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On P136 they define a map $h:H_\delta\,H_d(K)\to H_D(K)$ by some diagram chasing on the double complex $K$, and claim that associated chain $\phi + \phi_1 + \cdots + \phi_n$ is a $D$-cocycle (though there are some nuances been discussed here)

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However, my question is about the premise $H^{p+2,q-1}_{\delta}\,H_d(K) = 0$. In the context of Bott & Tu, they discuss a double complex $K$ with a nonzero row of $H_\delta\, H_d$-cohomology.

Why do they ignore that row in their construction of $h$?

I reason if $\phi$ is above that row, then when extended to $\phi + \phi_1 + \cdots$ step by step, it will finally meet the nonzero row and the last $\delta \phi_n$ cannot be cancelled out so that $\phi + \phi_1 + \cdots$ is not a true $D$-cocycle. Any help is appreciated.

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The proposition says $$H_D^m(K)\cong \bigoplus\limits_{a+b=m}H_\delta^aH_d^b(K)=H_\delta^pH_d^q(K),$$where $q$ refers exactly to the row that $H^p_dH^q_\delta(K)$ has some nontrivial element. Therefore to define $h:\bigoplus\limits_{a+b=m}H_\delta^aH_d^b(K)=H^p_dH^q_\delta(K)\to H_D^m(K)$, we just need to choose a representative $\phi$ of $(p,q)$ degree(i.e. $\phi\in K^{p,q}$) such that $[\phi]\in H^p_dH^q_\delta(K)$ is nontrivial and extend it all along the diagonal to the last block $K^{p+q,0}$. Obviously, in this construction we will never meet any nontrvial row, since we just start from there. Moreover, in this construction, we see the condition that only one row is nontrivial is essential!

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