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 | -7x+42y  19x-42y  -4x+28y  -46x+18y 26x-38y  16x-8y   -42x+24y 45x-12y |
              | 6x+34y   -32x-26y 47x-15y  -30x-24y 46x-45y  43x+44y  -42x-39y 28x+24y |
              | -2x-28y  44x-10y  -39x+30y -27x-50y 36x+5y   28x+2y   7x+15y   29x+39y |
              | 5x-5y    -22x+15y -13x+46y 7x+23y   -48x-30y -37x-47y -45x+35y -25x-3y |
              | -36x-47y 46x+13y  38x+17y  -25x+44y 47x+48y  14x-19y  49x+34y  12x+10y |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | 16  -26 -38 -33 -47 |)
               | 0 0 x 0 y 0 0 0 |  | 50  46  42  -25 -50 |
               | 0 0 0 y x 0 0 0 |  | -11 0   48  16  50  |
               | 0 0 0 0 0 x 0 y |  | 1   0   0   0   0   |
               | 0 0 0 0 0 0 y x |  | 22  -45 -20 -50 18  |

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 :