Abstract:
Reciprocity laws and Mennicke symbols for regular rings of arbitrary dimension are introduced and studied. The universal Mennicke symbol for the polynomial ring $k[X_1,\dots,X_n]$ relative to the principal ideal $(X_1^2-X_1)\cdots(X_n^2-X_n)$ is computed. The results obtained in this paper are applied to the problem of determining the precise value of the stable rank of polynomial rings and affine algebras over fields.
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