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+45y -31x-49y 11x+21y  14x+35y  -47x-8y -3x-27y  7x+9y   19x-17y  |
              | 21x+25y -26x+45y 13x-41y  -46x+17y -30x-8y -14x-48y 35x+47y 27x-18y  |
              | 30x+33y -45x-17y -2x+29y  23x+15y  27x-50y 44x+24y  9x+46y  18x+44y  |
              | 31x+4y  -22x+17y -50x-22y -5x-45y  -32x-7y 26x-39y  17x-32y -22x+33y |
              | -17y    47x-49y  -40x-23y 34x-33y  -x-13y  -10x-6y  4x+11y  25x+6y   |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | 34  -33 25  29  46  |)
               | 0 0 x 0 y 0 0 0 |  | -21 5   -41 -40 -10 |
               | 0 0 0 y x 0 0 0 |  | -18 12  -16 37  47  |
               | 0 0 0 0 0 x 0 y |  | 2   -7  31  -50 4   |
               | 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 :