I've got the following chain complex:

0->ZZ^2->ZZ^4->ZZ^3->0

With the boundarymaps given by

d0:(z1,z2,z3) |-> 0

d1:(z1,z2,z3,z4) |-> (-2(z1+z3+z4), 2(z1-z2), z2+z3+z4)

d2:(z1,z2)|-> (z1+z2, z1+z2,-z2,-z2)

Now I tried to compute the homology groups (e.g. H0 = ker d0 / im d1) using sage. One time manually via taking the quotients of the respective modules, one time using the ChainComplex() module. However, I don't really understand the output using the first method, and both methods seem to deliver different results...

Here's the respective worksheet: http://www.sagenb.org/home/pub/4890 - which should be self-explanatory. I've taken the transpose of d0, d1, d2 since I'm used working with leftmultiplication (i.e. d0(x) = d0*x), but sage seems to use rightmultiplication to determine the kernel resp. image of a matrix.