Singular

D.15.11 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.15.11.1 id		return a nxn identity Matrix
D.15.11.2 zero		return a nxm zero Matrix
D.15.11.3 freeModule		creating a graded free module
D.15.11.4 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.15.11.5 makeIdeal		creates an Ideal from an given ideal, is used to compute a resolution of the ideal
D.15.11.6 Target		return target of the Matrix
D.15.11.7 Source		return source of the Matrix
D.15.11.8 printMatrix		print a Matrix
D.15.11.9 printFreeModule		print a FreeModule
D.15.11.10 printResolution		print a Resolution
D.15.11.11 printModule		print a Module
D.15.11.12 printHom		print a Homomorphism
D.15.11.13 mRes		return a minimized graded Resolution
D.15.11.14 sRes		return a graded Resolution computet with Schreyer's method
D.15.11.15 Res		return a graded Resolution
D.15.11.16 Betti		return the Betti-Matrix of the Resolution
D.15.11.17 printBetti		prints the Betti-matrix of the Resolution
D.15.11.18 SetDeg		sets an own graduatuation for the monomials
D.15.11.19 Deg		same as deg, but can be used with an own graduation
D.15.11.20 Degree		return list with degrees of the module
D.15.11.21 Degrees		return list with degrees of the module
D.15.11.22 subquotient		return a Module, the subquotient of the two Matrices
D.15.11.23 coker		return a Module, the cokernel of the Matrix
D.15.11.24 image		return a Module, the image of the Matrix
D.15.11.25 Ker		return a Module, the kernel of the Matrix
D.15.11.26 compareModules		return 0 or 1, compares the two Modules up to isomorphism
D.15.11.27 addModules		return a Module, sum of the two Modules
D.15.11.28 homomorphism		creates a R-Modul-Homomorphism
D.15.11.29 target		return a Module, target of the Homomorphism
D.15.11.30 source		return a Module, source of the Homomorphism
D.15.11.31 compareMatrix		return 0 or 1, compares two Matrices
D.15.11.32 freeModule2Module		converts a FreeModule into a Module
D.15.11.33 makeVector		creates Vector in the given Module
D.15.11.34 netVector		prints Vector
D.15.11.35 netMatrix		prints Matrix
D.15.11.36 presentation		converts M as a Subquotient to the Coker of a matrix C
D.15.11.37 tensorMatrix		computes tensorproduct of two Matrices
D.15.11.38 tensorModule		computes tensorproduct of two Modules
D.15.11.39 tensorModFreemod		computes tensorproduct of Module and FreeModule
D.15.11.40 tensorFreemodMod		computes tensorproduct of FreeModule and Module
D.15.11.41 tensorFreeModule		computes tensorproduct ot two FreeModules
D.15.11.42 tensorProduct		computes tensorproduct
D.15.11.43 pruneModule		simplifies the presentation of a Module
D.15.11.44 hom		computes Hom(M,N)
D.15.11.45 kerHom		computes the kernel of a Homomorphism
D.15.11.46 interpret		interprets the Vector in some Module or abstract space
D.15.11.47 interpretInv		interprets a Vector or Homomorphism into the given Module
D.15.11.48 reduceIntChain		reduces a chain of interpretations to minimal size or # steps
D.15.11.49 interpretElem		interpret a Vector with # steps or until can't interpret further
D.15.11.50 interpretList		interpret a list of Vectors as far as possible
D.15.11.51 compareVectors		compares two Vectors with regard to the relations of their Module
D.15.11.52 simplePrune		simplify module