PrimaryDecomposition -- primary decomposition and associated primes routines for ideals and modules

Description

This package provides routines for computation involving components of ideals and modules, including associated primes and primary decompositions.

The following simple example illustrates the use of removeLowestDimension, topComponents, radical, minimalPrimes, associatedPrimes, and primaryDecomposition.

i1 : R = ZZ/32003[a..d];

i2 : I = monomialCurveIdeal(R,{1,3,4})

                        3      2     2    2    3    2
o2 = ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c)

o2 : Ideal of R

i3 : J = ideal(a^3,b^3,c^3-d^3)

             3   3   3    3
o3 = ideal (a , b , c  - d )

o3 : Ideal of R

i4 : I = intersect(I,J)

             4    3      3    3      4      3         3      4   6      3 2  
o4 = ideal (b  - a d, a*b  - a c, b*c  - a*c d - b*c*d  + a*d , c  - b*c d  -
     ------------------------------------------------------------------------
      3 3      5     5    2 3       2 3    2 4   2 4    3 3    3 3    2   3 
     c d  + b*d , a*c  - b c d - a*c d  + b d , a c  - a d  + b d  - a c*d ,
     ------------------------------------------------------------------------
      3 3    3 3     2 3    3   2    3   2      2 3   2   3    3 2     3 2   
     b c  - a d , a*b c  - a c*d  + b c*d  - a*b d , a b*c  - a c d + b c d -
     ------------------------------------------------------------------------
      2   3   3 3    3   2   4 2    3 2
     a b*d , a c  - a b*d , a c  - a b d)

o4 : Ideal of R

i5 : removeLowestDimension I

                        3      2     2    2    3    2
o5 = ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c)

o5 : Ideal of R

i6 : topComponents I

                        3      2     2    2    3    2
o6 = ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c)

o6 : Ideal of R

i7 : radical I

                          2    2    3    2    6    3 3    2 4      5
o7 = ideal (b*c - a*d, a*c  - b d, b  - a c, c  - c d  - b d  + b*d )

o7 : Ideal of R

i8 : minimalPrimes I

                         3      2     2    2    3    2                    
o8 = {ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c), ideal (c - d, b,
     ------------------------------------------------------------------------
                       2          2
     a), ideal (b, a, c  + c*d + d )}

o8 : List

i9 : associatedPrimes I

                         3      2     2    2    3    2                    
o9 = {ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c), ideal (c - d, b,
     ------------------------------------------------------------------------
                       2          2
     a), ideal (b, a, c  + c*d + d )}

o9 : List

i10 : primaryDecomposition I

                          3      2     2    2    3    2                   3 
o10 = {ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c), ideal (c - d, b ,
      -----------------------------------------------------------------------
       3           2          2   3   3
      a ), ideal (c  + c*d + d , b , a )}

o10 : List

References

Eisenbud-Huneke-Vasconcelos, Inventiones mathematicae, 110 207–235 (1992)
Shimoyama-Yokoyama, Journal of Symbolic Computation, 22(3) 247–277 (1996)

Authors

Version

This documentation describes version 2.0 of PrimaryDecomposition.

Source code

The source code from which this documentation is derived is in the file PrimaryDecomposition.m2. The auxiliary files accompanying it are in the directory PrimaryDecomposition/.

Exports

Functions and commands
- associatedPrimes -- find associated primes
- irreducibleDecomposition -- express a monomial ideal as an intersection of irreducible monomial ideals
- isPrimary -- determine whether a submodule is primary
- kernelOfLocalization -- the kernel of the localization map
- localize -- localize an ideal at a prime ideal
- primaryComponent -- find a primary component corresponding to an associated prime
- primaryDecomposition -- irredundant primary decomposition of an ideal
- regSeqInIdeal -- a regular sequence contained in an ideal
- removeLowestDimension -- remove components of lowest dimension
- topComponents -- compute top dimensional component of an ideal or module
Methods
- associatedPrimes(Ideal) -- see associatedPrimes -- find associated primes
- associatedPrimes(Module) -- see associatedPrimes -- find associated primes
- associatedPrimes(Ring) -- see associatedPrimes -- find associated primes
- irreducibleDecomposition(MonomialIdeal) -- see irreducibleDecomposition -- express a monomial ideal as an intersection of irreducible monomial ideals
- isPrimary(Ideal) -- see isPrimary -- determine whether a submodule is primary
- isPrimary(Ideal,Ideal) -- see isPrimary -- determine whether a submodule is primary
- isPrimary(Module,Module) -- see isPrimary -- determine whether a submodule is primary
- kernelOfLocalization(Module,Ideal) -- see kernelOfLocalization -- the kernel of the localization map
- localize(Ideal,Ideal) -- see localize -- localize an ideal at a prime ideal
- primaryComponent(Ideal,Ideal) -- see primaryComponent -- find a primary component corresponding to an associated prime
- primaryDecomposition(Ideal) -- see primaryDecomposition -- irredundant primary decomposition of an ideal
- primaryDecomposition(Module) -- irredundant primary decomposition of a module
- primaryDecomposition(Ring) -- see primaryDecomposition(Module) -- irredundant primary decomposition of a module
- 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
- removeLowestDimension(Module) -- see removeLowestDimension -- remove components of lowest dimension
- topComponents(Ideal) -- see topComponents -- compute top dimensional component of an ideal or module
- topComponents(Module) -- see topComponents -- compute top dimensional component of an ideal or module
Symbols
- Increment -- see primaryComponent -- find a primary component corresponding to an associated prime
- EisenbudHunekeVasconcelos -- see strategies for computing primary decomposition
- GTZ -- see strategies for computing primary decomposition
- ShimoyamaYokoyama -- see strategies for computing primary decomposition

For the programmer

The object PrimaryDecomposition is a package.