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Let $X$ be smooth Fano variety with $\operatorname{Pic}(X) = \mathbb{Z}$ of dimension $m$ with canonical class $K$, and $E_0,...,E_n$ is exceptional sequence of $(n+1)$ vector bundles in $D^b(Coh(X))$. Define inductively $E_{n+i} = R_{<E_{i+1},...,E_{n+i-1}>} E_i$ and $E_{-i} = L_{<E_{-i+1},...,E_{-i+n-1}>} E_{-i+n}$ where $R_{C} A$ and $L_C A$ is left and right mutations of object $A$ through category $C$. In "Representation of associative algebras and coherent sheaves" in theorem 4.2 Bondal states

Collection $E_0,...,E_n$ full iff $\forall i E_{i+n+1} = E_i \otimes K[-m]$

Is it sufficient to check only that $E_{n+1} = E_0 \otimes K[-m]$ or I really have to check every $i$? More vague question: how could I check that some exceptional collection of vector bundles is in fact full by finite number of computations?

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It is not enough to check only $E_0$, but checking $E_0,\dots,E_n$ is enough.

There are many ways of proving fullness, most of them require first to construct some more objects from the considered exceptional collection, and after that relate the variety with another one, where an exceptional collection is already known.

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