Let $\mathcal{A}, \mathcal{B}$ be two abelian categories with sufficiently many injective objects (in my case these are categories of sheaves of vector spaces on a manifold). Let $f_*\colon \mathcal{A}\to\mathcal{B}$ be a left exact functor which commutes with direct sums (in my case it is push-forward on sheaves). Let $Rf_*\colon D^+\mathcal{A}\to D^+\mathcal{B}$ be the derived functor between the derived categories. Let $F^\bullet$ be a bounded complex of objects of $\mathcal{A}$.

**Question 1.** Assume that $R^qf_*(F^p)=0$ for every $q\ne i$ where $i$ is a fixed number, and for every $p$. Is it true that $Rf_*F^\bullet$ is isomorphic in $D^+\mathcal{B}$ to the complex
$$\dots\to R^if_*(F^k)\to R^if_*(F^{k+1})\to R^if_*(F^{k+2})\to\dots \,?$$
(For $i=0$ this is well known to be true. I believe that it should be true in general. A reference would also be helpful.)

**Question 2.** Assume now that for a fixed integer $a$ one has
$$\begin{eqnarray*}
R^qf_*(F^a)=0 \mbox{ for every } q;\\
R^qf_*(F^p)=0 \mbox{ for every } q\ne 0, p>a;\\
R^qf_*(F^p)=0 \mbox{ for every } q\ne 1, p<a.
\end{eqnarray*}$$
Is it true that $Rf_*(F^\bullet)$ is isomorphic to the complex (with an appropriate cohomological shift)
$$\to R^1f_*F^{a-2}\to R^1f_*F^{a-1}
\overset{\Delta}{\to}R^0f_*F^{a+1}\to R^0f_*F^{a+1}\to \dots$$
where $\Delta$ is the only non-trivial differential of second order in the well known spectral sequence $E^{p.q}_r$ which converges to $R^{p+q}f_*(F^\bullet)$ and whose first term is equal to $E_1^{p,q}=R^qf_*(F^p)$? (The other arrows in the complex are induced in the obvious way by arrows in the complex $F^\bullet$.)

**Remark.** The second question is motivated by the fact that the two complexes have the same cohomology, as it follows from the spectral sequence considerations.