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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 | 29x-41y  -26x-38y -48x+5y  14y      -43x+14y 22x-29y 36x-46y  -44x-3y  |
              | 30x-33y  -50x-19y 41x-24y  32x      32x+29y  32x+18y 17x-24y  33x-12y  |
              | -45x+47y 33x-24y  -19x-26y 47x-38y  38x+22y  16x+22y -19x-19y 4x-47y   |
              | -26x-48y -38x+13y 2x-32y   -30x-38y -32x+15y 47x-26y 33x-21y  -32x-22y |
              | -2x+21y  -25x-12y 47x+32y  35x-48y  31x+17y  48x-31y -2x+3y   -27x+10y |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -43 0   -6  3   -19 |)
               | 0 0 x 0 y 0 0 0 |  | -5  -18 -21 36  -17 |
               | 0 0 0 y x 0 0 0 |  | 30  42  -26 -39 35  |
               | 0 0 0 0 0 x 0 y |  | 34  -25 -10 41  -47 |
               | 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 :