Hom(Module,ChainComplex) -- the Hom functor

i1 : R = QQ[a..d];

i2 : C = res coker vars R

      1      4      6      4      1
o2 = R  <-- R  <-- R  <-- R  <-- R  <-- 0
                                         
     0      1      2      3      4      5

o2 : ChainComplex

i3 : M = R^1/(a,b)

o3 = cokernel | a b |

                            1
o3 : R-module, quotient of R

i4 : C' = Hom(C,M)

o4 = 0  <-- cokernel {-4} | b a | <-- cokernel {-3} | b a 0 0 0 0 0 0 | <-- cokernel {-2} | b a 0 0 0 0 0 0 0 0 0 0 | <-- cokernel {-1} | b a 0 0 0 0 0 0 | <-- cokernel | b a |
                                               {-3} | 0 0 b a 0 0 0 0 |              {-2} | 0 0 b a 0 0 0 0 0 0 0 0 |              {-1} | 0 0 b a 0 0 0 0 |      
     -5     -4                                 {-3} | 0 0 0 0 b a 0 0 |              {-2} | 0 0 0 0 b a 0 0 0 0 0 0 |              {-1} | 0 0 0 0 b a 0 0 |     0
                                               {-3} | 0 0 0 0 0 0 b a |              {-2} | 0 0 0 0 0 0 b a 0 0 0 0 |              {-1} | 0 0 0 0 0 0 b a |
                                                                                     {-2} | 0 0 0 0 0 0 0 0 b a 0 0 |      
                                      -3                                             {-2} | 0 0 0 0 0 0 0 0 0 0 b a |     -1
                                                                             
                                                                            -2

o4 : ChainComplex

i5 : C'.dd_-1

o5 = {-2} | 0 0 0 0  |
     {-2} | c 0 0 0  |
     {-2} | 0 c 0 0  |
     {-2} | d 0 0 0  |
     {-2} | 0 d 0 0  |
     {-2} | 0 0 d -c |

o5 : Matrix

i6 : C'.dd^2 == 0

o6 = true