Derived categories of arithmetic schemes?

Question

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.

Answer 1

This "answer" is an example of natural occurence of derived categories of arithmetic schemes rather than a description of what is known in general on this kind of categories.

Let me give an example related to mirror symmetry.

The Fukaya category of a Liouville manifold (an exact symplectic manifold with good properties at "infinity") can be defined over $\mathbb{Z}$ (or any ring) because the operations are defined by counting holomorphic polygons and counting gives integers. So one expects the mirror of a Liouville manifold $Y$ to be an algebraic variety $X$ defined over $\mathbb{Z}$, such that $Fuk(Y)=Perf(X)$. In this context $X$ is general singular and one has to make a difference between $Perf(X)$ and $D(X)$ ($D(X)$ should be related to the wrapped Fukaya category of $Y$).

The above relation should be a version "at infinity" of the maybe more familiar mirror relation between smooth and compact manifolds. On the symplectic side, one can compactify the Liouville manifold and obtained a compact symplectic manifold with a Fukaya category defined on $\mathbb{Z}[[q]]$ being a deformation of the Fukaya catgeory of $Y$. Here $q$ is a formal parameter keeping track of the intersection number of the holomorphic disks with the divisor added to compactify. On the mirror side, one should have a variety over $\mathbb{Z}[[q]]$ realizing a smoothing of $X$. The mirror statement is now an equivalence of categories over $\mathbb{Z}[[q]]$. The usual version of the mirror symmetry between smooth compact manifolds should be recovered by inverting $q$.

The above picture has been made precise in the case of the elliptic curve by Lekili and Perutz in http://arxiv.org/abs/1211.4632 In this example, the algebraic variety over $\mathbb{Z}[[q]]$ is the Tate curve, a very familiar object in arithmetic geometry.