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Background definitions:

Let $D$ be a triangulated category arising in nature (for instance as the cohomology category of a dg category). An object $E$ in $D$ is called exceptional if $RHom(E,E)$ is as simple as possible: quasi-isomorphic to the ground field. A sequence of objects $E_1, E_2,...,E_r$ is called exceptional if each object is exceptional and $RHom(E_i,E_j) \simeq 0$ for $i > j$.

Desired example:

An exceptional collection with either

a) $RHom(E_i,E_j)$ having cohomology in more than one degree for some $i < j$

or

b) some multiplication map $Ext^{m}(E_i,E_j) \otimes Ext^{n}(E_j,E_k) \rightarrow Ext^{m+n}(E_i,E_k)$ that fails to be surjective.

Of course, examples where both a) and b) would be even better.

An ideal example would be of an exceptional triple of vector bundles on a Fano 3-fold or 5-fold satisfying a) and b). But examples coming from representation theory are also welcome.

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3 Answers

If you are happy with derived categories of dg-algebras then what's the problem? If you consider an upper triangular dg-algebra with units on the diagonal it will have an exceptional collection, and the Ext's will be given by out-of-diagonal cohomology. So, e.g. take $A_{1,1} = A_{2,2} = A_{3,3} = k$, $A_{1,2} = k \oplus k[-1]$, $A_{2,3} = k$, $A_{1,3} = k^2$ with zero differential and arbitrary multiplication.

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This does answer the question as I stated it, though perhaps I should have been more specific. I'd like to know if a) and b) happen in practice, not by fiat. As Arend points out below, b) seems to happen in practice. Presumably a) does as well under some mutation. I'll try to construct an example and comment later. –  Chris Brav Mar 5 '11 at 13:59
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As you probably know, a strong exceptional collection in general does not remain strong when you start mutating. (Strong meaning that after shifting the exceptional objects, every $\mathrm{RHom}(E_i, E_j)$ is concentrated in degree zero.) Once the exceptional collection is not strong, it seems rather unlikely that b) would still be satisfied.

For instance, if you consider the examples of non-strong exceptional collections given in [Bridgeland, Stern, arXiv:0909.1732] for $\mathbb{P}^1 \times \mathbb{P}^1$, then it seems to me that they all fail to satisfy b).

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Since you welcomed examples from representation theory:

In the setting of quasi-hereditary algebras, or more precisely BGG algebras, a) happens all the time. The exceptional sequence in question is the set of standard modules, for instance it can be the Verma modules in a block of category $O$. For "neighboring" indices $\mu$, $\lambda$, you have $\text{Ext}^n(\Delta(\mu),\Delta(\lambda))=0$ for all $n$ in one direction and $\text{Hom}(\Delta(\lambda),\Delta(\mu))\neq 0$, $\text{Ext}^1(\Delta(\lambda),\Delta(\mu))\neq 0$ in the other.

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