Tight closure and graded integral extensions
For a Noetherian \(\mathbb{N}\)-graded domain \(R\), let \(R^+\) be the integral closure of \(R\) in an algebraic closure of the fraction field of \(R\), and let \(R^{+ gr}\) be the \(\mathbb{N}\)-graded ring generated by homogeneous elements of integral degree in \(R^+\). For an ideal \(I\) of \(R\) generated by a homogeneous system of parameters, the tight closure of \(I\) is equal to \(IR^{+ gr} \cap R\). This is obtained as a consequence of an analogous result for excellent local domains due to the author [Invent. Math. 115, No. 1, 41-60 (1994; Zbl 0820.13007)].
scientific article; zbMATH DE number 783936