Let $(C^{p,q},d_h,d_v)$ be a double complex of modules and let $d_h,d_v$ be the horizontal and vertical differential respectively. Suppose that

(1) the horizontal rows are exact,

(2) the columns are exact except at $C^{p,0}$ for all $p$.

Is there a quick way to deduce whether the induced horizontal differential is exact on $H^i(C^{p,q},d_v)$? That is , is $H^j(H^i(C^{p,q},d_v),d_h)=0$ for all $i,j$? If not, is it possible to impose conditions on the double complex so that this is true?

I am aware that a counterexample exists if we only assume (1), see Double complex with exact rows, but the counterexample mentioned in the above link does not satisfy assumption (2).

Would boundedness ($C^{p,q}=0$ if $|p|>k$ for some $k$) imply my question? What about when the complex is unbounded (this is the case I am really interested in)?




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