On Regular Local Rings1
نویسنده
چکیده
This paper generalizes slightly a result of Kunz [l ] and Nakai [2]. If R>S are commutative rings with identity we introduce a module D*(R/S) defined as the quotient of the module D(R/S) oí S differentials of A by the submodule consisting of elements which are mapped to zero by every homomorphism of D(R/S) having values in a finitely generated A module. The characteristic exponent of a field is defined to be 1 if the field is of characteristic zero and to be p if the characteristic of the field is p. The result is then: If A is a local ring containing a field k of characteristic exponent p such that D*(R/kp) is finitely generated, then the following conditions are equivalent: (i) A is a regular local ring, (ii) D*(R/kp) is free and if x is an element of the completion of A such that xp = 0 then x = 0. (iii) D*(R/kp) is free and if x is an element of the form ring of A such that xp = 0 then x = 0. We remark that in characteristic zero regularity (under the finiteness condition) is equivalent to the freedom of D*(R/k) and in any case if the local ring is of the form A q where A is a finitely generated integral domain and g is a prime the second part of (ii) is automatically satisfied. (See Zariski and Samuel [4, p. 314].)
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