Geodesic
Reciprocity laws and the stable rank of polynomial rings
Izvestiya. Mathematics , Tome 15 (1980) no. 3, pp. 589-623.

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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. Bibliography: 19 titles. 
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A. A. Suslin. Reciprocity laws and the stable rank of polynomial rings. Izvestiya. Mathematics , Tome 15 (1980) no. 3, pp. 589-623. http://geodesic.mathdoc.fr/item/IM2_1980_15_3_a6/

[1] Bass X., Algebraicheskaya $K$-teoriya, Mir, M., 1973 | MR | Zbl

[2] Bass H., Tate J., “The Milnor ring of a global field”, Lect. Notes. Math., 342, 1973, 349–446 | MR | Zbl

[3] Burbaki N., Kommutativnaya algebra, Mir, M., 1971 | MR

[4] Vasershtein L. N., “Stabilnyi rang kolets i razmernost topologicheskikh prostranstv”, Funkts. analiz, 5:2 (1971), 17–27 | MR | Zbl

[5] Vasershtein L. N., Suslin A. A., “Problema Serra o proektivnykh modulyakh nad koltsami mnogochlenov i algebraicheskaya $K$-teoriya”, Izv. AN SSSR. Ser. matem., 40 (1976), 993–1054 | MR

[6] Grothendieck A., “Elements de geometrie algebrique”, Publ. Math. IHES, 20 (1964), 1–196

[7] Krusenmeyer M. I., “Fundamental groups, algebraic $K$-theory and a problem of Abhyankhar”, Invent. Math., 19:1 (1973), 15–27 | DOI | MR

[8] Keune F., “$(t^2-t)$-reciprocities on the affine line and the Matsumoto's theorem”, Invent. Math., 28:2 (1975), 185–192 | DOI | MR | Zbl

[9] Milnor Dzh., Vvedenie v algebraicheskuyu $K$-teoriyu, Mir, M., 1974 | MR | Zbl

[10] Milnor J., “Algebraic $K$-theory and quadratic forms”, Invent. Math., 9 (1970), 318–344 | DOI | MR | Zbl

[11] Serre J.-P., Corps locaux, Paris, 1962 | MR

[12] Cepp Zh.-P., “Lokalnaya algebra i teoriya kratnostei”, Matematika, 9:1 (1965), 54–126

[13] Suslin A. A., “O strukture spetsialnoi lineinoi gruppy nad koltsami mnogochlenov”, Izv. AN SSSR. Ser. matem., 41 (1977), 235–252 | MR | Zbl

[14] Shevalle K., Vvedenie v teoriyu algebraicheskikh funktsii ot odnoi peremennoi, Fizmatgiz, M., 1959

[15] Vasershtein L. N., “O stabilizatsii obschei lineinoi gruppy nad koltsom”, Matem. sb., 79(121):3 (1969), 405–424 | MR

[16] Vasershtein L. N., “O stabilizatsii dlya $K_2$-funktora Milnora”, Uspekhi matem. nauk, 30:1 (1975), 224 | MR

[17] Dennis R. K., “Surjective stability for the functor $K_2$”, Lect. Notes. Math., 353, 1973, 85–94 | MR | Zbl

[18] Suslin A. A., Tulenbaev M. S., “Teorema o stabilizatsii dlya $K_2$-funktora Milnora”, Zap. nauchn. sem. LOMI, 64, 1976, 131–152 | MR | Zbl

[19] Suslin A. A., “O stabilnom range kolets mnogochlenov i algebr nad polyami”, Tezisy soobsch. Vsesoyuznogo algebraich. simpoziuma, ch. 2, Gomel, 1975