  Let R be a locally finitely generated algebra over a discrete valuation ring
V of mixed characteristic. For any of the homological properties, the Direct
Summand Theorem, the Monomial Theorem, the Improved New Intersection Theorem,
the Vanishing of Maps of Tors and the Hochster-Roberts Theorem, we show that it
holds for R and possibly some other data defined over R, provided the residual
characteristic of V is sufficiently large in terms of the complexity of the
data, where the complexity is primarily given in terms of the degrees of the
polynomials over V that define the data, but possibly also by some additional
invariants.
