next | previous | forward | backward | up | top | index | toc | home

populateCechComplexCC(Ideal,List) -- Cech complex skeleton for the computation of the characteristic cycles of local cohomology modules

Synopsis

Description

For the ideal I=(f1,...,fk) the routine computes the characteristic cycles of the localized modules Mfi1,...,fik and places them in the corresponding places in the Cech complex.
i1 : W =  QQ[x_1..x_6, a_1..a_6];
i2 : I = minors(2, matrix{{x_1, x_2, x_3}, {x_4, x_5, x_6}});

o2 : Ideal of W
i3 : cc = {ideal W => 1};
i4 : B = populateCechComplexCC(I,cc)

o4 = MutableHashTable{...8...}

o4 : MutableHashTable
i5 : scan(keys B, k->print (k=>B#k)) -- CCs of Cech complex BEFORE pruning
{1, 2} => {0 => 1, ideal(x x  - x x ) => 1, ideal (x , x , x , x ) => 1, ideal(x x  - x x ) => 1, ideal (x x  - x x , x x  - x x , x x  - x x ) => 1, ideal (x , x ) => 1, ideal (x , x , x , x ) => 1, ideal (x , x , x x  - x x ) => 1}
                          3 5    2 6                6   5   3   2               3 4    1 6                3 5    2 6   3 4    1 6   2 4    1 5                6   3                6   4   3   1                6   3   2 4    1 5
{} => {0 => 1}
{0, 1} => {0 => 1, ideal(x x  - x x ) => 1, ideal (x , x , x , x ) => 1, ideal(x x  - x x ) => 1, ideal (x x  - x x , x x  - x x , x x  - x x ) => 1, ideal (x , x ) => 1, ideal (x , x , x , x ) => 1, ideal (x , x , x x  - x x ) => 1}
                          3 4    1 6                6   4   3   1               2 4    1 5                3 5    2 6   3 4    1 6   2 4    1 5                4   1                5   4   2   1                4   1   3 5    2 6
{0, 2} => {0 => 1, ideal(x x  - x x ) => 1, ideal (x , x , x , x ) => 1, ideal(x x  - x x ) => 1, ideal (x x  - x x , x x  - x x , x x  - x x ) => 1, ideal (x , x ) => 1, ideal (x , x , x , x ) => 1, ideal (x , x , x x  - x x ) => 1}
                          3 5    2 6                6   5   3   2               2 4    1 5                3 5    2 6   3 4    1 6   2 4    1 5                5   2                5   4   2   1                5   2   3 4    1 6
{0, 1, 2} => {0 => 1, ideal(x x  - x x ) => 1, ideal (x , x , x , x ) => 1, ideal(x x  - x x ) => 1, ideal (x x  - x x , x x  - x x , x x  - x x ) => 2, ideal (x , x ) => 1, ideal (x , x , x , x ) => 1, ideal (x , x , x x  - x x ) => 1, ideal(x x  - x x ) => 1, ideal (x , x ) => 1, ideal (x , x , x , x ) => 1, ideal (x , x , x x  - x x ) => 1, ideal (x , x ) => 1, ideal (x , x , x x  - x x ) => 1, ideal (x , x , x , x , x , x ) => 1}
                             3 5    2 6                6   5   3   2               3 4    1 6                3 5    2 6   3 4    1 6   2 4    1 5                6   3                6   4   3   1                6   3   2 4    1 5               2 4    1 5                5   2                5   4   2   1                5   2   3 4    1 6                4   1                4   1   3 5    2 6                6   5   4   3   2   1
{0} => {0 => 1, ideal(x x  - x x ) => 1, ideal (x , x , x , x ) => 1}
                       2 4    1 5                5   4   2   1
{1} => {0 => 1, ideal(x x  - x x ) => 1, ideal (x , x , x , x ) => 1}
                       3 4    1 6                6   4   3   1
{2} => {0 => 1, ideal(x x  - x x ) => 1, ideal (x , x , x , x ) => 1}
                       3 5    2 6                6   5   3   2

Caveat

The module has to be a regular holonomic complex-analytic module; while the holomicity can be checked by isHolonomic there is no algorithm to check the regularity.

See also