Macaulay2 » Documentation
Packages » Macaulay2Doc » ideals » Ideal
next | previous | forward | backward | up | index | toc

Ideal -- the class of all ideals

Description

For basic information about ideals in Macaulay2, see ideals.

Common ways to make an ideal: Common ways to get information about an ideal: Common operations on ideals: Gröbner bases, normal forms, free resolutions Numeric information about homogeneous ideals Primary decomposition and components of an ideal Ideals from geometry Common ways to use an ideal:

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

See also

Menu

Types of ideal:

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

Functions and methods returning an ideal:

Methods that use an ideal:

  • Number % Ideal -- see % -- a binary operator, usually used for remainder and reduction
  • Ideal * Module -- see * -- a binary operator, usually used for multiplication
  • Ideal * Vector -- see * -- a binary operator, usually used for multiplication
  • Ideal + Number -- see + -- a unary or binary operator, usually used for addition
  • Ideal + RingElement -- see + -- a unary or binary operator, usually used for addition
  • Number + Ideal -- see + -- a unary or binary operator, usually used for addition
  • RingElement + Ideal -- see + -- a unary or binary operator, usually used for addition
  • 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
  • comodule(Ideal) -- see comodule -- submodule to quotient module
  • quotient(Ideal) -- see comodule -- submodule to quotient module
  • decompose(Ideal)
  • degree(Ideal)
  • degrees(Ideal) -- see degrees(Ring) -- degrees of generators
  • dim(Ideal) -- compute the Krull dimension
  • Ideal == Ideal -- see equality and containment
  • Ideal == ZZ -- see equality and containment
  • ZZ == Ideal -- see equality and containment
  • euler(Ideal)
  • eulers(Ideal)
  • 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(Ring,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
  • Ext^ZZ(Ring,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)
  • generator(Ideal) -- see generator -- provide a single generator
  • Ideal _ ZZ -- see generators of rings, ideals, and modules
  • Ideal _* -- see generators of rings, ideals, and modules
  • generators(Ideal) -- the generator matrix of an ideal
  • genus(Ideal)
  • groebnerBasis(Ideal) -- see groebnerBasis -- Gröbner basis, as a matrix
  • hilbertFunction(Ideal) -- see hilbertFunction -- the Hilbert function
  • 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 of homomorphisms
  • Hom(Ideal,Module) -- see Hom -- module of homomorphisms
  • Hom(Ideal,Ring) -- see Hom -- module of homomorphisms
  • Hom(Module,Ideal) -- see Hom -- module of homomorphisms
  • Hom(Ring,Ideal) -- see Hom -- module of homomorphisms
  • 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 ^~
  • independentSets(Ideal) -- see independentSets -- some size-maximal independent subsets of variables modulo an ideal
  • installHilbertFunction(Ideal,RingElement) (missing documentation)
  • 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
  • isMember(Number,Ideal) -- see isMember(RingElement,Ideal) -- test membership in an ideal
  • isMember(RingElement,Ideal) -- test membership in an ideal
  • isMonomialIdeal(Ideal) -- see isMonomialIdeal -- whether something is a monomial ideal
  • isPrime(Ideal)
  • isSquareFree(Ideal) -- see isSquareFree -- whether something is square free monomial ideal
  • 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(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
  • Vector % 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
  • Module / Ideal -- see Module / Module -- quotient module
  • Module : Ideal
  • 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
  • normalCone(Ideal)
  • normalCone(Ideal,RingElement)
  • numgens(Ideal) -- number of generators of an ideal
  • part(List,Ideal) (missing documentation)
  • part(ZZ,Ideal) (missing documentation)
  • poincare(Ideal) -- see poincare -- assemble degrees of a ring, module, or ideal into a polynomial
  • quotient(Module,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
  • regularity(Ideal) -- see regularity -- compute the Castelnuovo-Mumford regularity
  • 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
  • saturate(Vector,Ideal) -- see saturate -- saturation of ideal or submodule
  • singularLocus(Ideal) -- see singularLocus -- singular locus
  • substitute(Ideal,Option) -- see substitute -- substituting values for variables
  • support(Ideal) -- list of variables occurring in the generators of an ideal
  • 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
  • Tor_ZZ(Ring,Ideal) -- see Tor_ZZ(Module,Module) -- compute a Tor module
  • truncate(ZZ,Ideal)

For the programmer

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


The source of this document is in Macaulay2Doc/doc_ideals.m2:110:0.