Fontaine rings and local cohomology
The author investigates the possibility of using Fontaine's construction to construct almost Cohen-Macaulay algebras in mixed characteristic. He also shows that the existence of almost Cohen-Macaulay algebras implies the Monomial conjecture and gives an example of how this procedure. For a ring \(R\), an \(R\)-algebra \(A\) is called almost Cohen-Macaulay if it satisfies following conditions: \((1)\) \(i\)th local cohomology module \(H ^{i}_{({\mathbf x})}(A)\) with respect to a system of parameters \(x_{1}, \ldots, x_{d}\) is almost zero for \(i=0, \ldots, d-1\). \((2)\) \(A/(x_{1}, \ldots,x_{d})A\) is not almost zero. The author shows that the Monomial Conjecture holds for a Noetherian local ring of mixed characteristic with almost Cohen-Macaulay algebra, and that the perfect closure \(R^{\infty}\) of a Noetherian integral domain \(R\) of positive characteristic is an almost Cohen-Macaulay algebra for \(R\). Let \(R\) be a ring obtained by adjoining \(p^{n}\)th roots of certain elements to a complete local domain of mixed characteristic \(R_{0}\). The author considers the perfect closure \(E_{0}^{\infty}\) of some ring \(E_{0}\) which is constructed by using the notion of Fontaine ring, and show a following theorem: Let \(T\) be a ring containing \(E_{0}^{\infty}\) and \(W(T)\) be a ring of Witt vector for \(T\). We suppose that \(P\) is not a zero-divisor in \(T\) and the local cohomology \(H^{i}_{(P, X_{2},\ldots, X_{d})}(T)\) is almost zero for \(i=0,\ldots,d-1\). Then the local cohomology \(H^{i}_{(p, X_{2},\ldots, X_{d})}(W(T)/(P-p))\) is almost zero for \(i=0,\ldots,d-1\).
scientific article; zbMATH DE number 5696948
scientific article