Exceptional collection

Remarks

Example: on the projective space $\mathbb{P}^n$ over $k$, the line bundles $(\mathcal{O}, \mathcal{O}(1), \dots, \mathcal{O}(n))$ form a full strong exceptional collection. It lives in the bounded derived category of coherent sheaves, and this is a theorem of Beilinson. Clause 2 holds since $\Hom(\mathcal{O}(j), \mathcal{O}(i)[l]) = H^l(\mathbb{P}^n, \mathcal{O}(i - j))$, and for $-n \le i - j < 0$ all cohomology of $\mathcal{O}(i - j)$ vanishes. Non-example: on an elliptic curve $C$ over $k$, no line bundle $L$ is exceptional, since $\Hom(L, L[1]) = H^1(C, \mathcal{O}_C) \cong k \ne 0$. A full exceptional collection gives a semiorthogonal decomposition $\mathcal{T} = \langle \mathcal{A}_1, \dots, \mathcal{A}_n \rangle$, where $\mathcal{A}_i$ is the subcategory generated by $E_i$, equivalent to the bounded derived category of finite-dimensional vector spaces over $k$.