Kawamata-viehweg Vanishing via Purity Properties of the Ultra-frobenius

Part 1 of this project is concerned with proving a relative version of the results in [1]. Consider the category of Lefschetz rings in which the objects are rings of equal characteristic zero which are realized as an ultraproduct of Noetherian rings of positive characteristic, and the morphisms between two such ultraproducts are given as ultraproducts of homomorphisms between the components. A Lefschetz ring D is in general no longer Noetherian, but it has a very important feature which will play a pivotal role. Namely, let D be the ultraproduct of the Dw where Dw has characteristic p(w) > 0. Note that in order for D to have characteristic zero, p(w) must grow unboundedly. On each Dw, choose an endomorphism φw which is a power of the Frobenius, that is to say, φw raises each element to its p(w)-th power, where e(w) is some positive integer. The ultraproduct of the φw is denoted by φ and is called an ultra-Frobenius on D. In the remainder of this proposal, (R,m) will denote a Noetherian local ring containing Q.