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Throughout this paper "ring" means commutative ring with identity and modules are unital. Our main interest is in local rings, i.e., Noetherian rings (R, m) with a unique maximal ideal m . In such a ring, xx, ... , xn e m is a system of parameters if m has a power in the ideal (x{, ... , xn)R and n is the Krull dimension of R. When R is complete and contains a field, this means that R is module-finite over a formal power series subring K((xx, ... , xn)) = A. i? is called Cohen-Macaulay if some (equivalently, every) system of parameters is a regular se­ quence in i?, which means that every xi+x is a nonzerodivisor on R/{xx, ... , xt)R, for 0 < i < n - 1. In the case where R is module-finite over the formal power series subring A , this means that R is a free ^4-module. For many theorems of commutative algebra and of algebraic geometry, the Cohen-Macaulay condition (possibly on the local rings of a variety) is just what is needed to make the theory work. Our main result, which is given below in an algebraic form in Theorem 1 and in a geometric form in The­ orem 2, asserts that, under mild conditions on a local ring R of positive prime characteristic p, one can "correct" the failure of the Cohen-Macaulay condition in R itself by passing to a very large integral extension of R. It is worth emphasizing that this is quite false in characteristic 0, and comes as a surprise, we believe, in characteristi c p. One can get an idea of how far these theorems are from the truth over a field of characteristic 0 from the following observation: In the situation of Theorem 2, when X is projectively normal, the maps of cohomology are always injective, by an easy trace argument. Despite the fact that both Theorem 1 and Theorem 2

Absolute integral closures are big Cohen-Macaulay algebras in characteristic 𝑃