Differential basis, $p$-basis, and smoothness in characteristic $p>0$
HTML articles powered by AMS MathViewer
- by Andrzej Tyc PDF
- Proc. Amer. Math. Soc. 103 (1988), 389-394 Request permission
Abstract:
It is shown that every differential basis of a Noetherian ring $R$ of prime characteristic $p$ over an arbitrary subring $k$ is a $p$-basis of $R$ over $k$. Moreover, if $k$ is a field, then $R$ is smooth over $k$, provided $R$ has a differential basis over $k$ and the ring $R{ \otimes _k}{k^{{p^{ - 1}}}}$ is reduced.References
- Michel André, Homologie des algèbres commutatives, Die Grundlehren der mathematischen Wissenschaften, Band 206, Springer-Verlag, Berlin-New York, 1974 (French). MR 0352220, DOI 10.1007/978-3-642-51449-4
- Robert Berger and Ernst Kunz, Über die Strucktur der Differentialmoduln von diskreten Bewertungsringen, Math. Z. 77 (1961), 314–338 (German). MR 130891, DOI 10.1007/BF01180183
- John Fogarty, Kähler differentials and Hilbert’s fourteenth problem for finite groups, Amer. J. Math. 102 (1980), no. 6, 1159–1175. MR 595009, DOI 10.2307/2374183
- Tetsuzo Kimura and Hiroshi Niitsuma, Differential basis and $p$-basis of a regular local ring, Proc. Amer. Math. Soc. 92 (1984), no. 3, 335–338. MR 759648, DOI 10.1090/S0002-9939-1984-0759648-8
- Tetsuzo Kimura and Hiroshi Niitsuma, Regular local ring of characteristic $p$ and $p$-basis, J. Math. Soc. Japan 32 (1980), no. 2, 363–371. MR 567425, DOI 10.2969/jmsj/03220363
- Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
- William C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Springer-Verlag, New York-Berlin, 1979. MR 547117, DOI 10.1007/978-1-4612-6217-6
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 103 (1988), 389-394
- MSC: Primary 13B99
- DOI: https://doi.org/10.1090/S0002-9939-1988-0943051-5
- MathSciNet review: 943051