×

The indecomposable \(K_ 3\) of fields. (English) Zbl 0705.19001

The author extends Hilbert’s Theorem 90 for \(K_ 2\) of fields - due to Merkur’ev-Suslin - to the relative \(K_ 2\) of semi-local principal ideal rings containing a field. This yields a computation of the torsion and cotorsion of the indecomposable \(K_ 3\) of a field. The main results of the paper have independently been obtained by A. S. Merkur’ev and A. A. Suslin [Izv. Akad. Nauk SSSR, Ser. Mat. 54, No.3, 522-545 (1990)] with a slightly different method.
Reviewer: M.Kolster

MSC:

19D50 Computations of higher \(K\)-theory of rings
13D15 Grothendieck groups, \(K\)-theory and commutative rings
19F99 \(K\)-theory in number theory
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] H. BASS and J. TATE , On the Milnor Ring of a Field , in Higher Algebraic K-Theory I, LNM, # 341, Springer, 1973 . MR 56 #449 · Zbl 0299.12013
[2] A. A. BEILINSON , Letter to C. Soulé 11/1/82.
[3] B. BIRCH , K2 of Global Fields (Proc. Symp. Pure Math., 20, 1969 , pp. 87-95). MR 48 #11052 | Zbl 0218.12010 · Zbl 0218.12010
[4] A. BOREL , Stable Real Cohomology of Arithmetic Groups , (Ann. Scient. Éc. Norm. Sup., T. 7, 1974 , pp. 235-272). Numdam | MR 52 #8338 | Zbl 0316.57026 · Zbl 0316.57026
[5] J. COATES and S. LICHTENBAUM , On l-Adic Zeta Functions (Ann. of Math. 98, 1973 , pp. 498-550). MR 48 #8445 | Zbl 0279.12005 · Zbl 0279.12005
[6] K. R. COOMBES , Relative Algebraic K-Theory , an appendix to Relative K-Theory and Vector Bundles on Certain Singular Varieties, by K. R. COOMBES and V. SRINIVAS, Invent. Math., Vol. 70, 1982 , pp. 13-25. MR 84e:14016 | Zbl 0507.14015 · Zbl 0507.14015
[7] R. K. DENNIS and M. R. STEIN , ”K2 of discrete valuation rings” , Adv. Math 18 # 2. MR 55 #10544 | Zbl 0318.13017 · Zbl 0318.13017
[8] W. G. DWYER and E. M. FRIEDLANDER , Algebraic and Étale K-Theory (Trans. Am. Math. Soc., Vol. 292, No. 1, 1985 , pp. 247-280). MR 87h:18013 | Zbl 0581.14012 · Zbl 0581.14012
[9] J. HURRELBRINK and M. KOLSTER , On the 2-Primary Part of the Birch-Tate Conjecture (Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory II, pp. 519-528, A.M.S. Contemporary Math. Series, Vol. 55, 1986 ). MR 88a:11114 | Zbl 0601.12016 · Zbl 0601.12016
[10] H. GILLET , Riemann-Roch Theorems for Higher Algebraic K-Theory (Adv. in Math., Vol. 40, 1981 , pp. 203-289). MR 83m:14013 | Zbl 0478.14010 · Zbl 0478.14010
[11] H. GILLET and D. GRAYSON , The Loop Space of the Q-Construction (Ill. J. Math., Vol. 31, No. 4, 1987 , pp. 574-597). MR 89h:18012 | Zbl 0628.55011 · Zbl 0628.55011
[12] U. JANNSEN , Continuous Etale Cohomology , preprint. · Zbl 0649.14011
[13] F. KEUNE , The Relativization of K2 (J. of Alg., Vol. 54, 1978 , pp. 159-177). MR 80a:18013 | Zbl 0403.18009 · Zbl 0403.18009
[14] R. LEE and R. SZCZARBA , The Group K3(\Bbb Z) is Cyclic of Order 48 (Ann. of Math., Vol. 104, 1976 , pp. 31-60). MR 56 #1309 | Zbl 0341.18008 · Zbl 0341.18008
[15] S. LICHTENBAUM , On the Values of Zeta and L-Functions : I (Ann. of Math., Vol. 96, 1972 , pp. 338-360). MR 50 #12975 | Zbl 0251.12002 · Zbl 0251.12002
[16] S. LICHTENBAUM , Values of Zeta Functions, Etale Cohomology, and Algebraic K-Theory , in Algebraic K-Theory II (LNM, # 342, 1973 , pp. 489-501). MR 53 #10765 | Zbl 0284.12005 · Zbl 0284.12005
