Ideal -- the class of all ideals

Description

For basic information about ideals in Macaulay2, 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
Ideal _* -- get the list of generators of an ideal
isSubset(Ideal,Ideal) -- whether one object is a subset of another
isPrime(Ideal) -- whether an ideal is prime

Common operations on ideals:

Ideal + Ideal -- sum of ideals
Ideal * Ideal -- product of ideals
Ideal == Ideal -- equality
Ideal == ZZ -- equality
Ideal ^ ZZ -- power of an ideal
trim(Ideal) -- minimize generators and relations

Gröbner bases, normal forms, free resolutions

gb -- compute a Gröbner basis
leadTerm -- get the greatest term
codim -- compute the codimension
dim -- compute the Krull dimension
Matrix % Ideal -- normal form of ring elements and matrices
resolution -- projective resolution
betti -- display or modify a Betti diagram

Numeric information about homogeneous ideals

degree
poincare -- assemble degrees of a ring, module, or ideal into a polynomial
hilbertFunction -- 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 primes of an ideal
radical -- the radical of an ideal
associatedPrimes -- find associated primes
primaryDecomposition -- irredundant primary decomposition of an ideal
topComponents -- compute top dimensional component of an ideal or module
saturate(Ideal,Ideal) -- saturation of ideal or submodule
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 -- the conductor of a finite ring map
"content(RingElement)" -- see content -- the content of a polynomial
"content(RingElement,RingElement)" -- see content -- the content of a polynomial
"expectedReesIdeal(Module)" -- see expectedReesIdeal -- symmetric algebra ideal plus jacobian dual
Fano(ZZ,Ideal) -- Fano scheme
Fano(ZZ,Ideal,Ring) -- Fano scheme
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
"icPIdeal(RingElement,RingElement,ZZ)" -- see icPIdeal -- compute the integral closure in prime characteristic of a principal ideal
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 * RingElement (missing documentation)
Ideal + Ideal -- sum of ideals
"Ideal + MonomialIdeal" -- see Ideal + Ideal -- sum of ideals
"MonomialIdeal + Ideal" -- see Ideal + Ideal -- sum of ideals
Ideal ^ ZZ -- power of an ideal
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(Number)" -- see ideal(RingElement) -- make an ideal
ideal(RingElement) -- make an ideal
ideal(String) -- make an ideal using classic Macaulay syntax
"integralClosure(Ideal)" -- see integralClosure(Ideal,RingElement,ZZ) -- integral closure of an ideal in an affine domain
"integralClosure(Ideal,RingElement)" -- see integralClosure(Ideal,RingElement,ZZ) -- integral closure of an ideal in an affine domain
"integralClosure(Ideal,ZZ)" -- see integralClosure(Ideal,RingElement,ZZ) -- integral closure of an ideal in an affine domain
"intersect(Ideal)" -- see intersect(Ideal,Ideal) -- compute an intersection of a sequence of ideals or modules
intersect(Ideal,Ideal) -- compute an intersection of a sequence of ideals or modules
kernel(RingMap) -- kernel of a ringmap
"lift(Ideal,type of RingElement)" -- see lift -- lift to another ring
"localize(Ideal,Ideal)" -- see localize -- localize an ideal at a prime ideal
minimalPresentation(Ideal) -- compute a minimal presentation of the quotient ring defined by an ideal
"prune(Ideal)" -- see minimalPresentation(Ideal) -- compute a minimal presentation of the quotient ring defined by an ideal
"minimalReduction(Ideal)" -- see minimalReduction -- Find a minimal reduction of an ideal
minors(ZZ,Matrix) -- ideal generated by minors
"Ideal ** Ring" -- see Module ** Ring -- tensor product
"Ring ** Ideal" -- see Module ** Ring -- tensor product
"permanents(ZZ,Matrix)" -- see permanents -- ideal generated by square permanents of a matrix
pfaffians -- ideal generated by Pfaffians
"preimage(RingMap,Ideal)" -- see preimage -- preimage of a map
"primaryComponent(Ideal,Ideal)" -- see primaryComponent -- find a primary component corresponding to an associated prime
"Ideal : Ideal" -- see quotient(Module,Module) -- ideal or submodule quotient
"Ideal : RingElement" -- see quotient(Module,Module) -- ideal or submodule quotient
"Module : Module" -- see quotient(Module,Module) -- ideal or submodule quotient
"quotient(Ideal,Ideal)" -- see quotient(Module,Module) -- ideal or submodule quotient
"quotient(Ideal,RingElement)" -- see quotient(Module,Module) -- ideal or submodule quotient
quotient(Module,Module) -- ideal or submodule quotient
"radical(Ideal)" -- see radical -- the radical of an ideal
"reesIdeal(Ideal)" -- see reesIdeal -- Compute the defining ideal of the Rees Algebra
"reesIdeal(Ideal,RingElement)" -- see reesIdeal -- Compute the defining ideal of the Rees Algebra
"reesIdeal(Module)" -- see reesIdeal -- Compute the defining ideal of the Rees Algebra
"reesIdeal(Module,RingElement)" -- see reesIdeal -- Compute the defining ideal of the Rees Algebra
"regSeqInIdeal(Ideal)" -- see regSeqInIdeal -- a regular sequence contained in an ideal
"regSeqInIdeal(Ideal,ZZ)" -- see regSeqInIdeal -- a regular sequence contained in an ideal
"regSeqInIdeal(Ideal,ZZ,ZZ,ZZ)" -- see regSeqInIdeal -- a regular sequence contained in 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
"Schubert" -- see Schubert(ZZ,ZZ,VisibleList) -- find the Plücker ideal of a Schubert variety
specialFiberIdeal -- Special fiber of a blowup
"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
"substitute(Ideal,RingFamily)" -- see substitute -- substituting values for variables
"symmetricAlgebraIdeal(Ideal)" -- see symmetricAlgebraIdeal -- Ideal of the symmetric algebra of an ideal or module
"symmetricAlgebraIdeal(Module)" -- see symmetricAlgebraIdeal -- Ideal of the symmetric algebra of an ideal or module
"symmetricKernel(Matrix)" -- see symmetricKernel -- Compute the Rees ring of the image of a matrix
"tangentCone" -- see tangentCone(Ideal)
"topComponents(Ideal)" -- see topComponents -- compute top dimensional component of an ideal or module
"trim(Ideal)" -- see trim -- minimize generators and relations
truncate(List,Ideal)
truncate(List,Ring)

