Good filtrations and \(F\)-purity of invariant subrings. (English) Zbl 1222.13006

Summary: Let \(k\) be an algebraically closed field of positive characteristic, \(G\) a reductive group over \(k\), and \(V\) a finite dimensional \(G\)-module. Let \(B\) be a Borel subgroup of \(G\), and \(U\) its unipotent radical. We prove that if \(S=\text{Sym}\,V\) has a good filtration, then \(S^U\) is \(F\)-pure.


13A50 Actions of groups on commutative rings; invariant theory
13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
14L24 Geometric invariant theory
Full Text: DOI arXiv


[1] S. Donkin, Rational Representations of Algebraic Groups, Lecture Notes in Math., 1140 , Springer, 1985. · Zbl 0586.20017
[2] F. D. Grosshans, The invariants of unipotent radicals of parabolic subgroups, Invent. Math., 73 (1983), 1-9. · Zbl 0501.14007
[3] M. Hashimoto, Good filtrations of symmetric algebras and strong \(F\)-regularity of invariant subrings, Math. Z., 236 (2001), 605-623. · Zbl 1034.13007
[4] M. Hochster and C. Huneke, Tight closure and strong \(F\)-regularity, Mém. Soc. Math. France, 38 (1989), 119-133. · Zbl 0699.13003
[5] M. Hochster and C. Huneke, Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc., 3 (1990), 31-116. · Zbl 0701.13002
[6] M. Hochster and C. Huneke, \(F\)-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc., 346 (1994), 1-62. · Zbl 0844.13002
[7] M. Hochster and J. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Adv. Math., 13 (1974), 115-175. · Zbl 0289.14010
[8] M. Hochster and J. Roberts, The purity of the Frobenius and local cohomology, Adv. Math., 21 (1976), 117-172. · Zbl 0348.13007
[9] J. C. Jantzen, Representations of Algebraic Groups, 2nd ed., Mathematical Surveys and Monographs, 107 , Amer. Math. Soc., 2003. · Zbl 1034.20041
[10] V. B. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. of Math. (2), 122 (1985), 27-40. · Zbl 0601.14043
[11] J. D. Vélez, Openness of the \(F\)-rational locus and smooth base change, J. Algebra, 172 (1995), 425-453. · Zbl 0826.13009
[12] K.-i. Watanabe, Characterizations of singularities in characteristic \(0\) via Frobenius map, Commutative Algebra, Algebraic Geometry, and Computational Methods (Hanoi 1996), Springer, 1999, pp.,155-169. · Zbl 0959.13003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.