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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 | 17x-30y  -11x-13y -41x+48y 13x-10y  -49x-29y 50x-41y  x-33y    -28x+16y |
              | -34x+44y -7x-24y  -18x-37y 26x      -9x-37y  15x+49y  38x+48y  -21x-45y |
              | -40x-41y 23x-40y  -8x-5y   -22x+6y  49x+47y  25x-6y   -8x-5y   -17x-38y |
              | 35x+35y  -50x-16y -32x-33y -13x-44y 14x+44y  -50x-21y -38x+18y 41x-15y  |
              | 34x-11y  -35x+28y 9x-26y   -5x-37y  -44x-33y -3x+26y  12x-2y   31x+2y   |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -18 -43 -7 -35 16  |)
               | 0 0 x 0 y 0 0 0 |  | 6   2   21 37  50  |
               | 0 0 0 y x 0 0 0 |  | -13 29  4  -21 -19 |
               | 0 0 0 0 0 x 0 y |  | 1   0   0  0   0   |
               | 0 0 0 0 0 0 y x |  | 46  -27 22 46  -38 |

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 :