Methods that use an ideal :

"Number % Ideal" -- see % -- a binary operator, usually used for remainder and reduction
"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
"Ideal == Module" -- see == -- equality
"Ideal == MonomialIdeal" -- see == -- equality
"Ideal == Ring" -- see == -- equality
"Ideal == ZZ" -- see == -- equality
"Module == Ideal" -- see == -- equality
"MonomialIdeal == Ideal" -- see == -- equality
"Ring == Ideal" -- see == -- equality
"ZZ == Ideal" -- see == -- equality
"analyticSpread(Ideal)" -- see analyticSpread -- Compute the analytic spread of a module or ideal
"analyticSpread(Ideal,RingElement)" -- see analyticSpread -- Compute the analytic spread of a module or ideal
"associatedPrimes(Ideal)" -- see associatedPrimes -- find associated primes
"basis(Ideal)" -- see basis -- basis or generating set of all or part of a ring, ideal or module
"basis(InfiniteNumber,InfiniteNumber,Ideal)" -- see basis -- basis or generating set of all or part of a ring, ideal or module
"basis(InfiniteNumber,List,Ideal)" -- see basis -- basis or generating set of all or part of a ring, ideal or module
"basis(InfiniteNumber,ZZ,Ideal)" -- see basis -- basis or generating set of all or part of a ring, ideal or module
"basis(List,Ideal)" -- see basis -- basis or generating set of all or part of a ring, ideal or module
"basis(List,InfiniteNumber,Ideal)" -- see basis -- basis or generating set of all or part of a ring, ideal or module
"basis(List,List,Ideal)" -- see basis -- basis or generating set of all or part of a ring, ideal or module
"basis(List,ZZ,Ideal)" -- see basis -- basis or generating set of all or part of a ring, ideal or module
"basis(ZZ,Ideal)" -- see basis -- basis or generating set of all or part of a ring, ideal or module
"basis(ZZ,InfiniteNumber,Ideal)" -- see basis -- basis or generating set of all or part of a ring, ideal or module
"basis(ZZ,List,Ideal)" -- see basis -- basis or generating set of all or part of a ring, ideal or module
"basis(ZZ,ZZ,Ideal)" -- see basis -- basis or generating set of all or part of a ring, ideal or module
"betti(Ideal)" -- see betti -- display or modify a Betti diagram
codim(Ideal) -- compute the codimension
"CoherentSheaf / Ideal" -- see CoherentSheaf / CoherentSheaf -- quotient of coherent sheaves
"comodule(Ideal)" -- see comodule -- submodule to quotient module
"quotient(Ideal)" -- see comodule -- submodule to quotient module
degree(Ideal)
"degrees(Ideal)" -- see degrees(Ring) -- degrees of generators
dim(Ideal) -- compute the Krull dimension
"distinguished(Ideal)" -- see distinguished -- Compute the distinguished subvarieties of a pullback, intersection or cone
"distinguished(Ideal,Ideal)" -- see distinguished -- Compute the distinguished subvarieties of a pullback, intersection or cone
"distinguished(RingMap,Ideal)" -- see distinguished -- Compute the distinguished subvarieties of a pullback, intersection or cone
"eliminate(List,Ideal)" -- see eliminate
"eliminate(RingElement,Ideal)" -- see eliminate
euler(Ideal) -- Euler characteristic
eulers(Ideal) -- list the sectional Euler characteristics
"expectedReesIdeal(Ideal)" -- see expectedReesIdeal -- symmetric algebra ideal plus jacobian dual
"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
"flattenRing(Ideal)" -- see flattenRing -- write a ring as a (quotient of a) polynomial ring
"gb(Ideal)" -- see gb -- compute a Gröbner basis
"gbRemove(Ideal)" -- see gbRemove -- remove Gröbner basis
"gbSnapshot(Ideal)" -- see gbSnapshot -- the Gröbner basis matrix as so far computed
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
