Originally Posted by

**MaximalIdeal** Hi,

How do we prove isomorphic complexes have the same cohomology?

If C. and D. are isomorphic, how do we prove

$\displaystyle \mbox{H}_{n}\mbox{(C.)}\cong\mbox{H}_{n}\mbox{(D.) } \ \forall\mbox{n}$

so we need an isomorphism for:

$\displaystyle \mbox{kerd}_{n}\mbox{/}\mbox{imd}_{n+1}\cong \mbox{kerd^\prime}_{n}{/}\mbox{imd}^\prime_{n+1}$

where $\displaystyle \mbox{d}_{n}$, $\displaystyle \mbox{d^\prime}_{n}$ are the differentiations for C. and D. respectively.

I think we must use the individual isomorphisms $\displaystyle \mbox{g:}\mbox{C}_{n}\rightarrow\mbox{D}_{n}$ to find some property/relation between the kernels in each of the exact sequences, and same for the images. I found such properties, just can't see how to finish by constructing the isomorphism!

Thanks in advance