Module -- the class of all modules

Module -- the class of all 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.

More general types (whose methods may also apply) :

Type -- the class of all types

MutableHashTable -- the class of all mutable hash tables

HashTable -- the class of all hash tables

See also:

ideals and modules

Common ways to make a module:

Ring ^ ZZ -- make a free module

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

cokernel Matrix -- cokernel of a map

image Matrix -- image of a map

kernel Matrix -- kernel of a map

Common ways to get information about modules:

ring Module -- get the associated ring

numgens Module -- the number of generators

degrees Module -- degrees of generators

generators Module -- matrix of generators

relations Module -- the defining relations

isFreeModule -- whether something is a free module

isHomogeneous Module -- test for homogeneity

rank -- compute the rank

ambient Module -- ambient free module of a subquotient, or ambient ring

cover Module -- get the covering free module

super Module -- get the ambient module

Common operations on modules:

Module + Module -- sum of submodules

Module / Module -- quotient module

Module == Module -- equality

Module ++ Module -- direct sum of modules

Module ** Module -- tensor product of modules

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

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

Numerical information about a module:

codim Module -- calculate the codimension

degree Module -- the degree

dim Module -- calculate the dimension

genera Module -- list the sectional arithmetic genera

hilbertSeries Module -- compute Hilbert series

hilbertFunction(ZZ,Module) -- Hilbert function of a module

poincare Module -- assemble degrees into polynomial

pdim Module -- calculate the projective dimension

regularity Module -- compute the regularity

rank Module -- compute the rank

Common computations on modules:

Module : Ideal -- a binary operator

annihilator Module -- the annihilator ideal

gb Module -- compute a Groebner basis

prune Module -- minimize generators and relations

resolution Module -- make a projective resolution

saturate(Module,Ideal) -- saturation of ideal or submodule

Hom -- module of homomorphisms

homomorphism Matrix -- get the homomorphism from element of Hom

Ext^ZZ(Module,Module) -- compute an Ext module

Tor_ZZ(Module,Module) -- compute a Tor module

homology(Matrix,Matrix) -- kernel modulo image

Common ways to use a module:

fittingIdeal(ZZ,Module) -- Fitting ideal of a module

isSubset(Module,Module) -- whether something is a subset of another

exteriorPower(ZZ,Module) -- exterior power

Functions and methods returning a a module :

ambient Module -- ambient free module of a subquotient, or ambient ring

ChainComplex ^ ZZ

ChainComplex _ ZZ -- get component

coimage Matrix

cokernel Matrix

cokernel RingElement

cover Module -- get the covering free module

dual Module -- dual module

Ext(Module,Module) -- compute an Ext module

Ext^ZZ(CoherentSheaf,CoherentSheaf)

Ext^ZZ(CoherentSheaf,SheafOfRings)

Ext^ZZ(CoherentSheaf,SumOfTwists)

Ext^ZZ(Module,Module) -- compute an Ext module

Ext^ZZ(SheafOfRings,CoherentSheaf)

Ext^ZZ(SheafOfRings,SheafOfRings)

Ext^ZZ(SheafOfRings,SumOfTwists)

exteriorPower(ZZ,Module)

GradedModule _ ZZ

HH^ZZ ChainComplex -- cohomology of a chain complex

HH^ZZ CoherentSheaf -- coherent sheaf cohomology

HH^ZZ Module -- local cohomology

HH^ZZ SheafOfRings

HH^ZZ SumOfTwists -- coherent sheaf cohomology

HH_ZZ ChainComplex -- homology of a chain complex

Hom(CoherentSheaf,CoherentSheaf)

Hom(Ideal,Ideal)

Hom(Ideal,Module)

Hom(Ideal,Ring)

Hom(Module,Ideal)

Hom(Module,Module)

Hom(Module,Ring)

Hom(Ring,Ideal)

Hom(Ring,Module)

homogenize(Module,RingElement)

homogenize(Module,RingElement,List)

homology(Matrix,Matrix) -- kernel modulo image

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 / Sequence

Module / List

Module / Ideal -- quotient module by an ideal

Module / Module -- quotient module

Module ^ ZZ -- make a direct sum of several copies of a module

module

module CoherentSheaf -- get the module defining a coherent sheaf

module Ideal -- turn an ideal into a module

module SheafOfRings

MonomialIdeal * Module

prune Module -- minimize generators and relations

pushForward(RingMap,Module)

pushForward1(RingMap,Module)

quotient(Module,Ideal)

quotient(Module,RingElement)

removeLowestDimension Module

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

Ring ^ ZZ -- make a free module

RingElement * Module

RingMap ** Module -- a binary operator, usually used for tensor product

RingMap Module -- blank operator for adjacent expressions

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,List)

substitute(Module,Matrix)

substitute(Module,Ring)

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

super Module

tensor(Module,Module)

top Module

Tor_ZZ(Module,Module)

trim Module

truncate(List,Module)

truncate(ZZ,Module)

Methods for using a module :

adjoint(Matrix,Module,Module)

adjoint1(Matrix,Module,Module)

ann' Module

annihilator Module -- the annihilator ideal

basis Module -- basis of a module

basis({...},Module) -- basis of the part of a module of a certain degree

basis(ZZ,Module)

betti Module

ChainComplex ** Module

ChainComplexMap ** Module

codim Module -- calculate the codimension

degrees Module -- degrees of generators

Ext(Ideal,Module)

Ext(Module,Ideal)

Ext(Module,Ring)

Ext^ZZ(Ideal,Module)

Ext^ZZ(Matrix,Module)

Ext^ZZ(Module,Ideal)

Ext^ZZ(Module,Matrix)

Ext^ZZ(Module,Ring)

fittingIdeal(ZZ,Module)

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

gb Module

genera Module

generators Module -- matrix of generators

GradedModule ** Module

GradedModule ++ Module

gradedModule Module

hilbertFunction(List,Module)

hilbertFunction(ZZ,Module)

hilbertPolynomial Module

hilbertSeries Module

Hom(ChainComplex,Module) -- Hom

Hom(ChainComplexMap,Module)

Hom(Matrix,Module)

Hom(Module,ChainComplex)

Hom(Module,ChainComplexMap)

Hom(Module,Matrix)

Ideal == 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 -- whether something is a free module

isHomogeneous Module

isIdeal Module -- whether something is an ideal

isModule Module

isQuotientModule Module -- whether a module is evidently a quotient of a free module

isSubmodule Module -- whether a module is evidently a submodule of a free 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,{...}) -- 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,Nothing,{...}) -- make a map

map(Module,Nothing,Matrix)

map(Module,RingElement) -- make a map

map(Module,ZZ)

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

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

Matrix % Module

Matrix ** Module -- tensor product

mingens Module

Module ** ChainComplex

Module ** ChainComplexMap

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 -- tensor power

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

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

Module _ ZZ -- get a generator

Module ~ -- make a coherent sheaf

new Module from Ring

quotient(Module,Module)

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

rank Module

regularity Module

relations Module

reshape(Module,Module,Matrix)

resolution Module -- make a projective resolution

Ring / Module

sheaf Module -- make a coherent sheaf

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

substitute(Module,Option)

symmetricAlgebra Module

tensorAssociativity(Module,Module,Module)

toExternalString Module

toString Module

wedgeProduct(ZZ,ZZ,Module)

ZZ * Module

ZZ == Module

ZZ _ Module