chap5.txt

  
  5 Homogeneous Matrices
  
  The  package  GradedRingForHomalg  defines the classes of graded rings, ring
  elements  and  homogeneous 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.
  
  
  5.1 Homogeneous Matrices: Category and Representations
  
  5.1-1 IsHomalgMatrixOverGradedRingRep
  
  IsHomalgMatrixOverGradedRingRep( A )  Representation
  Returns:  true or false
  
  The representation of homalg matrices with entries in a homalg graded ring.
  
  (It is a representation of the GAP category IsMatrixOverGradedRing.)
  
    Code  
    DeclareRepresentation( "IsHomalgMatrixOverGradedRingRep",
            IsMatrixOverGradedRing,
            [ ] );
  
  
  
  5.2 Homogeneous Matrices: Constructors
  
  5.2-1 MatrixOverGradedRing
  
  MatrixOverGradedRing( mat, S )  function
  Returns:  a matrix over a graded ring
  
  Creates  a  matrix  for  the  graded  ring  S,  where  mat  is a matrix over
  UnderlyingNonGradedRing(S).
  
  
  5.3 Homogeneous Matrices: Attributes
  
  5.3-1 DegreesOfEntries
  
  DegreesOfEntries( A )  attribute
  Returns:  a listlist of degrees/multi-degrees
  
  The matrix of degrees of the matrix A.
  
  5.3-2 NonTrivialDegreePerRow
  
  NonTrivialDegreePerRow( A[, col_degrees] )  attribute
  Returns:  a list of degrees/multi-degrees
  
  The list of first nontrivial degree per row of the matrix A.
  
  5.3-3 NonTrivialDegreePerColumn
  
  NonTrivialDegreePerColumn( A[, row_degrees] )  attribute
  Returns:  a list of degrees/multi-degrees
  
  The list of first nontrivial degree per column of the matrix A.
  
  
  5.4 Homogeneous Matrices: Operations and Functions
  
  5.4-1 UnderlyingNonGradedRing
  
  UnderlyingNonGradedRing( mat )  operation
  Returns:  a homalg ring
  
  The nongraded ring underlying HomalgRing(mat).
  
  5.4-2 SetMatElm
  
  SetMatElm( mat, i, j, r, R )  operation
  
  Changes  the  entry  (i,j)  of  the matrix mat to the value r. Here R is the
  graded homalg ring involved in these computations.
  
  5.4-3 AddToMatElm
  
  AddToMatElm( mat, i, j, r, R )  operation
  
  Changes  the entry (i,j) of the matrix mat by adding the value r to it. Here
  R is the (graded) homalg ring involved in these computations.
  
  5.4-4 MatElmAsString
  
  MatElmAsString( mat, i, j, R )  operation
  Returns:  a string
  
  Returns  the  entry  (i,j)  of  the  matrix  mat  as a string. Here R is the
  (graded) homalg ring involved in these computations.
  
  5.4-5 MatElm
  
  MatElm( mat, i, j, R )  operation
  Returns:  a graded ring element
  
  Returns  the  entry  (i,j)  of the matrix mat. Here R is the (graded) homalg
  ring involved in these computations.