Ideal -- the class of all ideals

Description

For basic information about ideals in Macaulay 2, see ideals.

Common ways to make an ideal:

ideal -- make an ideal
annihilator -- the annihilator ideal
content -- the content of a polynomial
fittingIdeal -- Fitting ideal of a module
kernel(RingMap) -- kernel of a ringmap
minors -- ideal generated by minors
pfaffians -- ideal generated by Pfaffians

Common ways to get information about an ideal:

generators(Ideal) -- the generator matrix of an ideal
isSubset(Ideal,Ideal) -- whether one object is a subset of another

Common operations on ideals:

Ideal + Ideal -- sum of ideals
Ideal * Ideal -- product of ideals
Ideal == Ideal -- equality
Ideal == ZZ
Ideal ^ ZZ -- power
trim(Ideal)

Groebner bases, normal forms, free resolutions

gb -- compute a Groebner basis
leadTerm -- get the greatest term
codim -- compute the codimension
dim -- compute the Krull dimension
Matrix % Ideal -- calculate the normal form of ring elements and matrices
resolution -- projective resolution
betti -- display degrees

Numeric information about homogeneous ideals

degree
poincare -- assemble degrees into polynomial
hilbertFunction -- compute the Hilbert function
hilbertPolynomial -- compute the Hilbert polynomial
hilbertSeries -- compute the Hilbert series
genera -- list of the successive linear sectional arithmetic genera
euler -- Euler characteristic

Primary decomposition and components of an ideal

minimalPrimes -- minimal associated primes of an ideal
radical -- the radical of an ideal
associatedPrimes -- find the associated primes of an ideal
primaryDecomposition -- irredundant primary decomposition of an ideal
topComponents -- compute top dimensional component
saturate -- saturation of ideal or submodule
quotient -- quotient or division
Ideal : Ideal -- ideal or submodule quotient
intersect -- compute an intersection

Ideals from geometry

Fano -- Fano scheme
Grassmannian -- the Grassmannian of linear subspaces of a vector space
monomialCurveIdeal -- make the ideal of a monomial curve
singularLocus -- singular locus

Common ways to use an ideal:

Ring / Ideal -- make a quotient ring

An ideal I is an immutable object, so if you want to cache information about it, put it in the hash table I.cache.

Types of ideal :

MonomialIdeal -- the class of all monomial ideals handled by the engine

Functions and methods returning an ideal :

Ideal * Ring, see * -- a binary operator, usually used for multiplication
MonomialIdeal * Ring, see * -- a binary operator, usually used for multiplication
Ring * Ideal, see * -- a binary operator, usually used for multiplication
Ring * MonomialIdeal, see * -- a binary operator, usually used for multiplication
RingElement * Ideal, see * -- a binary operator, usually used for multiplication
annihilator(CoherentSheaf), see annihilator -- the annihilator ideal
annihilator(Ideal), see annihilator -- the annihilator ideal
annihilator(Module), see annihilator -- the annihilator ideal
annihilator(RingElement), see annihilator -- the annihilator ideal
conductor(RingMap), see conductor -- compute the conductor of a finite ring map
content(RingElement) -- the content of a polynomial
Fano(ZZ,Ideal) -- Fano scheme
Fano(ZZ,Ideal,Ring) -- Fano scheme
fittingIdeal -- Fitting ideal of a module
fittingIdeal(ZZ,Module), see fittingIdeal -- Fitting ideal of a module
graphIdeal(RingMap) -- the ideal of the graph of the regular map corresponding to a ring map
Grassmannian, see Grassmannian(ZZ,ZZ) -- the Grassmannian of linear subspaces of a vector space
homogenize(Ideal,RingElement), see homogenize -- homogenize with respect to a variable
ideal -- make an ideal
Ideal * Ideal -- product of ideals
Ideal * MonomialIdeal, see Ideal * Ideal -- product of ideals
MonomialIdeal * Ideal, see Ideal * Ideal -- product of ideals
Ideal + Ideal -- sum of ideals
Ideal + MonomialIdeal, see Ideal + Ideal -- sum of ideals
MonomialIdeal + Ideal, see Ideal + Ideal -- sum of ideals
Ideal ^ ZZ -- power
ideal(List) -- make an ideal
ideal(Sequence), see ideal(List) -- make an ideal
ideal(Matrix) -- make an ideal
ideal(Module) -- converts a module to an ideal
ideal(RingElement) -- make an ideal
ideal(String) -- make an ideal using classic Macaulay syntax
kernel(RingMap) -- kernel of a ringmap
lift(Ideal,type of RingElement), see lift -- lift to another ring
localize(Ideal,Ideal) -- localize an ideal at a prime ideal
minors(ZZ,Matrix)
permanents(ZZ,Matrix), see permanents -- ideal generated by square permanents of a matrix
pfaffians -- ideal generated by Pfaffians
pfaffians(ZZ,Matrix), see pfaffians -- ideal generated by Pfaffians
primaryComponent(Ideal,Ideal) -- find a primary component corresponding to an associated prime
Ideal : Ideal, see quotient(Ideal,Ideal) -- ideal or submodule quotient
Ideal : RingElement, see quotient(Ideal,Ideal) -- ideal or submodule quotient
Module : Module, see quotient(Ideal,Ideal) -- ideal or submodule quotient
quotient(Ideal,Ideal) -- ideal or submodule quotient
quotient(Ideal,RingElement), see quotient(Ideal,Ideal) -- ideal or submodule quotient
quotient(Module,Module), see quotient(Ideal,Ideal) -- ideal or submodule quotient
radical(Ideal), see radical -- the radical of an ideal
removeLowestDimension(Ideal), see removeLowestDimension -- remove components of lowest dimension
RingMap Ideal, see RingMap RingElement -- apply a ring map
saturate(Ideal), see saturate -- saturation of ideal or submodule
saturate(Ideal,Ideal), see saturate -- saturation of ideal or submodule
saturate(Ideal,RingElement), see saturate -- saturation of ideal or submodule
saturate(MonomialIdeal,RingElement), see saturate -- saturation of ideal or submodule
Schubert, see Schubert(ZZ,ZZ,VisibleList) -- find the Pluecker ideal of a Schubert variety
substitute(Ideal,List), see substitute -- substituting values for variables
substitute(Ideal,Matrix), see substitute -- substituting values for variables
substitute(Ideal,Ring), see substitute -- substituting values for variables
tangentCone, see tangentCone(Ideal)
topComponents(Ideal) -- compute top dimensional component
trim(Ideal)
truncate(List,Ideal), see truncate -- truncate the module at a specified degree
truncate(ZZ,Ideal), see truncate -- truncate the module at a specified degree

