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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 | 46x-47y  -12x-18y -11x+8y -15x-31y 13x+3y   -10x-47y 18x-7y   -17x+14y |
              | 41x-24y  -16x-15y -32x-3y -34x-34y 44x-15y  -13x-11y 48x+26y  -28x-7y  |
              | -25x+27y 8x+22y   -14x+8y x+20y    -45x+14y -17x-33y -14x-31y -15x+9y  |
              | -x-3y    -24x-23y -29x+6y 8x-10y   -26x-49y 34x-40y  9x+48y   5x-22y   |
              | -26x-12y -5x-23y  22x-18y 23x+8y   48x+45y  22x+12y  -33x+9y  -17x-47y |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -19 -49 -48 -46 12  |)
               | 0 0 x 0 y 0 0 0 |  | -14 -23 17  49  40  |
               | 0 0 0 y x 0 0 0 |  | -46 40  -36 -35 -17 |
               | 0 0 0 0 0 x 0 y |  | 1   0   0   0   0   |
               | 0 0 0 0 0 0 y x |  | -41 19  13  -29 4   |

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 :