  
  [1X4 [33X[0;0YRing Maps[133X[101X
  
  [33X[0;0YA  [5Xhomalg[105X  ring  map is a data structure for maps between finitely generated
  rings.  [5Xhomalg[105X more or less provides the basic declarations and installs the
  generic  methods for ring maps, but it is up to other high level packages to
  install  methods  applicable  to  specific  rings.  For example, the package
  [5XSheaves[105X provides methods for ring maps of (finitely generated) affine rings.[133X
  
  
  [1X4.1 [33X[0;0YRing Maps: Category and Representations[133X[101X
  
  [1X4.1-1 IsHomalgRingMap[101X
  
  [29X[2XIsHomalgRingMap[102X( [3Xphi[103X ) [32X Category
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YThe [5XGAP[105X category of ring maps.[133X
  
  [1X4.1-2 IsHomalgRingSelfMap[101X
  
  [29X[2XIsHomalgRingSelfMap[102X( [3Xphi[103X ) [32X Category
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YThe [5XGAP[105X category of ring self-maps.[133X
  
  [33X[0;0Y(It is a subcategory of the [5XGAP[105X category [10XIsHomalgRingMap[110X.)[133X
  
  [1X4.1-3 IsHomalgRingMapRep[101X
  
  [29X[2XIsHomalgRingMapRep[102X( [3Xphi[103X ) [32X Representation
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YThe [5XGAP[105X representation of [5Xhomalg[105X ring maps.[133X
  
  [33X[0;0Y(It is a representation of the [5XGAP[105X category [2XIsHomalgRingMap[102X ([14X4.1-1[114X).)[133X
  
  
  [1X4.2 [33X[0;0YRing Maps: Constructors[133X[101X
  
  [1X4.2-1 RingMap[101X
  
  [29X[2XRingMap[102X( [3Ximages[103X, [3XS[103X, [3XT[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X ring map[133X
  
  [33X[0;0YThis  constructor  returns  a  ring map (homomorphism) of finitely generated
  rings/algebras.  It  is  represented  by  the  images  [3Ximages[103X  of the set of
  generators  of  the  source  [5Xhomalg[105X ring [3XS[103X in terms of the generators of the
  target  ring  [3XT[103X  (--> [14X3.2[114X). Unless the source ring is free [13Xand[113X given on free
  ring/algebra  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 ([14X4.3-1[114X). If source and target are identical objects, and
  only then, the ring map is created as a selfmap.[133X
  
  
  [1X4.3 [33X[0;0YRing Maps: Properties[133X[101X
  
  [1X4.3-1 IsMorphism[101X
  
  [29X[2XIsMorphism[102X( [3Xphi[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YCheck  if  [3Xphi[103X  is  a  well-defined  map,  i.e.  independent of all involved
  presentations.[133X
  
  [1X4.3-2 IsIdentityMorphism[101X
  
  [29X[2XIsIdentityMorphism[102X( [3Xphi[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YCheck if the [5Xhomalg[105X ring map [3Xphi[103X is the identity morphism.[133X
  
  [1X4.3-3 IsMonomorphism[101X
  
  [29X[2XIsMonomorphism[102X( [3Xphi[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YCheck if the [5Xhomalg[105X ring map [3Xphi[103X is a monomorphism.[133X
  
  [1X4.3-4 IsEpimorphism[101X
  
  [29X[2XIsEpimorphism[102X( [3Xphi[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YCheck if the [5Xhomalg[105X ring map [3Xphi[103X is an epimorphism.[133X
  
  [1X4.3-5 IsIsomorphism[101X
  
  [29X[2XIsIsomorphism[102X( [3Xphi[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YCheck if the [5Xhomalg[105X ring map [3Xphi[103X is an isomorphism.[133X
  
  [1X4.3-6 IsAutomorphism[101X
  
  [29X[2XIsAutomorphism[102X( [3Xphi[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YCheck if the [5Xhomalg[105X ring map [3Xphi[103X is an automorphism.[133X
  
  
  [1X4.4 [33X[0;0YRing Maps: Attributes[133X[101X
  
  [1X4.4-1 Source[101X
  
  [29X[2XSource[102X( [3Xphi[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X ring[133X
  
  [33X[0;0YThe source of the [5Xhomalg[105X ring map [3Xphi[103X.[133X
  
  [1X4.4-2 Range[101X
  
  [29X[2XRange[102X( [3Xphi[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X ring[133X
  
  [33X[0;0YThe target (range) of the [5Xhomalg[105X ring map [3Xphi[103X.[133X
  
  [1X4.4-3 DegreeOfMorphism[101X
  
  [29X[2XDegreeOfMorphism[102X( [3Xphi[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Yan integer[133X
  
  [33X[0;0YThe degree of the morphism [3Xphi[103X of graded rings.[133X
  [33X[0;0Y(no method installed)[133X
  
  [1X4.4-4 CoordinateRingOfGraph[101X
  
  [29X[2XCoordinateRingOfGraph[102X( [3Xphi[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ya [5Xhomalg[105X ring[133X
  
  [33X[0;0YThe coordinate ring of the graph of the ring map [3Xphi[103X.[133X
  
  
  [1X4.5 [33X[0;0YRing Maps: Operations and Functions[133X[101X
  
