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Kronecker :: decomposeModule

decomposeModule -- decompose a module into a direct sum of simple modules

Synopsis

Description

This function decomposes a module into a direct sum of simple modules, given some fairly strong assumptions on the ring which acts on the ring which acts on the module. This ring must only have two variables, and the square of each of those variables must kill the module.
i1 : Q = ZZ/101[x,y]

o1 = Q

o1 : PolynomialRing
i2 : R = Q/(x^2,y^2)

o2 = R

o2 : QuotientRing
i3 : M = coker random(R^5, R^8 ** R^{-1})

o3 = cokernel | -43x-17y -18x-42y -47x+40y 18x-13y 38x+14y  -19x-45y 46x+15y 35x-46y  |
              | 25x-39y  -46x-5y  -27x-34y 5x-15y  -12x+38y -28x+11y 20x+41y -3x+42y  |
              | 50x-22y  -38x+32y -6x-47y  44x-23y -21x+36y 12x-8y   14x+44y -17x+39y |
              | 35x-20y  44x+49y  x-43y    34x+27y -39x+19y 34x-31y  18x+2y  -28x+3y  |
              | 18x-12y  -7x-31y  22x+9y   48x+25y -32x-35y -45x+28y 2x+46y  9x+38y   |

                            5
o3 : R-module, quotient of R
i4 : (N,f) = decomposeModule M

o4 = (cokernel | y x 0 0 0 0 0 0 |, | 32  -45 46  5  34  |)
               | 0 0 x 0 y 0 0 0 |  | 16  -1  -37 19 -40 |
               | 0 0 0 y x 0 0 0 |  | 49  -1  20  36 20  |
               | 0 0 0 0 0 x 0 y |  | -11 -50 5   50 43  |
               | 0 0 0 0 0 0 y x |  | 1   0   0   0  0   |

o4 : Sequence
i5 : components N

o5 = {cokernel | y x |, cokernel | x 0 y |, cokernel | x 0 y |}
                                 | 0 y x |           | 0 y x |

o5 : List
i6 : ker f == 0

o6 = true
i7 : coker f == 0

o7 = true

Ways to use decomposeModule :