chap4.txt

  
  4 Graded Rings
  
  The  package  GradedRingForHomalg  defines the classes of graded rings, ring
  elements  and  matrices  over such rings. These three objects can be used as
  data structures defined in MatricesForHomalg on which the homalg project can
  rely to do homological computations over graded rings.
  
  The  graded  rings  most  prominently  can  be  used with methods known from
  general  homalg  rings. The methods for doing the computations are presented
  in the appendix (B), 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 MatricesForHomalg, 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 Graded Rings: Category and Representations
  
  4.1-1 IsHomalgGradedRingRep
  
  IsHomalgGradedRingRep( R )  Representation
  Returns:  true or false
  
  The representation of homalg graded rings.
  
  (It is a subrepresentation of the GAP representation
  IsHomalgRingOrFinitelyPresentedModuleRep.)
  
    Code  
    DeclareRepresentation( "IsHomalgGradedRingRep",
            IsHomalgGradedRing and
            IsHomalgGradedRingOrGradedModuleRep,
            [ "ring" ] );
  
  
  4.1-2 IsHomalgGradedRingElementRep
  
  IsHomalgGradedRingElementRep( r )  Representation
  Returns:  true or false
  
  The representation of elements of homalg graded rings.
  
  (It is a representation of the GAP category IsHomalgRingElement.)
  
    Code  
    DeclareRepresentation( "IsHomalgGradedRingElementRep",
            IsHomalgGradedRingElement,
            [ ] );
  
  
  
  4.2 Graded Rings: Constructors
  
  4.2-1 HomalgGradedRingElement
  
  HomalgGradedRingElement( numer, denom, R )  function
  HomalgGradedRingElement( numer, R )  function
  Returns:  a graded ring element
  
  Creates  the  graded  ring element numer/denom or in the second case numer/1
  for  the  graded  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 Graded Rings: Attributes
  
  4.3-1 DegreeGroup
  
  DegreeGroup( S )  attribute
  Returns:  a left ℤ-module
  
  The degree Abelian group of the commutative graded ring S.
  
  4.3-2 CommonNonTrivialWeightOfIndeterminates
  
  CommonNonTrivialWeightOfIndeterminates( S )  attribute
  Returns:  a degree
  
  The  common  nontrivial weight of the indeterminates of the graded ring S if
  it  exists. Otherwise an error is issued. WARNING: Since the DegreeGroup and
  WeightsOfIndeterminates  are  in some cases bound together, you MUST not set
  the  DegreeGroup  by hand and let the algorithm create the weights. Set both
  by  hand,  set only weights or use the method WeightsOfIndeterminates to set
  both. Never set the DegreeGroup without the WeightsOfIndeterminates, because
  it simply wont work!
  
  4.3-3 WeightsOfIndeterminates
  
  WeightsOfIndeterminates( S )  attribute
  Returns:  a list or listlist of integers
  
  The list of degrees of the indeterminates of the graded ring S.
  
  4.3-4 MatrixOfWeightsOfIndeterminates
  
  MatrixOfWeightsOfIndeterminates( S )  attribute
  Returns:  a homalg matrix
  
  A   homalg   matrix   where  the  list  (or  listlist)  of  degrees  of  the
  indeterminates of the graded ring S is stored.
  
  
  4.4 Graded Rings: Operations and Functions
  
  4.4-1 UnderlyingNonGradedRing
  
  UnderlyingNonGradedRing( R )  operation
  Returns:  a homalg ring
  
  Internally there is a ring, in which computations take place.
  
  4.4-2 UnderlyingNonGradedRing
  
  UnderlyingNonGradedRing( r )  operation
  Returns:  a homalg ring
  
  Internally there is a ring, in which computations take place.
  
  4.4-3 Name
  
  Name( r )  operation
  Returns:  a string
  
  The name of the graded ring element r.