chap4.txt

  
  4 Ring Maps
  
  A  homalg  ring  map is a data structure for maps between finitely generated
  rings.  homalg 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
  Sheaves provides methods for ring maps of (finitely generated) affine rings.
  
  
  4.1 Ring Maps: Category and Representations
  
  4.1-1 IsHomalgRingMap
  
  IsHomalgRingMap( phi )  Category
  Returns:  true or false
  
  The GAP category of ring maps.
  
  4.1-2 IsHomalgRingSelfMap
  
  IsHomalgRingSelfMap( phi )  Category
  Returns:  true or false
  
  The GAP category of ring self-maps.
  
  (It is a subcategory of the GAP category IsHomalgRingMap.)
  
  4.1-3 IsHomalgRingMapRep
  
  IsHomalgRingMapRep( phi )  Representation
  Returns:  true or false
  
  The GAP representation of homalg ring maps.
  
  (It is a representation of the GAP category IsHomalgRingMap (4.1-1).)
  
  
  4.2 Ring Maps: Constructors
  
  4.2-1 RingMap
  
  RingMap( images, S, T )  operation
  Returns:  a homalg ring map
  
  This  constructor  returns  a  ring map (homomorphism) of finitely generated
  rings/algebras.  It  is  represented  by  the  images  images  of the set of
  generators  of  the  source  homalg ring S in terms of the generators of the
  target  ring  T  (--> 3.2). Unless the source ring is free and given on free
  ring/algebra  generators the returned map will cautiously be indicated using
  parenthesis:  homomorphism. To verify if the result is indeed a well defined
  map  use IsMorphism (4.3-1). If source and target are identical objects, and
  only then, the ring map is created as a selfmap.
  
  
  4.3 Ring Maps: Properties
  
  4.3-1 IsMorphism
  
  IsMorphism( phi )  property
  Returns:  true or false
  
  Check  if  phi  is  a  well-defined  map,  i.e.  independent of all involved
  presentations.
  
  4.3-2 IsIdentityMorphism
  
  IsIdentityMorphism( phi )  property
  Returns:  true or false
  
  Check if the homalg ring map phi is the identity morphism.
  
  4.3-3 IsMonomorphism
  
  IsMonomorphism( phi )  property
  Returns:  true or false
  
  Check if the homalg ring map phi is a monomorphism.
  
  4.3-4 IsEpimorphism
  
  IsEpimorphism( phi )  property
  Returns:  true or false
  
  Check if the homalg ring map phi is an epimorphism.
  
  4.3-5 IsIsomorphism
  
  IsIsomorphism( phi )  property
  Returns:  true or false
  
  Check if the homalg ring map phi is an isomorphism.
  
  4.3-6 IsAutomorphism
  
  IsAutomorphism( phi )  property
  Returns:  true or false
  
  Check if the homalg ring map phi is an automorphism.
  
  
  4.4 Ring Maps: Attributes
  
  4.4-1 Source
  
  Source( phi )  attribute
  Returns:  a homalg ring
  
  The source of the homalg ring map phi.
  
  4.4-2 Range
  
  Range( phi )  attribute
  Returns:  a homalg ring
  
  The target (range) of the homalg ring map phi.
  
  4.4-3 DegreeOfMorphism
  
  DegreeOfMorphism( phi )  attribute
  Returns:  an integer
  
  The degree of the morphism phi of graded rings.
  (no method installed)
  
  4.4-4 CoordinateRingOfGraph
  
  CoordinateRingOfGraph( phi )  attribute
  Returns:  a homalg ring
  
  The coordinate ring of the graph of the ring map phi.
  
  
  4.5 Ring Maps: Operations and Functions