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D.15.21 modules_lib


J. Boehm, boehm@mathematik.uni-kl.de
D. Wienholz wienholz@mathematik.uni-kl.de
C. Koenen koenen@rhrk.uni-kl.de
M. Mayer mayer@mathematik.uni-kl.de

This library is used for the computation of graded free resolutions with an own graduation of the monomials. For these Resolution is a new class of modules needed. These modules, can be computed via the image, kernel, cokernel of a matrix or the subquotient of two matrices. The used matrices also have a free module as source and target, with graded generators if the matrix is homogenous. A matrix of this new form is created by a normal matrix, source, target and the graduatin, if the matrix is homogenous, are done automatically. With this matrices it is then possible to compute the new class of modules.
This library also offers the opppurtunity to create R-module-homomorphisms betweens two modules. For these homorphisms the kernel can be computed an will be returned as a module of the new class.

This is experimental work in progress!!!

Matrix the class of matrices with source and target in form of free modules FreeModule free modules representet with the ring and degree Resolution class of graded resolutions
Module modules represented by either the image, coker, kernelof a matrix or the subquotient of two matrices Vector element of a Module
Ideal same as ideal, but with it's own basering saved, used to compute resolutions Homomorphism class of R-module-homomormphisms


D.15.21.1 id  return a nxn identity Matrix
D.15.21.2 zero  return a nxm zero Matrix
D.15.21.3 freeModule  creating a graded free module
D.15.21.4 makeMatrix  creating a Matrix with graded target and source if the matrix is homogenous. If # is set to 1, makeMatrix ignores the grading of source & target.
D.15.21.5 makeIdeal  creates an Ideal from an given ideal, is used to compute a resolution of the ideal
D.15.21.6 Target  return target of the Matrix
D.15.21.7 Source  return source of the Matrix
D.15.21.8 printMatrix  print a Matrix
D.15.21.9 printFreeModule  print a FreeModule
D.15.21.10 printResolution  print a Resolution
D.15.21.11 printModule  print a Module
D.15.21.12 printHom  print a Homomorphism
D.15.21.13 mRes  return a minimized graded Resolution
D.15.21.14 sRes  return a graded Resolution computet with Schreyer's method
D.15.21.15 Res  return a graded Resolution
D.15.21.16 Betti  return the Betti-Matrix of the Resolution
D.15.21.17 printBetti  prints the Betti-matrix of the Resolution
D.15.21.18 SetDeg  sets an own graduatuation for the monomials
D.15.21.19 Deg  same as deg, but can be used with an own graduation
D.15.21.20 Degree  return list with degrees of the module
D.15.21.21 Degrees  return list with degrees of the module
D.15.21.22 subquotient  return a Module, the subquotient of the two Matrices
D.15.21.23 coker  return a Module, the cokernel of the Matrix
D.15.21.24 image  return a Module, the image of the Matrix
D.15.21.25 Ker  return a Module, the kernel of the Matrix
D.15.21.26 compareModules  return 0 or 1, compares the two Modules up to isomorphism
D.15.21.27 addModules  return a Module, sum of the two Modules
D.15.21.28 homomorphism  creates a R-Modul-Homomorphism
D.15.21.29 target  return a Module, target of the Homomorphism
D.15.21.30 source  return a Module, source of the Homomorphism
D.15.21.31 compareMatrix  return 0 or 1, compares two Matrices
D.15.21.32 freeModule2Module  converts a FreeModule into a Module
D.15.21.33 makeVector  creates Vector in the given Module
D.15.21.34 netVector  prints Vector
D.15.21.35 netMatrix  prints Matrix
D.15.21.36 presentation  converts M as a Subquotient to the Coker of a matrix C
D.15.21.37 tensorMatrix  computes tensorproduct of two Matrices
D.15.21.38 tensorModule  computes tensorproduct of two Modules
D.15.21.39 tensorModFreemod  computes tensorproduct of Module and FreeModule
D.15.21.40 tensorFreemodMod  computes tensorproduct of FreeModule and Module
D.15.21.41 tensorFreeModule  computes tensorproduct ot two FreeModules
D.15.21.42 tensorProduct  computes tensorproduct
D.15.21.43 pruneModule  simplifies the presentation of a Module
D.15.21.44 hom  computes Hom(M,N)
D.15.21.45 kerHom  computes the kernel of a Homomorphism
D.15.21.46 interpret  interpretes the Vector in some Module or abstract space
D.15.21.47 interpretInv  interpretes a Vector or Homomorphism into the given Module
D.15.21.48 reduceIntChain  reduces a chain of interpretations to minimal size or # steps
D.15.21.49 interpretElem  interpret a Vector with # steps or until can't interpret further
D.15.21.50 interpretList  interpret a list of Vectors as far as possible
D.15.21.51 compareVectors  compares two Vectors with regard to the relations of their Module
D.15.21.52 simplePrune  simplify module