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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 | 24x-36y  -29x-24y -18x-13y 45x-34y  39x+43y  40x+11y 2x+29y   27x-22y  |
              | -30x-29y -38x-16y -43x-15y -48x-47y -17x-11y 46x-28y -47x+15y 32x-9y   |
              | 19x+19y  39x+21y  -28x-47y 47x+19y  48x+36y  x-3y    -37x-13y -32x-20y |
              | -10x-29y 34x+19y  38x+2y   -16x+7y  35x+11y  22x-47y -10x+30y 24x-30y  |
              | -8x-22y  -47x-39y 16x+22y  15x-23y  -38x+33y -23x-7y -18x+39y -48x-15y |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -25 -49 -25 15  1   |)
               | 0 0 x 0 y 0 0 0 |  | 28  -45 29  30  -39 |
               | 0 0 0 y x 0 0 0 |  | -16 43  -43 -47 -24 |
               | 0 0 0 0 0 x 0 y |  | -14 -37 25  -19 26  |
               | 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 :

For the programmer

The object decomposeModule is a method function.