Methods that use an ideal :

Ideal * CoherentSheaf, see * -- a binary operator, usually used for multiplication
Ideal * Module, see * -- a binary operator, usually used for multiplication
Ideal * Vector, see * -- a binary operator, usually used for multiplication
Ideal + RingElement, see + -- a unary or binary operator, usually used for addition
Ideal == Ideal, see == -- equality
associatedPrimes(Ideal), see associatedPrimes -- find the associated primes of an ideal
basis(Ideal), see basis -- basis of all or part of a module or ring
basis(InfiniteNumber,InfiniteNumber,Ideal), see basis -- basis of all or part of a module or ring
basis(InfiniteNumber,List,Ideal), see basis -- basis of all or part of a module or ring
basis(InfiniteNumber,ZZ,Ideal), see basis -- basis of all or part of a module or ring
basis(List,Ideal), see basis -- basis of all or part of a module or ring
basis(List,InfiniteNumber,Ideal), see basis -- basis of all or part of a module or ring
basis(List,List,Ideal), see basis -- basis of all or part of a module or ring
basis(List,ZZ,Ideal), see basis -- basis of all or part of a module or ring
basis(ZZ,Ideal), see basis -- basis of all or part of a module or ring
basis(ZZ,InfiniteNumber,Ideal), see basis -- basis of all or part of a module or ring
basis(ZZ,List,Ideal), see basis -- basis of all or part of a module or ring
basis(ZZ,ZZ,Ideal), see basis -- basis of all or part of a module or ring
betti(Ideal) -- gives the degrees of generators.
codim(Ideal) -- compute the codimension
CoherentSheaf / Ideal, see CoherentSheaf / CoherentSheaf -- quotient of coherent sheaves
comodule(Ideal), see comodule -- submodule to quotient module
degree(Ideal)
degrees(Ideal) -- degrees of generators
dim(Ideal) -- compute the Krull dimension
eliminate(List,Ideal), see eliminate
eliminate(RingElement,Ideal), see eliminate
euler(Ideal) -- Euler characteristic
eulers(Ideal) -- list the sectional Euler characteristics
Ext(Ideal,Ideal), see Ext(Module,Module) -- total Ext module
Ext(Ideal,Module), see Ext(Module,Module) -- total Ext module
Ext(Ideal,Ring), see Ext(Module,Module) -- total Ext module
Ext(Module,Ideal), see Ext(Module,Module) -- total Ext module
Ext^ZZ(Matrix,Ideal), see Ext^ZZ(Matrix,Module) -- map between Ext modules
Ext^ZZ(Ideal,Matrix), see Ext^ZZ(Module,Matrix) -- map between Ext modules
Ext^ZZ(Ideal,Ideal), see Ext^ZZ(Module,Module) -- Ext module
Ext^ZZ(Ideal,Module), see Ext^ZZ(Module,Module) -- Ext module
Ext^ZZ(Ideal,Ring), see Ext^ZZ(Module,Module) -- Ext module
Ext^ZZ(Module,Ideal), see Ext^ZZ(Module,Module) -- Ext module
gb(Ideal), see gb -- compute a Groebner basis
genera(Ideal) -- list of the successive linear sectional arithmetic genera
generator(Ideal), see generator -- provide a single generator
Ideal _ ZZ, see generators of ideals and modules
generators(Ideal) -- the generator matrix of an ideal
genus(Ideal), see genus(CoherentSheaf) -- arithmetic genus
hilbertFunction(List,Ideal) -- compute the Hilbert function of the quotient of the ambient ring by an ideal
hilbertFunction(ZZ,Ideal), see hilbertFunction(List,Ideal) -- compute the Hilbert function of the quotient of the ambient ring by an ideal
hilbertPolynomial(Ideal) -- compute the Hilbert polynomial of the quotient of the ambient ring by the ideal
hilbertSeries(Ideal) -- compute the Hilbert series of the quotient of the ambient ring by the ideal
