sum(Complex) -- make the direct sum of all terms

Synopsis

Function: sum
Usage:

sum C

sum f
Inputs:
- C, a complex, or f, a map of complexes
Outputs:
- a module, or a matrix, if the input is a complex map

Description

This is the forgetful functor from the category of chain complexes to the category of modules. A chain complex $C$ is sent to the direct sum $\bigoplus_i C_i$ of its terms. A map of chain complexes $f \colon C \to D$ is sent to the direct sum $\bigoplus_i f_i \colon \bigoplus_i C_i \to \bigoplus_i D_i$.

i1 : S = ZZ/101[a,b,c];

i2 : C = koszulComplex {a,b,c}

      1      3      3      1
o2 = S  <-- S  <-- S  <-- S
                           
     0      1      2      3

o2 : Complex

i3 : sum C

      8
o3 = S

o3 : S-module, free, degrees {0..1, 2:1, 3:2, 3}

i4 : assert(rank sum C == 2^3)

i5 : f = randomComplexMap(C, C, InternalDegree => 1, Cycle => true)

          1                       1
o5 = 0 : S  <------------------- S  : 0
               | -5a-27b-40c |

          3                                                  3
     1 : S  <---------------------------------------------- S  : 1
               {1} | -5a-46b+42c -7b+24c    29b+9c      |
               {1} | 19a-8c      2a-27b-14c -29a-30c    |
               {1} | 19a+8b      -24a-26b   -14a+3b-40c |

          3                                                  3
     2 : S  <---------------------------------------------- S  : 2
               {2} | 2a-46b-30c -29a-10c     -29b+22c   |
               {2} | -24a-29b   -14a-36b+42c -38b+24c   |
               {2} | -16a-8b    39a-8c       24a+3b-14c |

          1                           1
     3 : S  <----------------------- S  : 3
               {3} | 24a-36b-30c |

o5 : ComplexMap

i6 : g = sum f

o6 = {0} | -5a-27b-40c 0           0          0           0         
     {1} | 0           -5a-46b+42c -7b+24c    29b+9c      0         
     {1} | 0           19a-8c      2a-27b-14c -29a-30c    0         
     {1} | 0           19a+8b      -24a-26b   -14a+3b-40c 0         
     {2} | 0           0           0          0           2a-46b-30c
     {2} | 0           0           0          0           -24a-29b  
     {2} | 0           0           0          0           -16a-8b   
     {3} | 0           0           0          0           0         
     ------------------------------------------------------------------------
     0            0          0           |
     0            0          0           |
     0            0          0           |
     0            0          0           |
     -29a-10c     -29b+22c   0           |
     -14a-36b+42c -38b+24c   0           |
     39a-8c       24a+3b-14c 0           |
     0            0          24a-36b-30c |

             8      8
o6 : Matrix S  <-- S

i7 : assert(g^2 === sum f^2)

i8 : assert(target g === sum target f)

i9 : assert(source g === sum source f)

i10 : h = sum dd^C

o10 = {0} | 0 a b c 0  0  0  0  |
      {1} | 0 0 0 0 -b -c 0  0  |
      {1} | 0 0 0 0 a  0  -c 0  |
      {1} | 0 0 0 0 0  a  b  0  |
      {2} | 0 0 0 0 0  0  0  c  |
      {2} | 0 0 0 0 0  0  0  -b |
      {2} | 0 0 0 0 0  0  0  a  |
      {3} | 0 0 0 0 0  0  0  0  |

              8      8
o10 : Matrix S  <-- S

i11 : assert(h^2 == 0)

Ways to use this method: