Module -- the class of all modules

Functions and methods returning a module:

• Ideal * Module -- see * -- a binary operator, usually used for multiplication
• RingElement * Module -- see * -- a binary operator, usually used for multiplication
• ambient(Module) -- ambient free module
• coimage(Matrix) -- see coimage -- coimage of a map
• cokernel(Matrix) -- cokernel of a map of modules
• cokernel(RingElement) -- see cokernel(Matrix) -- cokernel of a map of modules
• comodule(Ideal) -- see comodule -- submodule to quotient module
• comodule(Module) -- see comodule -- submodule to quotient module
• quotient(Ideal) -- see comodule -- submodule to quotient module
• quotient(Module) -- see comodule -- submodule to quotient module
• cover(Module) -- get the covering free module
• dual(Module) -- dual module
• 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(Module,Module) -- total Ext module
• Ext(Module,Ring) -- see Ext(Module,Module) -- total Ext module
• Ext(Ring,Ideal) -- see Ext(Module,Module) -- total Ext module
• Ext(Ring,Module) -- see Ext(Module,Module) -- total Ext module
• Ext(Ring,Ring) -- see Ext(Module,Module) -- total Ext module
• 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(Module,Module) -- Ext module
• Ext^ZZ(Module,Ring) -- see Ext^ZZ(Module,Module) -- Ext module
• Ext^ZZ(Ring,Ideal) -- see Ext^ZZ(Module,Module) -- Ext module
• Ext^ZZ(Ring,Module) -- see Ext^ZZ(Module,Module) -- Ext module
• Ext^ZZ(Ring,Ring) -- see Ext^ZZ(Module,Module) -- Ext module
• exteriorPower(ZZ,Module) -- exterior power of a module
• HH^ZZ Module -- local cohomology of a module
• 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(Module,Module) -- see Hom -- module of homomorphisms
• Hom(Module,Ring) -- see Hom -- module of homomorphisms
• Hom(Ring,Ideal) -- see Hom -- module of homomorphisms
• Hom(Ring,Module) -- see Hom -- module of homomorphisms
• Hom(Ring,Ring) -- see Hom -- module of homomorphisms
• homogenize(Module,RingElement) -- see homogenize -- homogenize with respect to a variable
• homogenize(Module,RingElement,List) -- see homogenize -- homogenize with respect to a variable
• homology(Matrix,Matrix) -- homology of a pair of maps
• Ideal / Ideal -- quotient module
• image(Matrix) -- image of a map
• image(RingElement) -- see image(Matrix) -- image of a map
• intersect(Module) -- see intersect(Ideal,Ideal) -- compute an intersection of a sequence of ideals or modules
• intersect(Module,Module) -- see intersect(Ideal,Ideal) -- compute an intersection of a sequence of ideals or modules
• kernel(Matrix) -- kernel of a matrix
• kernel(RingElement) -- see kernel(Matrix) -- kernel of a matrix
• lift(Module,type of RingElement) -- see lift -- lift to another ring
• minimalPresentation(Module) -- minimal presentation of a module
• prune(Module) -- see minimalPresentation(Module) -- minimal presentation of a module
• Module ** Ring -- tensor product
• Ring ** Module -- see Module ** Ring -- tensor product
• Module + Module -- sum of submodules
• Module ++ Module -- direct sum of modules
• Module / Ideal -- see Module / Module -- quotient module
• Module / List -- see Module / Module -- quotient module
• Module / Module -- quotient module
• Module / RingElement -- see Module / Module -- quotient module
• Module / Sequence -- see Module / Module -- quotient module
• Module / Vector -- see Module / Module -- quotient module
• Module : Ideal
• Module : Number (missing documentation)
• Module : RingElement
• Module ^ ZZ -- direct sum
• Module ^** ZZ -- tensor power
• module(Ideal) (missing documentation)
• module(Ring) -- make or get a module
• part(List,Ideal) (missing documentation)
• part(List,Module) (missing documentation)
• part(List,Ring) (missing documentation)
• part(ZZ,Ideal) (missing documentation)
• part(ZZ,Module) (missing documentation)
• part(ZZ,Ring) (missing documentation)
• pullback(List) (missing documentation)
• pullback(Matrix,Matrix) (missing documentation)
• pushForward(RingMap,Module) -- compute the pushforward of a module along a ring map
• pushout(List) (missing documentation)
• pushout(Matrix,Matrix) (missing documentation)
• quotient(Module,Ideal)
• quotient(Module,Number) (missing documentation)
• quotient(Module,RingElement)
• Ring ^ List -- make a free module
• RingFamily ^ List -- see Ring ^ List -- make a free module
• Ring ^ ZZ -- make a free module
• RingFamily ^ ZZ -- see Ring ^ ZZ -- make a free module
• RingMap Module -- see RingMap RingElement -- apply a ring map
• saturate(Module) -- see saturate -- saturation of ideal or submodule
• saturate(Module,Ideal) -- see saturate -- saturation of ideal or submodule
• saturate(Module,RingElement) -- see saturate -- saturation of ideal or submodule
• saturate(Vector) -- see saturate -- saturation of ideal or submodule
• saturate(Vector,Ideal) -- see saturate -- saturation of ideal or submodule
• saturate(Vector,RingElement) -- see saturate -- saturation of ideal or submodule
• source(MutableMatrix) -- see source(Matrix) -- find the source module of matrix
• subquotient -- make a subquotient module
• substitute(Module,List) -- see substitute -- substituting values for variables
• substitute(Module,Matrix) -- see substitute -- substituting values for variables
• substitute(Module,Ring) -- see substitute -- substituting values for variables
• substitute(Module,RingFamily) -- see substitute -- substituting values for variables
• super(Module) -- see super -- get the ambient module
• symmetricPower(ZZ,Module) -- see symmetricPower -- symmetric power
• target(MutableMatrix) -- see target(Matrix) -- find the target module of matrix
• Module ** Module -- see tensor(Module,Module) -- tensor product of modules
• tensor(Module,Module) -- tensor product of modules
• RingMap ** Module -- see tensor(RingMap,Matrix) -- tensor product via a ring map
• tensor(RingMap,Module) -- see tensor(RingMap,Matrix) -- tensor product via a ring map
• 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(Module,Module) -- compute a Tor module
• Tor_ZZ(Module,Ring) -- see Tor_ZZ(Module,Module) -- compute a Tor module
• Tor_ZZ(Ring,Ideal) -- see Tor_ZZ(Module,Module) -- compute a Tor module
• Tor_ZZ(Ring,Module) -- see Tor_ZZ(Module,Module) -- compute a Tor module
• Tor_ZZ(Ring,Ring) -- see Tor_ZZ(Module,Module) -- compute a Tor module
• trim(Module) -- see trim -- minimize generators and relations
• truncate(List,Module)