Hom(Ideal,Ideal), see Hom(Module,Module) -- module of homomorphisms
Hom(Ideal,Module), see Hom(Module,Module) -- module of homomorphisms
Hom(Ideal,Ring), see Hom(Module,Module) -- module of homomorphisms
Hom(Module,Ideal), see Hom(Module,Module) -- module of homomorphisms
Hom(Ring,Ideal), see Hom(Module,Module) -- module of homomorphisms
Ideal / Ideal -- quotient module
Ideal _ List -- missing documentation
independentSets(Ideal), see independentSets -- some maximal independent subsets of variables modulo an ideal
irreducibleCharacteristicSeries(Ideal), see irreducibleCharacteristicSeries -- irreducible characteristic series of an ideal
isHomogeneous(Ideal), see isHomogeneous -- whether something is homogeneous (graded)
isIdeal(Ideal), see isIdeal -- whether something is an ideal
isMonomialIdeal(Ideal), see isMonomialIdeal -- whether something is a monomial ideal
isPrime(Ideal), see isPrime -- whether a integer, polynomial, or ideal is prime
isSubset(Ideal,Ideal) -- whether one object is a subset of another
isSubset(Ideal,Module), see isSubset(Module,Module) -- whether one object is a subset of another
isSubset(Module,Ideal), see isSubset(Module,Module) -- whether one object is a subset of another
jacobian(Ideal) -- the Jacobian matrix of the generators of an ideal
leadTerm(Ideal) -- get the ideal of greatest terms
leadTerm(ZZ,Ideal) -- get the ideal of lead polynomials
Matrix % Ideal, see methods for normal forms and remainder -- calculate the normal form of ring elements and matrices
RingElement % Ideal, see methods for normal forms and remainder -- calculate the normal form of ring elements and matrices
mingens(Ideal), see mingens(Module) -- minimal generator matrix
minimalPresentation(Ideal) -- compute a minimal presentation of the quotient ring defined by an ideal
minimalPrimes(Ideal), see minimalPrimes -- minimal associated primes of an ideal
Module / Ideal, see Module / Module -- quotient module
module(Ideal) -- turn an ideal into a module
monomialIdeal(Ideal) -- monomial ideal of lead terms of a Groebner basis
monomialSubideal(Ideal), see monomialSubideal -- find the largest monomial ideal in an ideal
numgens(Ideal) -- number of generators of an ideal
poincare(Ideal) -- assemble degrees of the quotient of the ambient ring by an ideal into a polynomial
preimage(RingMap,Ideal), see preimage -- preimage of an ideal under a ring map
primaryDecomposition(Ideal), see primaryDecomposition -- irredundant primary decomposition of an ideal
Module : Ideal, see quotient(Ideal,Ideal) -- ideal or submodule quotient
quotient(Module,Ideal), see quotient(Ideal,Ideal) -- ideal or submodule quotient
regularity(Ideal), see regularity -- compute the Castelnuovo-Mumford regularity
resolution(Ideal) -- compute a projective resolution of (the quotient ring corresponding to) an ideal
ring(Ideal), see ring -- Get the associated ring of an object
Ring / Ideal -- make a quotient ring
saturate(Module,Ideal), see saturate -- saturation of ideal or submodule
singularLocus(Ideal), see singularLocus -- singular locus
substitute(Ideal,Option), see substitute -- substituting values for variables
tangentCone(Ideal)
variety(Ideal) -- the closed projective subvariety defined by an ideal

For the programmer

The object Ideal is a type, with ancestor classes HashTable < Thing.