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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+29y  -5x-39y  -40x+4y  -10x+29y -26x+48y 26x-32y  8x+20y   -48x+18y |
              | 12x+31y  43x-34y  35x+38y  -8x+15y  -32x+37y 45x-44y  41x+46y  -7x+9y   |
              | 15x+39y  47x-21y  50x+27y  17x+13y  -46x+29y 19x+30y  40x-14y  -13x+36y |
              | -40x+5y  -10x-25y -48x+23y 16x-37y  -25x-18y 6x+18y   17x-42y  -9x-9y   |
              | -40x-17y 19x+50y  -44x+30y 49x-15y  13x-38y  -26x-25y -39x-23y -19x-19y |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | 46  33 10  4   -20 |)
               | 0 0 x 0 y 0 0 0 |  | -45 28 -36 48  -21 |
               | 0 0 0 y x 0 0 0 |  | 22  15 26  -48 11  |
               | 0 0 0 0 0 x 0 y |  | -47 -3 16  7   -26 |
               | 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 :