A survey of test ideals (Q2895440)

In this paper the authors give a survey of test ideals and their applications. Originally introduced by Hochster and Huneke in their theory of tight closure, the test ideal associated with each ring \(R\) of prime characteristic \(p\) is a measure of the singularities of \(R\). The paper introduces the test ideal from the point of view of theory of Frobenius splittings. Let \(R\) be an integral domain essentially of finite type over a perfect field \(k\) of prime positive characteristic \(p\). For \(e>0\) and a non-zero \(R\)-linear map \(\phi: R^{1/p^e} \to R\), the test ideal \(\tau(R, \phi)\) is the unique smallest non-zero ideal \(J \subseteq R\) such that \(\phi(J^{1/p^e}) \subseteq J\). The test ideal \(\tau (R)\) is then defined to be the unique smallest non-zero ideal \(J \subseteq R\) such that \(\phi(J^{1/p^e}) \subseteq J\) for all \(e > 0\) and all \(\phi \in \text{Hom}_R(R^{1/p^e}, R)\). In the literature, the test ideal \(\tau(R)\) defined as above is often referred to as the \textit{big (or non-finitistic) test ideal} of \(R\). The authors also adopt the convention of referring to the classical \textit{test ideal} introduced by Hochster and Huneke as the \textit{finitistic test ideal} \(\tau_{\text{fg}}(R)\).NEWLINENEWLINEIn addition to the exposition of test ideals, the paper contains sections on the connections with algebraic geometry (explaining the relation between the test ideal and the multiplier ideal), tight closure and applications of test ideals, test ideals for ``pairs'', and other measures of singularities in characteristic \(p\) (\(F\)-rationality and Hilbert-Kunz multiplicity). The text also contains a large number of exercises. As noted by the authors, the intended audience of the paper consists of readers working in characteristic \(p> 0\) commutative algebra (tight closure) who wish to understand connections between test ideals and algebraic geometry, readers working with Frobenius splittings who wish to understand the methods from tight closure, as well as readers with a background in complex analytic and algebraic geometry who want to familiarize themselves with the characteristic \(p>0\) methods.NEWLINENEWLINEFor the entire collection see [Zbl 1237.13006].