ag.algebraic geometry - Families of sheaves on arithmetic varieties

When doing moduli theory over $mathbb Z$, or another base scheme, one works with sheaves that are flat over the base; this implies that all the discrete invariants, such as the Hilbert polynomial, are constant in the fibers. Stability is defined fiber by fiber; i.e., a sheaf is (semi)stable when it it (semi)stable on all the fibers. The Quot schemes of sheaves with fixed Hilbert polynomial are defined and projective over $mathbb Z$; then the standard boundedness results all generalize. Thus one obtains stacks of stable, or semistable, bundles, which are defined over $mathbb Z$. When the existence of (quasi)projective moduli spaces is obtained via GIT, this also works over $mathbb Z$ (a result of Seshadri, see Geometric reductivity over arbitrary base, Adv. Math. 26 (1977), 225–274).

There is an issue of when the fiber of one of these moduli spaces over a prime $p$ is the moduli space of the corresponding sheaves on the fiber of $X$ over $p$; this is not automatic, because the formation of moduli spaces in positive or mixed characteristic does not, in general, commute with non-flat base chage. If this comes up, it has to be analyzed case by case.

I hope this is what you what. If the question is the construction of a space whose points corresponds to global sheaves on $X$ with metric at infinity, defining stability by some kind of Arakelov-theoretic Hilbert polynomial, then I don't have a clue.

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