Methods that use a module:

• adjoint'(Matrix,Module,Module) -- see adjoint -- the tensor-Hom adjunction maps
• adjoint(Matrix,Module,Module) -- see adjoint -- the tensor-Hom adjunction maps
• annihilator(Module) -- see annihilator -- the annihilator ideal
• basis(InfiniteNumber,InfiniteNumber,Module) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(InfiniteNumber,List,Module) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(InfiniteNumber,ZZ,Module) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(List,InfiniteNumber,Module) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(List,List,Module) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(List,Module) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(List,ZZ,Module) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(Module) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(ZZ,InfiniteNumber,Module) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(ZZ,List,Module) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(ZZ,Module) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(ZZ,ZZ,Module) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• betti(Module) -- see betti -- display or modify a Betti diagram
• codim(Module) -- codimension of the support of a module
• components(Module) -- see components -- list the components of a direct sum
• compose(Module,Module,Module) -- see compose -- composition as a pairing on Hom-modules
• coverMap(Module) -- the surjective map from a free module to a module corresponding to the generators
• degree(Module) -- get the degree of a module
• degrees(Module) -- see degrees(Ring) -- degrees of generators
• describe(Module) -- see describe -- real description
• dim(Module) -- compute the Krull dimension
• directSum(Module) -- see directSum -- direct sum of modules or maps
• Module == Module -- see equality and containment of modules
• euler(Module)
• eulers(Module)
• Ext^ZZ(Matrix,Module) -- map between Ext modules
• Ext^ZZ(Module,Matrix) -- map between Ext modules
• factor(Module) -- factor a ZZ-module
• fittingIdeal(ZZ,Module) -- see fittingIdeal -- Fitting ideal of a module
• flip(Module,Module) -- see flip -- isomorphism map of commutativity of tensor product
• formation(Module) -- see formation -- recover the methods used to make a module
• gb(Module) -- see gb -- compute a Gröbner basis
• gbRemove(Module) -- see gbRemove -- remove Gröbner basis
• gbSnapshot(Module) -- see gbSnapshot -- the Gröbner basis matrix as so far computed
• genera(Module)
• generator(Module) -- see generator -- provide a single generator
• Module _ ZZ -- see generators of rings, ideals, and modules
• Module _* -- see generators of rings, ideals, and modules
• generators(Module) -- the generator matrix of a module
• genus(Module)
• gradedModule(Module)
• groebnerBasis(Module) -- see groebnerBasis -- Gröbner basis, as a matrix
• hilbertFunction(List,Module) -- see hilbertFunction -- the Hilbert function
• hilbertFunction(Module) -- see hilbertFunction -- the Hilbert function
• hilbertFunction(ZZ,Module) -- see hilbertFunction -- the Hilbert function
• hilbertPolynomial(Module) -- compute the Hilbert polynomial of the module
• hilbertSeries(Module) -- compute the Hilbert series of the module
• Hom(Matrix,Module) -- see Hom(Module,Matrix) -- induced map on Hom
• Hom(Module,Matrix) -- induced map on Hom
• ideal(Module) -- convert a module to an ideal
• inducedMap(Module,Module) -- compute the map induced by the identity
• inducedMap(Module,Module,Matrix) -- compute the induced map
• inducedMap(Module,Nothing,Matrix) -- see inducedMap(Module,Module,Matrix) -- compute the induced map
• inducedMap(Nothing,Module,Matrix) -- see inducedMap(Module,Module,Matrix) -- compute the induced map
• inducesWellDefinedMap(Module,Module,Matrix) -- see inducesWellDefinedMap -- whether a map is well defined
• inducesWellDefinedMap(Module,Nothing,Matrix) -- see inducesWellDefinedMap -- whether a map is well defined
• inducesWellDefinedMap(Nothing,Module,Matrix) -- see inducesWellDefinedMap -- whether a map is well defined
• installHilbertFunction(Module,RingElement) (missing documentation)
• isDirectSum(Module) -- see isDirectSum -- whether something is a direct sum
• isFreeModule(Module) -- see isFreeModule -- whether something is a free module
• isHomogeneous(Module) -- see isHomogeneous -- whether something is homogeneous (graded)
• isIdeal(Module) -- see isIdeal -- whether something is an ideal
• isModule(Module) -- see isModule -- whether something is a module
• isMonomialIdeal(Module) -- see isMonomialIdeal -- whether something is a monomial ideal
• isQuotientModule(Module) -- see isQuotientModule -- whether something is evidently a quotient of a free module
• isSquareFree(Module) -- see isSquareFree -- whether something is square free monomial ideal
• isSubmodule(Module) -- see isSubmodule -- whether a module is evidently a submodule of a free module
• isSubquotient(Module,Module) -- check whether a module is a subquotient 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
