I am reading Bott and Tu.
https://www.maths.ed.ac.uk/~v1ranick/papers/botttu.pdf
On page 140 of that book, the exactness of co-boundary operator $\delta$ is used to prove that the double complex of forms with compact support has only one non zero column. "By the exactness of the rows, $H_{\delta}(K)$ is:" and then we are shown a figure with all but column zero filled with zeros. The zero column is $\Omega_c^i$ where $i=0, 1..$
When I look at the exactness and the definition of $H_{\delta}(k)$, I conclude that all columns have to be zero including the zero column. After all exactness means kernel=image and thus the cohomology should all be zero. What am I missing? Why is there a non-zero column at all?
ps: The answers in comments below have confused me even more :), so if anyone can give a comprehensive answer, I shall be grateful.