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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 | -36x-y   42x-14y  49x-33y -9x-42y  28x+31y  -23x+44y -25x+25y 26x-26y  |
              | 26x-32y  25x-5y   -8x+2y  -42x-11y -3x-42y  8x+y     -43x+41y -44x-7y  |
              | 22x+8y   -22x-25y 14x-13y 14x-41y  44x-y    8x-20y   -46x-24y -7x+38y  |
              | -29x+32y -29x-24y 45x-9y  -18x+8y  -48x-12y -36x-36y -8x-49y  -47x+26y |
              | -48x+23y 47x-34y  23x+14y 33x+25y  -48x+19y 31x-42y  -16x+42y 2x+31y   |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -23 36  -35 -45 38  |)
               | 0 0 x 0 y 0 0 0 |  | -13 49  46  34  -49 |
               | 0 0 0 y x 0 0 0 |  | -6  -13 -30 -30 25  |
               | 0 0 0 0 0 x 0 y |  | -15 27  23  -10 2   |
               | 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 :