I do not know much about moduli of sheaves and I wanted to shows that for a smooth (projective) family over a discrete valuation ring of mixed characteristics (relative dimension 3), the locally free stable sheaves (of rank 2) were parametrized by a open subscheme of the moduli of stable coherent torsion free sheaves. The (local) existence of a universal sheaf would be sufficient to conclude.

According to Simpson (here theorem 1.21 (4)) there is, locally in étale topology, such universal sheaf in characteristic zero. But I do not know if there is a similar statement in the mixed characteristic case. Langer (p.3) says that, locally in fppf topology, the quotient giving the moduli space of stable sheaves is just PGL(m)-bundle (maybe there is no connection at all with my question). But I do not know if it is sufficient.


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