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 | 9x+32y  -48x+10y 40x+19y  12x+38y  5x-47y   50x-29y  46x-9y  -29x+5y  |
              | 23x+7y  29x+25y  -26x-30y -13x+12y -8x+16y  -13x+26y 8x-42y  -23x-16y |
              | 24x-28y -27x-24y 25x+27y  40x+38y  -39x-14y 43x+16y  -13x    -13x+2y  |
              | 8x-13y  -47x+40y -10x-17y 19x+3y   10x+44y  37x+25y  45x-y   20x+21y  |
              | 34x-30y 35x+28y  -4x-8y   50x+26y  30x+9y   -37x+5y  45x-37y 30x+21y  |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -26 35  -36 46  29  |)
               | 0 0 x 0 y 0 0 0 |  | 34  -42 -1  29  15  |
               | 0 0 0 y x 0 0 0 |  | -19 43  -44 -5  47  |
               | 0 0 0 0 0 x 0 y |  | 33  -44 -28 -49 -45 |
               | 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 :