extracting information from chain complexes

extracting information from chain complexes

Let's make a chain complex.

i1 : R = ZZ/101[x,y,z];

i2 : C = res coker matrix {{x,y^2,z^3}};

Some simple functions for discovering the shape of C.

length ChainComplex -- length of a chain complex or graded module

max ChainComplex -- maximum index in a chain complex

min ChainComplex -- minimum index in a chain complex

i3 : length C



o3 = 3

i4 : max C



o4 = 4

i5 : min C



o5 = 0

In order to see the matrices of the differential maps in a chain complex, examine C.dd.

i6 : C.dd



          1                   3

o6 = 0 : R  <--------------- R  : 1

               | x y2 z3 |



          3                           3

     1 : R  <----------------------- R  : 2

               {1} | -y2 -z3 0   |

               {2} | x   0   -z3 |

               {3} | 0   x   y2  |



          3                   1

     2 : R  <--------------- R  : 3

               {3} | z3  |

               {4} | -y2 |

               {5} | x   |



          1

     3 : R  <----- 0 : 4

               0



o6 : ChainComplexMap

If C is a chain complex, then C_i will produce the i-th module in the complex, C^i will produce the -i-th module in it, and C.dd_i will produce the differential whose source is C_i.

i7 : C_1



      3

o7 = R



o7 : R-module, free, degrees {1, 2, 3}

i8 : C^-1



      3

o8 = R



o8 : R-module, free, degrees {1, 2, 3}

i9 : C.dd_2



o9 = {1} | -y2 -z3 0   |

     {2} | x   0   -z3 |

     {3} | 0   x   y2  |



             3       3

o9 : Matrix R  <--- R

The function betti can be used to display the ranks of the free modules in C, together with the distribution of the basis elements by degree, at least for resolutions of homgeneous modules.

i10 : betti C



o10 = total: 1 3 3 1

          0: 1 1 . .

          1: . 1 1 .

          2: . 1 1 .

          3: . . 1 1

The ranks are displayed in the top row, and below that in row i column j is displayed the number of basis elements of degree i+j in C_j.