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 | -30x-9y  -36x+10y 45x-43y  38x-49y  14x+10y  -9x+10y  34x-33y  -18x+34y |
              | -13x-42y -21x+24y x-12y    14x+18y  -25x+23y 50x+44y  -15x-45y -36x-y   |
              | -48x+23y -8x-3y   -21x-13y -41x+41y -45x+32y -25x+45y -45x-26y -48x-6y  |
              | -39x-2y  27x+10y  43x+36y  9x-14y   -41x+48y 45x+13y  -20x-20y -x+50y   |
              | -30x-20y -x+34y   -35x+45y -34x-2y  24x+23y  -18x-12y -32x-30y 20x+3y   |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -6  8   40  31  -12 |)
               | 0 0 x 0 y 0 0 0 |  | -5  35  45  -9  11  |
               | 0 0 0 y x 0 0 0 |  | -15 -48 -43 38  37  |
               | 0 0 0 0 0 x 0 y |  | 22  38  -37 -11 -30 |
               | 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 :