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 nonstandard
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.
Keywords
Homological conjectures, mixed characteristic, big
Cohen–Macaulay algebra, Ax–Kochen–Ershov, improved new
intersection theorem, vanishing of maps of tors
