×

On the K-theory of local fields. (English) Zbl 0548.12009

The author completes his proof of the Lichtenbaum conjecture on the algebraic K-theory of an algebraically closed field. In [Invent. Math. 73, 241-245 (1983; Zbl 0514.18008)] he showed that it suffices to take one field of each characteristic. Here, by considering local rings, he shows that one needs only one field altogether. He computes the K-theory of the complex numbers (also of the real numbers), and so makes the proof independent of computations for fields of finite characteristic.
Reviewer: R.Steiner

MSC:

11S70 \(K\)-theory of local fields
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
13D15 Grothendieck groups, \(K\)-theory and commutative rings

Citations:

Zbl 0514.18008
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bass, H., Algebraic \(K\)-theory (1968), Benjamin: Benjamin New York · Zbl 0174.30302
[2] Bourbaki, N., Algébre Commutative (1965), Hermann: Hermann Paris · Zbl 0141.03501
[3] Charney, R., A note on excision in algebraic \(K\)-theory, (Lecture Notes in Math., 1046 (1984), Springer: Springer Berlin), 47-54
[4] O. Gabber, On the \(K\); O. Gabber, On the \(K\)
[5] Grayson, D., Higher \(K\)-theory II, (Lecture Notes in Math., 551 (1976), Springer: Springer Berlin), 218-240
[6] Gromoll, D.; Klingenberg, W.; Meyer, W., Riemannsche Geometrie im Grossen (1968), Springer: Springer Berlin · Zbl 0155.30701
[7] Husemoller, D., Fiber Bundles (1966), McGraw-Hill: McGraw-Hill New York
[8] van der Kellen, W., Homology stability for general linear groups, Inventiones Math., 60, 3, 269-295 (1980) · Zbl 0415.18012
[9] Lang, S., Introduction to Algebraic Geometry, (Interscience Tracts, 5 (1958), Wiley: Wiley New York) · Zbl 0095.15301
[10] May, J. P., Simplicial Objects in Algebraic Topology (1967), Van Nostrand: Van Nostrand New York · Zbl 0165.26004
[11] Milnor, J., On the homology of Lie groups made discrete, Comment. Math. Helvetici, 58, 72-85 (1983) · Zbl 0528.20033
[12] Neisendorfer, J., Primary homotopy theory, Memoirs AMS, 25, 232 (1980) · Zbl 0446.55002
[13] Quillen, D., On the cohomology and \(K\)-theory of the general linear group over a finite field, Ann. of Math., 96, 3, 552-585 (1972) · Zbl 0249.18022
[14] Quillen, D., Higher algebraic \(K\)-theory, (Lecture Notes in Math., 341 (1972), Springer: Springer Berlin), 85-147 · Zbl 0292.18004
[15] Raynaud, M., Anneaux locaux henséliens, (Lecture Notes in Math., 169 (1970), Springer: Springer Berlin) · Zbl 0203.05102
[16] Spanier, E. H., Algebraic Topology (1966), McGraw-Hill: McGraw-Hill New York · Zbl 0145.43303
[17] Suslin, A., Stability in algebraic \(K\)-theory, (Lecture Notes in Math., 966 (1982), Springer: Springer Berlin), 344-356
[18] Suslin, A., On the \(K\)-theory of algebraically closed fields, Inventiones Math., 73, 241-245 (1983) · Zbl 0514.18008
[19] A. Suslin, Homology of \(GL_nK\); A. Suslin, Homology of \(GL_nK\)
[20] Weibel, C. A., Algebraic \(K\)-theory and the Adams \(e\)-invariant, (Lecture Notes in Math., 1046 (1984), Springer: Springer Berlin), 451-464
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.