Search

Abstract: Let R be a local ring essentially of finite type over a field k of characteristic . In the paper [M. Furuya, H. Niitsuma, Regularity criterion of Noetherian local rings of prime characteristic, J. Algebra 247 (2002) 219–230], we constructed some regularity criterion for such a local ring R in terms of the higher differential algebra and the -basis. In this paper, we introduce the concept of a reduced index of a Noetherian ring and we show the sharpened result of the above criterion and further we also give a geometric regularity criterion in terms of the higher differential algebra and the -basis. The latter criterion yield the sharpened result of a part of Orbanz''s theorem [U. Orbanz, Höhere Derivationen und Regularität, J. Reine. Angew. Math. 262/263 (1973) 194–204, 4.2]. [Copyright &y& Elsevier]

Abstract: Let G be a connected reductive algebraic group over an algebraically closed field of characteristic . Assume that p is good for G. Pommerening''s theorem [K. Pommerening, Über die unipotenten Klassen reduktiver Gruppen, J. Algebra 49 (1977) 525–536; K. Pommerening, Über die unipotenten Klassen reduktiver Gruppen, II, J. Algebra 65 (1980) 373–398] asserts that any distinguished nilpotent element in the Lie algebra of G is a Richardson element for a distinguished parabolic subgroup of G. This theorem implies the Bala–Carter theorem in good characteristic. In this paper we give a short proof of Pommerening''s theorem, which is a further simplification of Premet''s first uniform proof [A. Premet, Nilpotent orbits in good characteristic and the Kempf–Rousseau theory, J. Algebra 260 (2003) 338–366]. We also simplify Premet''s proof of the existence theorem for good transverse slices to the nilpotent -orbits in . [Copyright &y& Elsevier]

Abstract: This paper pursues Takeuchi''s Hopf algebraic approach [M. Takeuchi, A Hopf algebraic approach to the Picard–Vessiot theory, J. Algebra 122 (1989) 481–509] to the Picard–Vessiot (PV) theory for differential equations, to involve the PV extensions of difference equations. Differential fields and total difference rings in the standard PV theory are unified here by artinian simple (AS) module algebras over a cocommutative, pointed smooth Hopf algebra. [Copyright &y& Elsevier]