"groebnerBasis(Ideal)" -- see groebnerBasis -- Gröbner basis, as a matrix
"hilbertFunction(List,Ideal)" -- see hilbertFunction -- the Hilbert function
"hilbertFunction(ZZ,Ideal)" -- see hilbertFunction -- the Hilbert function
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 * ZZ (missing documentation)
Ideal + Number (missing documentation)
/// Function \ Ideal /// -- see Ideal / Function -- apply a function to generators of an ideal
Ideal / Function -- apply a function to generators of an ideal
Ideal / Ideal -- quotient module
Ideal ^ Array -- bracket power of an ideal
Ideal _* -- get the list of generators of an ideal
"idealizer(Ideal,RingElement)" -- see idealizer -- compute Hom(I,I) as a quotient ring
"independentSets(Ideal)" -- see independentSets -- some size-maximal independent subsets of variables modulo an ideal
installHilbertFunction(Ideal,RingElement) (missing documentation)
integralClosure(Ideal,RingElement,ZZ) -- integral closure of an ideal in an affine domain
"intersectInP(Ideal,Ideal)" -- see intersectInP -- Compute distinguished varieties for an intersection in A^n or P^n
"inverseSystem(Ideal)" -- see inverseSystem -- Inverse systems with equivariance
"inverseSystem(ZZ,Ideal)" -- see inverseSystem -- Inverse systems with equivariance
"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
"isLinearType(Ideal)" -- see isLinearType -- Determine whether module has linear type
"isLinearType(Ideal,RingElement)" -- see isLinearType -- Determine whether module has linear type
"isMonomialIdeal(Ideal)" -- see isMonomialIdeal -- whether something is a monomial ideal
"isPrimary(Ideal)" -- see isPrimary -- determine whether a submodule is primary
"isPrimary(Ideal,Ideal)" -- see isPrimary -- determine whether a submodule is primary
isPrime(Ideal) -- whether an ideal is prime
"isReduction(Ideal,Ideal)" -- see isReduction -- Determine whether an ideal is a reduction
"isReduction(Ideal,Ideal,RingElement)" -- see isReduction -- Determine whether an ideal is a reduction
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
"isSupportedInZeroLocus(Ideal,GradedModule)" -- see isSupportedInZeroLocus -- whether support of a module is contained in the zero locus of the (irrelevant) ideal
"isSupportedInZeroLocus(Ideal,Ideal)" -- see isSupportedInZeroLocus -- whether support of a module is contained in the zero locus of the (irrelevant) ideal
"isSupportedInZeroLocus(Ideal,Module)" -- see isSupportedInZeroLocus -- whether support of a module is contained in the zero locus of the (irrelevant) ideal
jacobian(Ideal) -- the Jacobian matrix of the generators of an ideal
"kernelOfLocalization(Module,Ideal)" -- see kernelOfLocalization -- the kernel of the localization map
leadTerm(Ideal) -- get the ideal of greatest terms
leadTerm(ZZ,Ideal) -- get the ideal of lead polynomials
"lift(Ideal,type of QQ)" -- see lift -- lift to another ring
"lift(Ideal,type of ZZ)" -- see lift -- lift to another ring
"Matrix % Ideal" -- see methods for normal forms and remainder -- normal form of ring elements and matrices
"RingElement % Ideal" -- see methods for normal forms and remainder -- normal form of ring elements and matrices
"mingens(Ideal)" -- see mingens(Module) -- minimal generator matrix
"minimalBetti(Ideal)" -- see minimalBetti -- minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module
"decompose(Ideal)" -- see minimalPrimes -- minimal primes of an ideal
"minimalPrimes(Ideal)" -- see minimalPrimes -- minimal primes of an ideal
"Module / Ideal" -- see Module / Module -- quotient module
