4X[25Xgap>[125X [27XSource( psi );[127X[104X
    [4X[28X<A free left module of rank 3 on free generators>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YTo construct a map between not yet specified free left modules[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xchi := HomalgMap( mat ); ## or chi := HomalgMap( mat, "l" );[127X[104X
    [4X[28X<A homomorphism of left modules>[128X[104X
    [4X[25Xgap>[125X [27XSource( chi );[127X[104X
    [4X[28X<A free left module of rank 3 on free generators>[128X[104X
    [4X[25Xgap>[125X [27XRange( chi );[127X[104X
    [4X[28X<A free left module of rank 4 on free generators>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YTo construct a map between not yet specified free right modules[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xkappa := HomalgMap( mat, "r" );[127X[104X
    [4X[28X<A homomorphism of right modules>[128X[104X
    [4X[25Xgap>[125X [27XSource( kappa );[127X[104X
    [4X[28X<A free right module of rank 4 on free generators>[128X[104X
    [4X[25Xgap>[125X [27XRange( kappa );[127X[104X
    [4X[28X<A free right module of rank 3 on free generators>[128X[104X
  [4X[32X[104X
  
  [1X8.2-2 HomalgZeroMap[101X
  
  [33X[1;0Y[29X[2XHomalgZeroMap[102X( [3XM[103X, [3XN[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X map[133X
  
  [33X[0;0YThe  constructor returns the zero map between the source [5Xhomalg[105X module [3XM[103X and
  the target [5Xhomalg[105X module [3XN[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZZ := HomalgRingOfIntegers( );;[127X[104X
    [4X[25Xgap>[125X [27XM := HomalgMatrix( "[ 2, 3, 4,   5, 6, 7 ]", 2, 3, ZZ );[127X[104X
    [4X[28X<A 2 x 3 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XM := LeftPresentation( M );[127X[104X
    [4X[28X<A non-torsion left module presented by 2 relations for 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XN := HomalgMatrix( "[ 2, 3, 4, 5,   6, 7, 8, 9 ]", 2, 4, ZZ );[127X[104X
    [4X[28X<A 2 x 4 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XN := LeftPresentation( N );[127X[104X
    [4X[28X<A non-torsion left module presented by 2 relations for 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XHomalgZeroMap( M, N );[127X[104X
    [4X[28X<The zero morphism of left modules>[128X[104X
  [4X[32X[104X
  
  [1X8.2-3 HomalgIdentityMap[101X
  
  [33X[1;0Y[29X[2XHomalgIdentityMap[102X( [3XM[103X, [3XN[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X map[133X
  
  [33X[0;0YThe constructor returns the identity map of the [5Xhomalg[105X module [3XM[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZZ := HomalgRingOfIntegers( );;[127X[104X
    [4X[25Xgap>[125X [27XM := HomalgMatrix( "[ 2, 3, 4,   5, 6, 7 ]", 2, 3, ZZ );[127X[104X
    [4X[28X<A 2 x 3 matrix over an internal ring>[128X[104X
    [4X[25Xgap>[125X [27XM := LeftPresentation( M );[127X[104X
    [4X[28X<A non-torsion left module presented by 2 relations for 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XHomalgIdentityMap( M );[127X[104X
    [4X[28X<The identity morphism of a non-zero left module>[128X[104X
  [4X[32X[104X
  
  
  [1X8.3 [33X[0;0YMaps: Properties[133X[101X
  
  
  [1X8.4 [33X[0;0YMaps: Attributes[133X[101X
  
  
  [1X8.5 [33X[0;0YMaps: Operations and Functions[133X[101X
  
  [1X8.5-1 HomalgRing[101X
  
  [33X[1;0Y[29X[2XHomalgRing[102X( [3Xphi[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X ring[133X
  
  [33X[0;0YThe [5Xhomalg[105X ring of the [5Xhomalg[105X map [3Xphi[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZZ := HomalgRingOfIntegers( );[127X[104X
    [4X[28XZ[128X[104X
    [4X[25Xgap>[125X [27Xphi := HomalgIdentityMap( 2 * ZZ );[127X[104X
    [4X[28X<The identity morphism of a non-zero left module>[128X[104X
    [4X[25Xgap>[125X [27XR := HomalgRing( phi );[127X[104X
    [4X[28XZ[128X[104X
    [4X[25Xgap>[125X [27XIsIdenticalObj( R, ZZ );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X8.5-2 PreInverse[101X
  
  [33X[1;0Y[29X[2XPreInverse[102X( [3Xphi[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X map, [10Xfalse[110X, or [10Xfail[110X[133X
  
  [33X[0;0YCompute  a  pre-inverse  of  the  morphism  [3Xphi[103X  in  case  one exists. For a
  pre-inverse  to  exist  [3Xphi[103X  must  be  an epimorphism. For [13Xcommutative[113X rings
  [5Xhomalg[105X  has an algorithm installed which decides the existence and returns a
  pre-inverse  in  case one exists. If a pre-inverse does not exist then [10Xfalse[110X
  is  returned.  The  algorithm  finds  a  particular  solution  of a two-side
  inhomogeneous  linear  system over [22XR :=[122X[10XHomalgRing[110X[22X( [3Xphi[103X )[122X. For [13Xnon[113Xcommutative
  rings  a heuristic method is installed. If it finds a pre-inverse it returns
  it,  otherwise it returns [10Xfail[110X (--> [14X1.1-3[114X). The operation [10XPreInverse[110X is used
  to   install   a   method   for  the  property  [2XIsSplitEpimorphism[102X  ([14Xhomalg:
  IsSplitEpimorphism[114X).[133X
  
  [33X[0;0Y[10XPreInverse[110X checks if it can decide the projectivity of [10XRange[110X[22X( [3Xphi[103X )[122X.[133X
  
