chap1.txt

  
  1 Introduction
  
  
  1.1 What is the role of the MatricesForHomalg package in the homalg project?
  
  
  1.1-1 MatricesForHomalg provides ...
  
  The package MatricesForHomalg provides:
  
      rings
  
      ring elements
  
      ring maps
  
      matrices
  
  
  1.1-2 homalg delegates ...
  
  The package homalg delegates all matrix operations as it treats matrices and
  their  rings  as black boxes. homalg comes with a single predefined class of
  rings  and  a  single  predefined  class of matrices over these rings -- the
  so-called  internal  matrices (--> 5.1-2) over so-called internal rings (-->
  3.1-4).  An  internal  matrix  (resp. ring) is simply a wrapper containing a
  GAP-builtin  matrix  (resp.  ring).  homalg  allows other packages to define
  further  classes  or  extend existing classes of rings and matrices together
  with their operations. For example:
  
      The  homalg  subpackage  ResidueClassRingForHomalg  (-->  Appendix  D)
        defines  the  classes  of  residue  class  rings,  residue  class ring
        elements,  and  matrices  over  residue  class rings. Such a matrix is
        defined  by  a  matrix  over  the  ambient  ring which is nevertheless
        interpreted  modulo  the ring relations, i.e. modulo the generators of
        the defining ideal.
  
      The  package  GaussForHomalg  extends  the  class of internal matrices
        enabling  it  to  wrap  sparse matrices provided by the package Gauss.
        GaussForHomalg delegates the essential part of the matrix creation and
        all matrix operations to Gauss.
  
      The  package  HomalgToCAS  defines  the  classes of so-called external
        rings  and  matrices  and  the  package  RingsForHomalg  delegates the
        essential  part  of  the  matrix creation and all matrix operations to
        external  computer  algebra  systems  like  Singular, Macaulay2, Sage,
        Macaulay2,  MAGMA,  Maple,  ... . The package homalg accesses external
        matrices via pointers. The pointer of an external matrix is simply its
        name in the external system. HomalgToCAS chooses these names.
  
      The  package  LocalizeRingForHomalg defines the classes of local(ized)
        rings,  local ring elements, and local matrices. A homalg local matrix
        contains  a  homalg matrix as a numerator and an element of the global
        ring as a denominator.
  
  The  matrix  operations are divided into two classes called Tools and Basic.
  The   Tools   operations   include  addition,  subtraction,  multiplication,
  extracting  certain  rows or columns, stacking, and augmenting matrices (-->
  Appendix B). The Basic operations include the two basic operations in linear
  algebra   needed   to   solve  an  inhomogeneous  linear  system  XA=B  with
  coefficients in a not necessarily commutative ring R (--> Appendix A):
  
      Effectively  reducing  B  modulo A, i.e. effectively deciding if a row
        (or a set of rows) B lies in the R-span of the rows of the matrix A.
  
      Computing  an R-generating set of row syzygies (=R-relations among the
        rows)  of  A, i.e. computing an R-generating set of the left kernel of
        A.  This  generating  set  is then given as the rows of a matrix Y and
        YA=0.
  
  The  first  operation  is  nothing  but  deciding  the  solvability  of  the
  inhomogeneous  system  XA=B and if solvable to compute a particular solution
  X,  while  the  second is to compute an R-generating set for the homogeneous
  solution  space, i.e. the solution space of the homogeneous system YA=0. The
  above is of course also valid for the column convention.
  
  
  1.1-3 The black box concept
  
  Now we address the following concerns: Wouldn't the idea of using algorithms
  like the Gröbnerbasis algorithm(s) as a black box (--> 1.1-2) contradict the
  following facts?
  
      It  is  known  that an efficient Gröbnerbasis algorithm depends on the
        ring  R  under  consideration.  For  example the implementation of the
        algorithm depends on the ground ring (or field) k.
  
      Often  enough  highly  specialized implementations are used to address
        specific  types  of  linear systems of equations (occuring in specific
        homological  problems)  in  order  to increase the speed or reduce the
        space needed for the computations.
  
  The following should clarify the above concerns.
  
      Since  each  ring  comes  with  its  own black box, the first point is
        automatically resolved.
  
      Allow  the  black  box  coming with each ring to contain the different
        available  implementations  and  make  them  accessible  to homalg via
        standarized  names, independent of the computer algebra system used to
        perform computations.
  
  
  1.2 This manual
  
  Chapter 2 describes the installation of this package. The remaining chapters
  are  each  devoted  to one of the MatricesForHomalg objects (--> 1.1-1) with
  its constructors, properties, attributes, and operations.