"Ideal _ List" -- see Module _ List -- map from free module to some generators
module(Ideal) (missing documentation)
monomialIdeal(Ideal) -- monomial ideal of lead monomials of a Gröbner basis
"monomialSubideal(Ideal)" -- see monomialSubideal -- find the largest monomial ideal in an ideal
"multidegree(Ideal)" -- see multidegree -- multidegree
"multiplicity(Ideal)" -- see multiplicity -- Compute the Hilbert-Samuel multiplicity of an ideal
"multiplicity(Ideal,RingElement)" -- see multiplicity -- Compute the Hilbert-Samuel multiplicity of an ideal
"normalCone(Ideal)" -- see normalCone -- The normal cone of a subscheme
"normalCone(Ideal,RingElement)" -- see normalCone -- The normal cone of a subscheme
Number + Ideal (missing documentation)
numgens(Ideal) -- number of generators of an ideal
"poincare(Ideal)" -- see poincare -- assemble degrees of a ring, module, or ideal into a polynomial
"primaryDecomposition(Ideal)" -- see primaryDecomposition -- irredundant primary decomposition of an ideal
"Module : Ideal" -- see quotient(Module,Module) -- ideal or submodule quotient
"quotient(Module,Ideal)" -- see quotient(Module,Module) -- ideal or submodule quotient
"radicalContainment(Ideal,Ideal)" -- see radicalContainment -- whether an element is contained in the radical of an ideal
"radicalContainment(RingElement,Ideal)" -- see radicalContainment -- whether an element is contained in the radical of an ideal
"random(List,Ideal)" -- see random(ZZ,Ideal) -- get a random homogeneous element from a graded ideal
random(ZZ,Ideal) -- get a random homogeneous element from a graded ideal
"randomKRationalPoint(Ideal)" -- see randomKRationalPoint -- Pick a random K rational point on the scheme X defined by I
"reductionNumber(Ideal,Ideal)" -- see reductionNumber -- Reduction number of one ideal with respect to another
"reesAlgebra(Ideal)" -- see reesAlgebra -- Compute the defining ideal of the Rees Algebra
"reesAlgebra(Ideal,RingElement)" -- see reesAlgebra -- Compute the defining ideal of the Rees Algebra
"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
RingElement + Ideal (missing documentation)
"saturate(Module,Ideal)" -- see saturate -- saturation of ideal or submodule
"saturate(Vector,Ideal)" -- see saturate -- saturation of ideal or submodule
"singularLocus(Ideal)" -- see singularLocus -- singular locus
"specialFiber(Ideal)" -- see specialFiber -- Special fiber of a blowup
"specialFiber(Ideal,RingElement)" -- see specialFiber -- Special fiber of a blowup
"specialFiberIdeal(Ideal)" -- see specialFiberIdeal -- Special fiber of a blowup
"specialFiberIdeal(Ideal,RingElement)" -- see specialFiberIdeal -- Special fiber of a blowup
"substitute(Ideal,Option)" -- see substitute -- substituting values for variables
support(Ideal) -- list of variables occurring in the generators of an ideal
tangentCone(Ideal)
"toDual(ZZ,Ideal)" -- see toDual -- finds the inverse system to an ideal up to a given degree
Tor_ZZ(Ideal,Matrix) (missing documentation)
Tor_ZZ(Matrix,Ideal) (missing documentation)
"Tor_ZZ(Ideal,Ideal)" -- see Tor_ZZ(Module,Module) -- compute a Tor module
"Tor_ZZ(Ideal,Module)" -- see Tor_ZZ(Module,Module) -- compute a Tor module
"Tor_ZZ(Ideal,Ring)" -- see Tor_ZZ(Module,Module) -- compute a Tor module
"Tor_ZZ(Module,Ideal)" -- see Tor_ZZ(Module,Module) -- compute a Tor module
truncate(ZZ,Ideal)
variety(Ideal) -- the closed projective subvariety defined by an ideal
Vector % Ideal (missing documentation)
"versalEmbedding(Ideal)" -- see versalEmbedding -- Compute a versal embedding
"whichGm(Ideal)" -- see whichGm -- Largest Gm satisfied by an ideal

For the programmer