• isSubset(Module,Module) -- whether one object is a subset of another
• length(Module) -- compute the length of a module
• map(Module) -- identity map
• map(Module,Module,Function) -- create a matrix by specifying a function that gives each entry
• map(Module,Module,List) -- create a matrix by giving a sparse or dense list of entries
• map(Module,Module,Matrix) -- create the matrix induced on generators by a given matrix
• map(Module,Module,Number) -- see map(Module,Module,RingElement) -- construct the map induced by multiplication by a ring element on the generators
• map(Module,Module,RingElement) -- construct the map induced by multiplication by a ring element on the generators
• map(Module,Module,ZZ) -- see map(Module,Module,RingElement) -- construct the map induced by multiplication by a ring element on the generators
• map(Module,ZZ,ZZ) -- see map(Module,Module,RingElement) -- construct the map induced by multiplication by a ring element on the generators
• map(Module,Module,RingMap,List) -- see map(Module,Module,RingMap,Matrix) -- homomorphism of modules over different rings
• map(Module,Module,RingMap,Matrix) -- homomorphism of modules over different rings
• map(Module,Nothing,RingMap,List) -- see map(Module,Module,RingMap,Matrix) -- homomorphism of modules over different rings
• map(Module,Nothing,RingMap,Matrix) -- see map(Module,Module,RingMap,Matrix) -- homomorphism of modules over different rings
• map(Module,RingMap) -- see map(Module,Module,RingMap,Matrix) -- homomorphism of modules over different rings
• map(Module,Nothing,List) -- create a matrix by giving a doubly nested list of ring elements
• map(Module,Nothing,Matrix) -- recast a matrix to have a new target, and a free module as source
• map(Module,ZZ,Function) -- create a matrix from a free module by specifying a function that gives each entry
• map(Module,ZZ,List) -- create a matrix by giving a sparse or dense list of entries
• Matrix % Module -- see methods for normal forms and remainder -- normal form of ring elements and matrices
• mingens(Module) -- minimal generator matrix
• minimalBetti(Module) -- see minimalBetti -- minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module
• Module : Module
• Module ^ Array -- projection onto summand
• Module ^ List -- projection onto summand
• Module ^~
• Module _ Array -- inclusion from summand
• Module _ List -- map from free module to some generators
• Module Array -- make a chain complex from a module
• monomialIdeal(Module) -- see monomialIdeal(Ideal) -- monomial ideal of lead monomials of a Gröbner basis
• multidegree(Module) -- see multidegree -- multidegree
• numgens(Module) -- number of generators of a module
• pdim(Module) -- compute the projective dimension of a module
• poincare(Module) -- see poincare -- assemble degrees of a ring, module, or ideal into a polynomial
• preimage(Matrix,Module) -- see preimage -- preimage of a map
• presentation(Module) -- presentation of a module
• quotient(Module,Module)
• random(List,Module) -- get a random vector in the module
• random(Module) -- see random(List,Module) -- get a random vector in the module
• random(ZZ,Module) -- see random(List,Module) -- get a random vector in the module
• random(Module,Module) -- get a random map of module
• rank(Module) -- see rank -- compute the rank
• regularity(Module) -- see regularity -- compute the Castelnuovo-Mumford regularity
• relations(Module) -- see relations -- the defining relations
• reshape(Module,Module,Matrix) -- see reshape -- reshape a matrix
• ring(Module) -- see ring -- get the associated ring of an object
• Ring / Module -- see Ring / Ideal -- make a quotient ring
• schreyerOrder(Module) -- obtain Schreyer order information
• subquotient(Module,Matrix,Matrix) -- see subquotient -- make a subquotient module
• subquotient(Module,Matrix,Nothing) -- see subquotient -- make a subquotient module
• subquotient(Module,Nothing,Matrix) -- see subquotient -- make a subquotient module
• subquotient(Module,Nothing,Nothing) -- see subquotient -- make a subquotient module
• substitute(Module,Option) -- see substitute -- substituting values for variables
• symmetricAlgebra(Module) -- see symmetricAlgebra -- the symmetric algebra of a module
• Matrix ** Module -- see tensor(Matrix,Module) -- tensor product
• Module ** Matrix -- see tensor(Matrix,Module) -- tensor product
• tensor(Matrix,Module) -- tensor product
• tensor(Module,Matrix) -- see tensor(Matrix,Module) -- tensor product
• tensorAssociativity(Module,Module,Module) -- see tensorAssociativity -- associativity isomorphisms for tensor products
• truncate(ZZ,Module)
• vector(Module,List) -- see vector -- make a vector
• vector(Module,Matrix) -- see vector -- make a vector
• vector(Module,Number) -- see vector -- make a vector
• vector(Module,RingElement) -- see vector -- make a vector
• wedgeProduct(ZZ,ZZ,Module) -- the exterior multiplication map
• ZZ _ Module -- get the zero vector in a module

For the programmer

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

The source of this document is in Macaulay2Doc/doc_module.m2:139:0.