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\(F\)-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
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