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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 | -50x-4y  -19x+20y -16x+40y -42x+24y -4x-21y  5x-29y   46x-4y   34x-39y  |
              | -27x-y   -14y     40x+15y  45x-32y  -29x-16y -36x+28y -31x+3y  -28x+37y |
              | -22x+32y -41x+22y -27x-8y  28x-19y  -50x-29y 34x-26y  -47x-11y 19x-16y  |
              | -41x-14y 19x-13y  -11x-2y  -48x+34y -28x+16y 35x-47y  12x-24y  -2x+12y  |
              | 12x+5y   30x-42y  33x-17y  12x-46y  39x+2y   -11x-28y -39x-30y -13x+28y |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | 41 46  14  20  5   |)
               | 0 0 x 0 y 0 0 0 |  | -2 -27 39  45  -31 |
               | 0 0 0 y x 0 0 0 |  | -5 -27 21  12  42  |
               | 0 0 0 0 0 x 0 y |  | 48 26  -40 -42 36  |
               | 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 :