Module

Module -- the class of all modules which are handled by the engine.

The most general module M is represented as a submodule of a quotient module of a free module F. The matrix of relations used to produce the quotient module is stored as 

     M.relations
and the matrix of generators is stored as M.generators .

Functions which create modules:

  • Ring ^ ZZ
  • cokernel
  • homology
  • ideal
  • image
  • kernel

    • Tests:

  • isFreeModule
  • isIdeal
  • isModule
  • isQuotientModule
  • isSubmodule

    • Operations on modules:

  • ==
  • M_i
  • Module + Module -- the sum of submodules I, J
  • M ++N -- direct sum
  • M **N -- tensor product
  • M :N -- the submodule quotient M : N
  • M /N -- the cokernel module (M+N)/N
  • ambient
  • annihilator M -- the annihilator of M
  • codim
  • cover
  • degree
  • degrees
  • dim
  • dual
  • End
  • euler
  • fittingIdeal(i,m) -- the i-th Fitting ideal of the module M
  • Ext^i(M,N) -- the ext module
  • gcdDegree
  • genera
  • generators
  • poincare(M,t) -- the numerator of the Hilbert series of M
  • hilbertFunction(d,M) -- the Hilbert function of a module.
  • hilbertPolynomial(M) -- the Hilbert polynomial of a module
  • hilbertSeries(M) -- the Hilbert series of a module.
  • Hom(M,N) -- the module of homomorphisms
  • intersect(I,J) -- intersection of modules
  • lcmDegree
  • numgens
  • mingens
  • pdim
  • presentation M -- a presentation matrix for M
  • prune M -- a minimal presentation for M
  • pruneAndMap
  • quotient
  • rank
  • relations
  • removeLowestDimension
  • resolutionM -- a finite free resolution of M
  • super
  • top
  • Tor_i(M,N) -- the tor module
  • trim -- replace generators and relations by minimal sets
  • truncate

    • Operations on elements of modules:

  • +
  • -
  • *
  • components
  • leadCoefficient
  • leadMonomial

    • See also Vector.

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