An application of the algebra of differentials of infinite rank
HTML articles powered by AMS MathViewer
- by William C. Brown PDF
- Proc. Amer. Math. Soc. 35 (1972), 9-15 Request permission
Abstract:
Let k denote an arbitrary field and let R be an affine local domain over k. Let $({\Omega _k}(R),\delta _k^R)$ be the universal algebra of k-higher differentials over R. Let K be the quotient field of R and L the residue class field of R. If K is a separable extension of k and L is a separable algebraic extension of k, then it is shown that R is a regular local ring if and only if ${\Omega _k}(R)$ is a free R-algebra. If both K and L are separable extensions of k and R has a separating residue class field, then R is a regular local ring if and only if ${\Omega _k}(R)$ is a free emphR-algebra.References
- W. C. Brown, The algebra of differentials of infinite rank, Canadian J. Math. 25 (1973), 141–155. MR 314814, DOI 10.4153/CJM-1973-013-4
- William C. Brown and Wei-eihn Kuan, Ideals and higher derivations in commutative rings, Canadian J. Math. 24 (1972), 400–415. MR 294319, DOI 10.4153/CJM-1972-033-1
- Nickolas Heerema, Higher derivations and automorphisms of complete local rings, Bull. Amer. Math. Soc. 76 (1970), 1212–1225. MR 266916, DOI 10.1090/S0002-9904-1970-12609-X
- Yoshikazu Nakai, On the theory of differentials in commutative rings, J. Math. Soc. Japan 13 (1961), 63–84. MR 125131, DOI 10.2969/jmsj/01310063
- Yoshikazu Nakai, High order derivations. I, Osaka Math. J. 7 (1970), 1–27. MR 263804
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 35 (1972), 9-15
- MSC: Primary 13B10
- DOI: https://doi.org/10.1090/S0002-9939-1972-0300999-0
- MathSciNet review: 0300999