## $$F$$-rational rings have rational singularities.(English)Zbl 0910.13004

Here is the author’s abstract: “It is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudo-rational. A key point in the proof is a characterization of $$F$$-rational local rings as those Cohen-Macaulay local rings $$(R,m)$$ in which the local cohomology module $$H^d_m(R)$$ (where $$d$$ is the dimension of $$R)$$ have no submodules stable under the natural action of the Frobenius map. An analog for finitely generated algebras over a field of characteristic zero is developed, which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over $$\mathbb{C}$$, without reference to a desingularization”.
Reviewer: M.Roitman (Haifa)

### MSC:

 13B22 Integral closure of commutative rings and ideals 13A35 Characteristic $$p$$ methods (Frobenius endomorphism) and reduction to characteristic $$p$$; tight closure 14B05 Singularities in algebraic geometry
Full Text: