An application of the algebra of differentials of infinite rank

Author:
William C. Brown

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

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Abstract | References | Similar Articles | Additional Information

Abstract: Let *k* denote an arbitrary field and let *R* be an affine local domain over *k*. Let 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 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 is a free

emphR-algebra.

**[1]**W. C. Brown,*The algebra of differentials of infinite rank*(to appear). MR**0314814 (47:3364)****[2]**W. C. Brown and W. E. Kuan,*Ideals and higher derivations in commutative rings*, Canad. J. Math. (to appear). MR**0294319 (45:3388)****[3]**N. Heerema,*Higher derivations and automorphisms of complete local rings*, Bull. Amer. Math. Soc.**76**(1970), 1212-1225, MR**42**#1818. MR**0266916 (42:1818)****[4]**Y. Nakai,*On the theory of differentials in commutative rings*, J. Math. Soc. Japan**13**(1961), 63-84. MR**23**#A2437. MR**0125131 (23:A2437)****[5]**-,*Higher order derivations*. I, Osaka J. Math.**7**(1970), 1-27. MR**41**#8404. MR**0263804 (41:8404)**

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DOI:
https://doi.org/10.1090/S0002-9939-1972-0300999-0

Keywords:
*k*-higher derivations,
algebra of *k*-higher differentials,
separating residue class field,
separating representatives

Article copyright:
© Copyright 1972
American Mathematical Society