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Macaulay2Doc :: minimalPresentation(Module)

minimalPresentation(Module) -- minimal presentation of a module

Synopsis

Description

If the Module M is graded then the module N is a minimal presentation of M. If not, then an attempt is made to improve the presentation of M. An example follows.
i1 : R = ZZ/32003[a..d];
i2 : M = coker matrix {{a,1,b},{c,3,b+d}}

o2 = cokernel | a 1 b   |
              | c 3 b+d |

                            2
o2 : R-module, quotient of R
i3 : N = minimalPresentation M

o3 = cokernel | b+16001d a-10668c |

                            1
o3 : R-module, quotient of R
i4 : peek N.cache

o4 = CacheTable{cache => MutableHashTable{}}
                pruningMap => | -10668 |
                              | 0      |
i5 : g = N.cache.pruningMap

o5 = | -10668 |
     | 0      |

o5 : Matrix
i6 : g^-1

o6 = | -3 1 |

o6 : Matrix
This function also works when M is a graded module, a chain complex, or a coherent sheaf, by acting on the modules and maps within it.
i7 : I = ideal(a^2,b^3,c^4,d^7)

             2   3   4   7
o7 = ideal (a , b , c , d )

o7 : Ideal of R
i8 : X = Proj R

o8 = X

o8 : ProjectiveVariety
i9 : J = (module I)~

o9 = image | a2 b3 c4 d7 |

                                         1
o9 : coherent sheaf on X, subsheaf of OO
                                        X
i10 : minimalPresentation J

         1
o10 = OO
        X

o10 : coherent sheaf on X, free

See also