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F-rational Rings Have Rational Singularities
| Content Provider | Semantic Scholar |
|---|---|
| Author | Smith, Karen E. |
| Copyright Year | 1997 |
| 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 pseu-dorational. 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 nitely generated algebras over a eld of characteristic zero is developed, which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over C , without reference to a desingularization. With the development of the theory of tight closure by M. Hochster and C. Huneke HH1], a natural question arose. What information does this powerful new tool provide about the structure of the singularities of an algebraic variety? The main theorem of this paper is the following: Theorem 3.1. If an excellent local ring has the property that all ideals generated by a system of parameters are tightly closed, then the ring is pseudorational. Pseudorationality (Deenition 1.8) is a desingularization-free analog of the notion of rational singularities which makes sense for any scheme. Theorem 3.1 has a characteristic zero version (Theorem 4.3) which can be used to test for rational singularities of a complex algebraic variety. Recently, A. Conca and J. Herzog have used Theorem 3.1 to prove that an interesting class of varieties generalizing Schubert varieties, called the ladder determinantal varieties, have rational singularities CH]. Theorem 3.1 is not unexpected. Striking similarities had suggested a connection between rings with rational singularities and rings in which all (or certain) ideals are tightly closed. Both are preserved upon passing to direct summands Bo], HH1]. Both are natural settings for the \Briann con-Skoda theorems," relating powers of ideals to integral closures their powers in a uniform way LT], HH1]. In the graded case, both force strong restrictions on the degrees of non-vanishing elements in local cohomology modules (the so-called a-invariant must be negative; see FW]). Prior to this paper, several special cases of Theorem 3.1 had been proved before, |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.math.lsa.umich.edu/~kesmith/finalration.ps |
| Language | English |
| Access Restriction | Open |
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| Content Type | Text |
| Resource Type | Article |
