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 | -27x-47y -2x-9y   23x-27y  27x+33y -21x+13y 31x-47y 19x+3y   28x+37y |
              | -39x+14y -2x+26y  -49x-30y -8y     -23x+17y 26x-12y -26x+8y  39x-39y |
              | 21x+34y  -24x+22y x+25y    26x-20y -30x+4y  -31y    -x+40y   -9x-37y |
              | -40x+49y -35x+45y -38x+46y 38x+27y -12x+42y 20x-49y -34x-35y -43x-8y |
              | -25x-32y 39x-11y  31x+28y  24x-48y x+10y    8x+6y   31x-19y  5x-45y  |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | 27  -21 41 5   -31 |)
               | 0 0 x 0 y 0 0 0 |  | 10  12  50 5   10  |
               | 0 0 0 y x 0 0 0 |  | 27  6   10 -29 -1  |
               | 0 0 0 0 0 x 0 y |  | 1   0   0  0   0   |
               | 0 0 0 0 0 0 y x |  | -16 -46 12 -49 -41 |

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 :