Suppose that we are given a smooth projective variety $X$ with a *full exceptional collection* of vector bundles $(F_1, F_2, \ldots, F_k)$ in $D^b(X)$ and two vector bundles $E_1$, $E_2$ on $X$. Consider the following statement:
$$ \text{If }H^i(X, E_1\otimes F_j) = H^i(X, E_2\otimes F_j)\text{ for all }i, j\text{, then } E_1=E_2. $$

**Question 1.** Is the above true?

If no,

**Question 2.** What if we assume that the collection is strong?

**Question 3.** What if we consider all twists by the ample sheaf $\mathcal{O}(1)$, i. e.
$H^i(X, E_1\otimes F_j(t)) = H^i(X, E_2\otimes F_j(t))$ for $i, j$ and all $t$?

**Question 4.** Are there any known cohomological criteria for isomorphism?

If yes,

**Question 5.** Does it suffice to assume that $(F_1, F_2, \ldots, F_k)$ generate the derived category, *without* them forming an exceptional collection?

**Remark.** Take $X=\mathbb{P}^n$ and $F_i = \mathcal{O}(i)$ or $F_i = \Omega^i(i)$. Then by the Beilinson spectral sequence, the cohomology groups $H^i(X, E\otimes F_i)$
*would* determine $E$ unambiguously *if* we knew the differentials.