Separable integral extensions and plus closure
Let \(R\) be an excellent domain of characteristic \(p\), \(K\) the quotient field of \(R\), \(\overline{K}\) an algebraic closure of \(K\) and \(L\) the maximal separable field extension of \(K\) in \(\overline{K}\). Let \(R^+\) (respectively \(R^{+\operatorname {sep}})\) be the integral closure of \(R\) in \(\overline{K}\) (respectively \(L\)) and, for an ideal \(I\) of \(R\), \(I^+= IR^+\cap R\). Using an equational construction, the author shows that for every element \(z\in I^+\) there exists an integral domain \(S\), which is a separable module-finite extension of \(R\), such that \(z\in IS\). Consequently, he shows that \(I^+= IR^{+\operatorname {sep}}\cap R\), for every ideal \(I\) of \(R\). He also proves that, when \(R\) is local, \(R^{+\operatorname {sep}}\) is a big Cohen-Macaulay \(R\)-algebra [\textit{M. Hochster} and \textit{C. Huneke} have shown in Ann. Math., II. Ser. 135, 53-89 (1992; Zbl 0753.13003) that \(R^+\) is a big Cohen-Macaulay algebra]. In the last section, it is shown that \(xyz\in (x^2, y^2, z^2)^+\) in the cubic hypersurface ring \(R = k[X, Y, Z]/ (X^3+ Y^3+ Z^3)\), where \(K\) is a field of characteristic \(p\equiv 1\pmod 3\).
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