Hom(Module,ChainComplex)

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

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