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.

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    $\begingroup$ I saw an abstract of a talk by Efimov where he studied derived categories of Grassmannians over the integers... I don't know if there is a paper about this. $\endgroup$ Nov 5 '13 at 8:18
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    $\begingroup$ My impression, from far far far away, is that the derived category is not the best thing to look at over Q or Z. I thought that due to some badness (non-unique arrows) in derived categories, they do not behave very well for descent. Presumably, if you want to say something meaningful over Q, you would want to treat algebraic extensions of Q as well. Hence the activity in modern fancy category/topos/homotopy theory to handle things better than D(X). $\endgroup$
    – Marty
    Nov 5 '13 at 11:04
  • $\begingroup$ Sure, one can do some sort of dg enhancements of derived category; it is also likely that there should be some Arakelov flavor to it, rather than a straightforward coherent sheaves, but I don't really have any feel for it. $\endgroup$ Nov 5 '13 at 12:00
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    $\begingroup$ For (3) and (4), there is no difference between $\mathbb{C}$ and $\mathbb{Q}$. $\endgroup$
    – naf
    Nov 7 '13 at 6:25

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.


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