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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 | 44x+25y  -4x+4y   -7x-17y  x+14y    -35x+4y  -20x+3y  -20x+11y 23x+9y   |
              | -15x+17y x-44y    42x+37y  -22x+11y 26x+18y  -2x-13y  -24x-6y  28x+45y  |
              | -6y      -28x+41y -17x+26y 13x-43y  41x+26y  24x-28y  -33x+24y -29x+18y |
              | 4y       -x+40y   39x+44y  33x-48y  -26x+29y -37x+21y -31x-43y -27x-43y |
              | 48x+15y  45x+14y  x+3y     7x-2y    x-30y    48x-43y  44x+30y  18x-13y  |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -45 -28 -15 -27 37 |)
               | 0 0 x 0 y 0 0 0 |  | 31  49  -2  -7  21 |
               | 0 0 0 y x 0 0 0 |  | 18  -30 24  -3  6  |
               | 0 0 0 0 0 x 0 y |  | -14 41  -18 28  16 |
               | 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 :