1. Introduction In this paper we use Fontaine rings to study properties of local cohomology for rings of mixed characteristic. Fontaine rings, which we deene and discuss in more detail in Section 3, give a method of constructing a ring of positive characteristic from one of mixed characteristic in such a way that,

We study a question raised by Eisenbud, Mustaţǎ, and Stillman regarding the injectivity of natural maps from Ext modules to local cohomology modules. We obtain some positive answers to this question which extend earlier results of Lyubeznik. In the process, we also prove a vanishing theorem for local cohomology modules which connects theorems previously known in the case of positive characteris...