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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 | 27x-36y 34x-26y  36x+44y 33x-19y 39x-2y   21x+48y  -17x-40y -34x-19y |
              | 39x-35y -8x+20y  19x+41y 4x+3y   27x+17y  29x-29y  -25x-15y 26x-23y  |
              | 10x-34y 16x-33y  23x     16x-20y -7x-6y   -27x+14y -26x+24y 18x+y    |
              | 47x-42y -13x+15y -33y    49x-45y 5x-11y   -18x-30y 4x+19y   -11x+7y  |
              | 14x+35y 4x+4y    38y     -4x     -14x+21y 40x-12y  8x+43y   24x-41y  |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -15 3  -49 -36 37  |)
               | 0 0 x 0 y 0 0 0 |  | 40  49 32  -30 21  |
               | 0 0 0 y x 0 0 0 |  | -39 20 10  42  26  |
               | 0 0 0 0 0 x 0 y |  | 1   0  0   0   0   |
               | 0 0 0 0 0 0 y x |  | 35  35 9   -45 -30 |

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 :