We consider the problem of determining the dimension of the space of bivariate splines C k( ), for all k. This problem is closely related to the question of whether C(
̂) is a free R-module. The main result is that C(
̂) is free if and only if j j has genus zero and C k( ) has the expected dimension for k = r + 1 (and hence for all k). We also obtain several interesting corollaries, including the...

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...