Let $X$ be a smooth scheme of finite type over $\mathbb Q$ or $\mathbb Z$. It is natural to consider $D(X)$, the derived category of coherent sheaves on $X$. Beyond the definition, have there been any interesting statements about it? In particular, are there any analogs of the following results over $\mathbb C$?
(1) $D(X)$ describes boundary conditions of type IIB nonlinear sigma model with target $X$, at least when $X$ is Calabi-Yau.
(2) There is a space of Bridgeland stability conditions on $D(X)$ with walls in codimension one.
(3) If canonical or anticanonical class of $X$ is ample, then $X$ is determined by $D(X)$.
(4) On occasion, $D^b(X)$ can be generated by an exceptional collection of objects.
(5) For a Calabi-Yau $X$ there is often a mirror partner such that $D^b(X)$ is, at least conjecturally equivalent to derived Fukaya category of the mirror (Kontsevich's homological mirror symmetry).
I want to point out that rational points on $Pic(X)$ give autoequivalences of $D(X)$, so an alternative description of $D(X)$, if it were to involve $L$-functions, would be relevant to BSD conjecture.