  In this paper, various Homological Conjectures are studied for local rings
which are locally finitely generated over a discrete valuation ring $V$ of
mixed characteristic. Typically, we can only conclude that a particular
Conjecture holds for such a ring 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 such as
(homological) multiplicity. Thus asymptotic versions of the Improved New
Intersection Theorem, the Monomial Conjecture, the Direct Summand Conjecture,
the Hochster-Roberts Theorem and the Vanishing of Maps of Tors Conjecture are
given.
  That the results only hold asymptotically, is due to the fact that
non-standard arguments are used, relying on the Ax-Kochen-Ershov Principle, to
infer their validity from their positive characteristic counterparts. A key
role in this transfer is played by the Hochster-Huneke canonical construction
of big Cohen-Macaulay algebras in positive characteristic via absolute integral
closures.
