Derived category of coherent sheaves

Remarks

$D^b(X)$ is a triangulated category with shift $F^\bullet \mapsto F^\bullet[1]$. It is equivalent to the full subcategory of the derived category of all $\mathcal{O}_X$-modules whose objects are the complexes with coherent cohomology sheaves, nonzero in finitely many degrees. Example: for $X = \operatorname{Spec} k$ with $k$ a field, coherent sheaves are finite-dimensional vector spaces, and every object of $D^b(X)$ is isomorphic to $\bigoplus_i H^i(F^\bullet)[-i]$, its cohomology placed in each degree. Non-example: on $X = \mathbb{A}^1_k = \operatorname{Spec} k[x]$, the complex $k[x] \xrightarrow{x} k[x]$ in degrees $-1, 0$ maps to $k[x]/(x)$ in degree $0$ by a quasi-isomorphism. In the homotopy category of complexes, before quasi-isomorphisms are inverted, these two complexes are not isomorphic, since there is no nonzero map of modules $k[x]/(x) \to k[x]$. In $D^b(X)$ they are isomorphic. Theorems on $D^b$: Beilinson showed that $D^b(\mathbb{P}^n)$ is generated by $\mathcal{O}, \mathcal{O}(1), \ldots, \mathcal{O}(n)$, and Mukai showed that an abelian variety and its dual have equivalent derived categories.