[17] S. LICHTENBAUM , Values of Zeta-Functions at Non-Negative Integers , in Number Theory (LNM, # 1068, 1984 , pp. 127-138). MR 756089 | Zbl 0591.14014 · Zbl 0591.14014
[18] S. LICHTENBAUM , The Construction of Weight Two Arithmetic Cohomology (Inv. Math., Vol. 88, 1987 , pp. 183-215). MR 88d:14011 | Zbl 0615.14004 · Zbl 0615.14004
[19] J. L. LODAY , Cohomologie et groupe de Steinberg relatifs (J. Algebra, Vol. 54, 1978 , pp. 178-202). MR 80b:18013 | Zbl 0391.20040 · Zbl 0391.20040
[20] B. MAZUR and A. WILES , Class Fields of Abelian Extensions of \Bbb Q (Invent. Math., Vol. 76, 1984 , pp. 179-330). MR 85m:11069 | Zbl 0545.12005 · Zbl 0545.12005
[21] A. S. MERKURJEV , K2 of Fields and the Brauer Group , in Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory, Part II (AMS Contempory Math. Series, # 55, 1986 , pp. 529-546). MR 88a:11120 | Zbl 0608.12028 · Zbl 0608.12028
[22] A. S. MERKURJEV and A. A. SUSLIN , K-Cohomology of Severi-Brauer Varieties and the Norm Residue Homomorphism (Math. of the U.S.S.R.-Izvestiya (trans.), Vol. 21, 1983 , pp. 307-340). Zbl 0525.18008 · Zbl 0525.18008
[23] A. S. MERKURJEV and A. A. SUSLIN , On the K3 of a Field , LOMI, preprint, M-2-07, 1987 .
[24] D. QUILLEN , Higher Algebraic K-Theory I , in Algebraic K-Theory I (LNM, # 341, Springer, 1972 , pp. 85-147). MR 49 #2895 | Zbl 0292.18004 · Zbl 0292.18004
[25] D. QUILLEN , On the Cohomology and K-Theory of the General Linear Group Over a Finite Field (Ann. Math., Vol. 96, 1972 , pp. 552-586). MR 47 #3565 | Zbl 0249.18022 · Zbl 0249.18022
[26] D. QUILLEN , Finite Generation of the Groups Ki of Rings of Algebraic Integers , in Algebraic K-Theory I (LNM, # 341, Springer, 1972 , pp. 179-198). MR 50 #2305 | Zbl 0355.18018 · Zbl 0355.18018
[27] C. C. SHERMAN , K-Cohomology of Regular Schemes (Comm. Algebra, 7, No. 10, 1979 , pp. 999-1029). MR 80h:14004 | Zbl 0425.18010 · Zbl 0425.18010
[28] J. P. SERRE , Cohomologie des Groupes Descrets (Ann. of Math. Studies, Vol. 70, pp. 77-169, Princeton Univ. Press, 1971 ). MR 52 #5876 | Zbl 0235.22020 · Zbl 0235.22020
[29] C. SOULÉ , K-théorie des anneaux d’entiers de corps de nombres et cohomologie étale (Inv. Math., Vol. 55, 1979 , pp. 251-295). MR 81i:12016 | Zbl 0437.12008 · Zbl 0437.12008
[30] R. SWAN , K-Theory of Quadric Hypersurfaces (Ann. of Math., Vol. 122, No. 1, 1985 ). MR 87g:14006 | Zbl 0601.14009 · Zbl 0601.14009
[31] A. A. SUSLIN , Torsion in K2 of Fields , K-theory 1, No. 1, 1987 , pp. 5-30. MR 89a:11123 | Zbl 0635.12015 · Zbl 0635.12015
[32] J. TATE , Relations Between K2 and Galois Cohomology (Invent. Math., Vol. 36, 1976 , pp. 257-274). MR 55 #2847 | Zbl 0359.12011 · Zbl 0359.12011
[33] J. TATE , Symbols in Arithmetic (Proc. of the International Congress at Nice, 1970 ).
[34] F. WALDHAUSEN , Algebraic K-Theory of Generalized Free Products I and II (Ann. of Math., Vol. 108, 1978 , pp. 135-256). MR 58 #16845a | Zbl 0397.18012 · Zbl 0397.18012
[35] C. WEIBEL , A Survey of Products in Algebraic K-Theory , (LNM, # 854, 1981 , pp. 494-517). MR 82k:18011 | Zbl 0482.18004 · Zbl 0482.18004
[36] C. WEIBEL , Mennicke-Type Symbols for Relative K2 (LNM, # 1046, 1984 ). MR 86c:13015 | Zbl 0528.13013 · Zbl 0528.13013
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.