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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 | -11x-46y -48x+11y -35x-42y 4x+12y   -27x+33y 18x+5y   -48x+16y 38x+23y |
              | 44x+9y   34x-40y  48x+23y  -41x+2y  23x+24y  -7x-24y  38x-43y  37x+23y |
              | -41x-5y  19x-5y   -20x+50y 10x-24y  -4x-32y  -44x-27y 12x-37y  39x-50y |
              | 4x+6y    3x+46y   -30x-15y -17x+50y 24x-17y  -15x+22y -11x+37y 33x-22y |
              | -35x+9y  -24x-42y 49x+49y  18y      -2x+48y  -28x+6y  -23x+46y 48x-18y |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -6  22  -33 17  -30 |)
               | 0 0 x 0 y 0 0 0 |  | 34  -32 -46 9   -46 |
               | 0 0 0 y x 0 0 0 |  | -5  21  -32 -3  -48 |
               | 0 0 0 0 0 x 0 y |  | 1   0   0   0   0   |
               | 0 0 0 0 0 0 y x |  | -33 -39 -7  -16 31  |

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 :