# Exceptional collections and cohomological criteria for isomorphism

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.

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You convinced me, except for my Question 3. Actually, in my case the bundle $E_1$ is the Frobenius push-forward of $\mathcal{O}_X$. There are easy cohomological criteria for $E_1 = \mathcal{O}_X$. What about $E_1 = F^s_\ast \mathcal{O}_X$? –  Piotr Achinger Feb 26 '10 at 20:17
If the isomorphism above is as dg-modules over the Ext dg-algebra of $\bigoplus F_i$, then the answer to question 1 is yes. This is a general statement about generating sets in derived categories. The reason is that we can get $E_i^*$ by tensoring $\bigoplus F_i$ with the cohomologies over the dg-Ext algebra.