A characterization of rational singularities
Let \((R, {\mathfrak m})\) be a local ring which is essentially of finite type over a field \(k\) of characteristic zero. One says that \(R\) is of \(F\)-rational type if the top local cohomology of ``almost all'' reductions modulo \(p\) do not have proper Frobenius stable submodules. \textit{Karen E. Smith} [Am. J. Math. 119, No. 1, 159-180 (1997; Zbl 0910.13004)] proved that if \(R\) has \(F\)-rational type, then \(R\) is rational. She moreover conjectured the converse. In this paper the authors give a proof of this conjecture. This result was also obtained earlier but independently by \textit{N. Hara} [cf. Am. J. Math. 120, No. 5, 981-996 (1998)]. In fact the authors prove a more general result: They prove that any local ring \(R\) as above is of so-called resolved \(F\)-rational type.
scientific article; zbMATH DE number 1143926