Families of sheaves on arithmetic varieties - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T13:23:17Z http://mathoverflow.net/feeds/question/20913 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/20913/families-of-sheaves-on-arithmetic-varieties Families of sheaves on arithmetic varieties TonyS 2010-04-10T11:26:19Z 2010-04-10T12:56:52Z <p>Given an arithmetic variety $f: X \rightarrow Spec(\mathbb{Z})$.</p> <p>Is there a notion of boundedness for families of sheaves on $X$?</p> <p>I only found the notion for families on the fibers of $f$. But i am interested in sheaves defined on $X$.</p> <p>All definitions / theorems i found only work when $X$ is defined over some field $k$, where one has the Hilbert polynomial, slope etc, which we don't have in this case. Is there some substitute for these terms? </p> <p>Or are there even results about moduli spaces of sheaves on arithmetic varieties? </p> <p>Edit: According to <a href="http://arxiv.org/abs/math/0612268" rel="nofollow">http://arxiv.org/abs/math/0612268</a> there is a notion of arithmetic (semi)stability. One even has a Harder Narasimhan filtration. Can one define the notion of boundedness in the Arakelov setting? Are there any results on moduli spaces of vector bundles in Arakelov theory?</p> http://mathoverflow.net/questions/20913/families-of-sheaves-on-arithmetic-varieties/20917#20917 Answer by Angelo for Families of sheaves on arithmetic varieties Angelo 2010-04-10T12:08:38Z 2010-04-10T12:08:38Z <p>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 <em>Geometric reductivity over arbitrary base</em>, Adv. Math. 26 (1977), 225–274). </p> <p>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.</p> <p>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.</p>