An introduction to tight closure (Q2717187)

The paper under review is an expanded version of the author's lecture at the Conference in Commutative Algebra and Algebraic Geometry, Messina Italy, June 1999. Its purpose is to give a brief introduction to the subject of tight closure, aimed at commutative algebraists who have not studied this topic before. Recall that for an ideal \(I\) generated by \(y_1,\dots ,y_r\) in the Noetherian ring \(R\) of prime characteristic \(p>0\), the tight closure \(I^*\) of \(I\) consists of all \(z\in R\) for which there exists \(c\in R\) not in any minimal prime of \(R\), such that \(c^{p^e}\in (y_1^{p^e},\dots ,y_r^{p^e})\) for all \(e\gg 0\); \(I\) is tightly closed if \(I=I^*\) and \(R\) is called weakly F-regular if all ideals are tightly closed. NEWLINENEWLINENEWLINEThe first part of the paper is focused on the definition and basic properties: A regular ring is (weakly) F-regular, elements mapped to \(I\) after integral extension are in \(I^*\), colon capturing, relationship to integral closure and persistence of tight closure. Some algebraic applications are included. NEWLINENEWLINENEWLINEThe second part is focused on applications to algebraic geometry: Fujita's freeness conjecture, Kodaira's vanishing theorem and singularities.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00042].