Regular local rings essentially of finite type over fields of prime characteristic

Let R be a local ring essentially of finite type over a field k of characteristic p>0. 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 pn-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 pn-basis. The latter criterion yield the sharpened result of a part of Orbanzʹs theorem [U. Orbanz, Höhere Derivationen und Regularität, J. Reine. Angew. Math. 262/263 (1973) 194–204, 4.2].