A survey of test ideals

TL;DR

This survey reviews the development, properties, and diverse applications of test ideals, highlighting their significance in tight closure theory, Frobenius splittings, and complex analytic geometry.

Contribution

It provides a comprehensive overview of test ideals, summarizing recent advances and broadening their context beyond initial algebraic frameworks.

Findings

Test ideals are central to tight closure theory.

They have applications in complex analytic geometry.

The survey summarizes recent developments in the field.

Abstract

Test ideals were first introduced by Mel Hochster and Craig Huneke in their celebrated theory of tight closure, and since their invention have been closely tied to the theory of Frobenius splittings. Subsequently, test ideals have also found application far beyond their original scope to questions arising in complex analytic geometry. In this paper we give a contemporary survey of test ideals and their wide-ranging applications.