A two-row hypercohomology spectral sequence

For $r=2$ the only possible nonzero arrows are ${\tau }_p$ from row one to row zero. Hence $E_3^{p,0}=\mathrm{coker}{\tau }_{p-2}$ and $E_3^{p,1}=\ker {\tau }_p$. For $r\ge 3$, every outgoing arrow from either row has negative second coordinate and every incoming arrow starts above row one. These positions stay zero on successive pages, so $E_3=E_{\mathrm{\infty }}$.

In total degree $n$, the only possible filtration quotients are at $p=n$ and $p=n-1$. F1's zero/full endpoints identify the first as a subobject of the target and the second as its quotient, giving the displayed exact sequence. At $n=0$ it reduces to ${\mathbb{R}}^{0}F(K)=F(A)$; at $n=1$ its subobject is $R^{1}F(A)$ and its quotient is $\ker {\tau }_0$. Below zero there are no surviving quotients, so the finite target filtration forces vanishing. This proves convergence and identifies the unresolved extension, rather than assuming a splitting.

A fully numerical specialization takes abelian groups, $F$ the identity, and $K^0=\mathbb{Z}\mathrm{/}2$, $K^1=\mathbb{Z}\mathrm{/}3$ with zero differential and supplied replacements. Exactness of identity means its positive derived objects vanish, since applying it preserves the exact resolution. Thus $E_2^{0,0}=\mathbb{Z}\mathrm{/}2$, $E_2^{0,1}=\mathbb{Z}\mathrm{/}3$, all other entries are zero, and every ${\tau }_p$ is zero. Each total degree has one nonzero quotient: the target is $\mathbb{Z}\mathrm{/}2$ in degree zero, $\mathbb{Z}\mathrm{/}3$ in degree one and zero otherwise. The upper edges are the identity under the augmentation identification. This specialization has no extension ambiguity and uses no choice beyond the supplied-data convention.