### D.4.16 modules_lib

Library:
  modules.lib
Purpose:
  Modules

Authors:
  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

Overview:
  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 homogeneous. A matrix of this new form is created by a normal matrix, source, target and the graduatin, if the matrix is homogeneous, 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!!!

Types:
  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

Procedures:

> D.4.16.1 id |  | return a nxn identity Matrix
> D.4.16.2 zero |  | return a nxm zero Matrix
> D.4.16.3 freeModule |  | creating a graded free module
> D.4.16.4 makeMatrix |  | creating a Matrix with graded target and source if the matrix is homogeneous. If # is set to 1, makeMatrix ignores the grading of source & target.
> D.4.16.5 makeIdeal |  | creates an Ideal from an given ideal, is used to compute a resolution of the ideal
> D.4.16.6 Target |  | return target of the Matrix
> D.4.16.7 Source |  | return source of the Matrix
> D.4.16.8 printMatrix |  | print a Matrix
> D.4.16.9 printFreeModule |  | print a FreeModule
> D.4.16.10 printResolution |  | print a Resolution
> D.4.16.11 printModule |  | print a Module
> D.4.16.12 printHom |  | print a Homomorphism
> D.4.16.13 mRes |  | return a minimized graded Resolution
> D.4.16.14 sRes |  | return a graded Resolution computet with Schreyer's method
> D.4.16.15 Res |  | return a graded Resolution
> D.4.16.16 Betti |  | return the Betti-Matrix of the Resolution
> D.4.16.17 printBetti |  | prints the Betti-matrix of the Resolution
> D.4.16.18 SetDeg |  | sets an own graduatuation for the monomials
> D.4.16.19 Deg |  | same as deg, but can be used with an own graduation
> D.4.16.20 Degree |  | return list with degrees of the module
> D.4.16.21 Degrees |  | return list with degrees of the module
> D.4.16.22 subquotient |  | return a Module, the subquotient of the two Matrices
> D.4.16.23 coker |  | return a Module, the cokernel of the Matrix
> D.4.16.24 image |  | return a Module, the image of the Matrix
> D.4.16.25 Ker |  | return a Module, the kernel of the Matrix
> D.4.16.26 compareModules |  | return 0 or 1, compares the two Modules up to isomorphism
> D.4.16.27 addModules |  | return a Module, sum of the two Modules
> D.4.16.28 homomorphism |  | creates a R-Modul-Homomorphism
> D.4.16.29 target |  | return a Module, target of the Homomorphism
> D.4.16.30 source |  | return a Module, source of the Homomorphism
> D.4.16.31 compareMatrix |  | return 0 or 1, compares two Matrices
> D.4.16.32 freeModule2Module |  | converts a FreeModule into a Module
> D.4.16.33 makeVector |  | creates Vector in the given Module
> D.4.16.34 netVector |  | prints Vector
> D.4.16.35 netMatrix |  | prints Matrix
> D.4.16.36 presentation |  | converts M as a Subquotient to the Coker of a matrix C
> D.4.16.37 tensorMatrix |  | computes tensorproduct of two Matrices
> D.4.16.38 tensorModule |  | computes tensorproduct of two Modules
> D.4.16.39 tensorModFreemod |  | computes tensorproduct of Module and FreeModule
> D.4.16.40 tensorFreemodMod |  | computes tensorproduct of FreeModule and Module
> D.4.16.41 tensorFreeModule |  | computes tensorproduct ot two FreeModules
> D.4.16.42 tensorProduct |  | computes tensorproduct
> D.4.16.43 pruneModule |  | simplifies the presentation of a Module
> D.4.16.44 hom |  | computes Hom(M,N)
> D.4.16.45 kerHom |  | computes the kernel of a Homomorphism
> D.4.16.46 interpret |  | interprets the Vector in some Module or abstract space
> D.4.16.47 interpretInv |  | interprets a Vector or Homomorphism into the given Module
> D.4.16.48 reduceIntChain |  | reduces a chain of interpretations to minimal size or # steps
> D.4.16.49 interpretElem |  | interpret a Vector with # steps or until can't interpret further
> D.4.16.50 interpretList |  | interpret a list of Vectors as far as possible
> D.4.16.51 compareVectors |  | compares two Vectors with regard to the relations of their Module
> D.4.16.52 simplePrune |  | simplify module

User manual for Singular version 4.4.1, 2025, generated by texi2html.