next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
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 | x+33y    31x-41y  38x-22y  -21x-47y -3x+8y   -24x+45y 23x-10y  23x+45y  |
              | -20x-44y 50x+23y  13x-34y  -6x+47y  26x-33y  -6x-24y  -15x+22y -45x-7y  |
              | -25x+23y 24x+23y  -21x-47y -48x-43y -28x-39y -19x-36y -35x+41y -9x+40y  |
              | 43x-41y  -16x+13y -28x-45y -23x-50y -20x+y   -15x+11y -36x+34y -14x-10y |
              | -7x-22y  46x+33y  41x+50y  45x+6y   29x+32y  44x-19y  -7x+47y  11x+4y   |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | 13 -9  -41 -1  18  |)
               | 0 0 x 0 y 0 0 0 |  | 22 -46 1   28  -47 |
               | 0 0 0 y x 0 0 0 |  | 21 -5  -31 21  -17 |
               | 0 0 0 0 0 x 0 y |  | 36 42  31  -45 46  |
               | 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 :