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 | -36x+18y 20x-36y  3x+6y    -8x+42y  -46x+17y 38x+41y 43x-29y  21x+19y  |
              | -33x-34y -8x-38y  25x-43y  -4x-20y  -10x+39y -6x-45y 19x-29y  -24x+44y |
              | -19x+3y  45x-12y  -19x+50y -23x+26y 11x+46y  31x-43y 30x-21y  40x+y    |
              | -50x+36y -21x-9y  9x+44y   -28x-y   42y      4x-50y  -41x-48y -20x-39y |
              | 8x+18y   -19x-38y 25x-19y  31x+17y  -x-27y   11x+6y  -2x+4y   -49x-39y |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | 3   39  -47 -37 32  |)
               | 0 0 x 0 y 0 0 0 |  | 32  -1  -6  23  17  |
               | 0 0 0 y x 0 0 0 |  | -33 -29 -22 21  -15 |
               | 0 0 0 0 0 x 0 y |  | 1   0   0   0   0   |
               | 0 0 0 0 0 0 y x |  | 46  13  -7  5   -14 |

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 :