Abstract
Let A be a regular local ring and let R be a polynomial extension of A in a finite number of variables. Several attempts have been made to study the structure of projective modules over R and to find the minimal numbers of generators of ideals of R; to wit, [Z, V.4; 3; 61. The methods employed in these references depend very much on the structure of A; the cases dealt with are formal power series rings over fields and local rings at smooth points of algebraic varieties. Cohen’s structure theorem tells us precisely what regular local rings are formal power series rings. We present here the following characterization for a regular local ring to be a locality over a field :
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