Module -- the class of all modules

Module -- the class of all modules

The most general module M is represented as a submodule of a quotient module of a free module F. The matrix of relations used to produce the quotient module is stored as M.relations (if there is a nonzero relation) and the matrix of generators is stored as M.generators (if the submodule is smaller than the free module).

Elements of modules are implemented as instances of the class Vector.

See also:

modules

The type Module is a member of the class Type. Each object of class Module is called a module. Each module is also a member of class Type.

Making a module :

ChainComplex ^ ZZ

ChainComplex _ ZZ -- get component

Ext(Module,Module)

Ext(ZZ,Module,Module)

GradedModule _ ZZ

HH^ZZ ChainComplex -- cohomology of a chain complex

HH^ZZ CoherentSheaf -- coherent sheaf cohomology

HH^ZZ Module -- local cohomology

HH_Matrix Matrix -- kernel modulo image

HH_ZZ ChainComplex -- homology of a chain complex

Hom(Ideal,Ideal)

Hom(Ideal,Module)

Hom(Ideal,Ring)

Hom(Module,Ideal)

Hom(Module,Module)

Hom(Module,Ring)

Hom(Ring,Ideal)

Hom(Ring,Module)

Ideal * Module

Ideal / Ideal -- quotient module

Module ** Module -- tensor product of modules

Module ** Ring -- tensor product

Module + Module -- sum of submodules

Module ++ Module -- direct sum of modules

Module / (...)

Module / Ideal -- quotient module by an ideal

Module / Module -- quotient module

Module / RingElement

Module / Vector

Module / {...}

Module : Ideal

Module : RingElement

Module ^ ZZ

Ring ^ ZZ -- make a free module

Ring ^ {...} -- make a free module

RingElement * Module

RingMap Module

Tor(ZZ,Module,Module)

ambient Module

coimage Matrix

cokernel Matrix

cokernel RingElement

cover Module

dual Module -- dual module

exteriorPower(ZZ,Module)

homogenize(Module,RingElement)

homogenize(Module,RingElement,{...})

image Matrix

image RingElement

kernel Matrix

kernel RingElement

module CoherentSheaf -- get the module defining a coherent sheaf

module Ideal -- turn an ideal into a module

prune Module

pushForward(RingMap,Module)

pushForward1(RingMap,Module)

quotient(Module,Ideal)

quotient(Module,RingElement)

removeLowestDimension Module

saturate Module

saturate Vector

saturate(Module,Ideal)

saturate(Module,RingElement)

subquotient -- make a subquotient module

subquotient(Matrix,Matrix)

subquotient(Matrix,Nothing)

subquotient(Nothing,Matrix)

substitute(Module,Matrix)

substitute(Module,Ring)

substitute(Module,{...})

sum ChainComplex -- direct sum of the components of a chain complex

super Module

top Module

trim Module

truncate(ZZ,Module)

truncate({...},Module)

Methods for using a module :

ChainComplex ** Module

Ext(ZZ,Ideal,Module)

Ext(ZZ,Matrix,Module)

Ext(ZZ,Module,Ideal)

Ext(ZZ,Module,Matrix)

Ext(ZZ,Module,Ring)

Ext(ZZ,Ring,Module)

GradedModule ** Module

GradedModule ++ Module

Hom(ChainComplex,Module) -- Hom

Hom(ChainComplexMap,Module)

Hom(Matrix,Module)

Hom(Module,ChainComplex)

Hom(Module,ChainComplexMap)

Hom(Module,Matrix)

Ideal == Module

Matrix % Module

Matrix ** Module -- tensor product

Module ** ChainComplex

Module ** GradedModule

Module ** Matrix

Module ++ GradedModule

Module : Module

Module == Ideal

Module == Module -- equality

Module == ZZ

Module [...] -- make a chain complex from a module

Module ^ [...] -- projection onto some factors of a direct sum module

Module ^ {...} -- projection map from a free module

Module _ ZZ -- get a generator

Module _ [...] -- get inclusion map into direct sum

Module _ {...} -- map from free module to some generators

Ring / Module

ZZ * Module

ZZ == Module

ZZ _ Module

annihilator Module

basis Module

basis(ZZ,Module)

basis({...},Module)

betti Module

codim Module

components Module

degree Module

degrees Module

dim Module

euler Module

fittingIdeal(ZZ,Module)

flip(Module,Module) -- matrix of commutativity of tensor product

gb Module

genera Module

generators Module

gradedModule Module

hilbertFunction(ZZ,Module)

hilbertFunction({...},Module)

hilbertPolynomial Module

hilbertSeries Module

ideal Module

inducedMap(Module,Module)

inducedMap(Module,Module,Matrix)

inducedMap(Module,Nothing,Matrix)

inducedMap(Nothing,Module,Matrix)

inducesWellDefinedMap(Module,Module,Matrix)

inducesWellDefinedMap(Module,Nothing,Matrix)

inducesWellDefinedMap(Nothing,Module,Matrix)

isDirectSum Module

isFreeModule Module

isHomogeneous Module

isIdeal Module

isModule Module

isQuotientModule Module

isSubmodule Module

isSubset(Ideal,Module)

isSubset(Module,Ideal)

isSubset(Module,Module)

map Module -- make a map

map(Module,Matrix) -- make a map

map(Module,Module) -- make a map

map(Module,Module,Function) -- make a map

map(Module,Module,Matrix) -- make a map

map(Module,Module,RingElement) -- make a map

map(Module,Module,ZZ) -- make a map

map(Module,Module,{...}) -- make a map

map(Module,Nothing,Matrix)

map(Module,Nothing,{...}) -- make a map

map(Module,RingElement) -- make a map

map(Module,ZZ)

map(Module,ZZ,Function) -- make a map

map(Module,ZZ,{...}) -- make a map

mingens Module

net Module

new Module from Ring

numgens Module

pdim Module

poincare Module

presentation Module

quotient(Module,Module)

random(Module,Module) -- make a random module map

rank Module

regularity Module

relations Module

resolution Module -- make a projective resolution

sheaf Module -- make a coherent sheaf

sheaf(Module,Variety) -- make a coherent sheaf

substitute(Module,Option)

symmetricAlgebra Module

tensorAssociativity(Module,Module,Module)

toExternalString Module

toString Module

wedgeProduct(ZZ,ZZ,Module)