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K-theory, \(\lambda\)-rings, and formal groups. (English) Zbl 0646.18004

Let F be a one-dimensional formal group over \({\mathbb{Z}}\). Let R be a commutative ring and let I be a nilpotent ideal. Then a group \(K_{2,F}(R,I)\) is defined using a presentation similar to the presentation of Maazen and Stienstra for \(K_ 2(R,I)\) [H. Maazen and J. Stienstra, J. Pure Appl. Algebra 10, 271-294 (1977; Zbl 0393.18013)]. If F is the multiplicative formal group then we obtain the Maazen-Stienstra presentation. If F is the additive formal group one obtains \(K_{2,L}(R,I)=\Omega_{R,I}/dI\) (the latter being the cyclic homology group of (R,I)). The main result is that there is a homomorphism \(L_ F: K_{2,F}(R,I)^{top}\to K_{2,L}(R,I)^{top}\) (the topology being relative to a suitable ideal J; the topology is introduced to ensure convergence of the formal power series needed in the proofs). This result generalizes that obtained by the author in “The K-groups of \(\lambda\)-rings. I” [Compos. Math. 61, 295-328 (1987; Zbl 0626.18008)], where only the multiplicative formal group was used. The homomorphism \(L_ F\) is defined using F-twisted versions of the \(\lambda\)-operations used in the author’s previous paper. The group \(K_{2,L}(R,I)\) is more readily computed than \(K_{2,F}(R,I)\), so the theorem enables one to prove that elements in \(K_{2,F}(R,I)\) are non-zero.
Reviewer: L.G.Roberts

MSC:

18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
13D15 Grothendieck groups, \(K\)-theory and commutative rings
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References:

[1] F. Clauwens , The K-groups of \lambda -rings, Part I, Construction of the logarithmic invariant , Comp. Math. 61 (1987) 295-328. · Zbl 0626.18008
[2] B. Ditters , Fonctions lexoides et produits euleriens , C.R. Acad. Sci. Paris 276 (1973) 531-534. · Zbl 0275.10017
[3] M. Hazewinkel , Formal groups and Applications , Academic Press, New York (1978). · Zbl 0454.14020
[4] F. Keune , The relativisation of K2 , J. of Alg. 54 (1978) 159-177. · Zbl 0403.18009 · doi:10.1016/0021-8693(78)90024-8
[5] J.-L. Loday and D. Quillen , Cyclic homology and the Lie algebra homology of matrices , Comm. Math. Helv. 59 (1984) 565-591. · Zbl 0565.17006 · doi:10.1007/BF02566367
[6] H. Maazen and J. Stienstra , A presentation for K2 of split radical pairs , J. of Pure and Appl. Algebra 10 (1977) 271-294 · Zbl 0393.18013 · doi:10.1016/0022-4049(77)90007-X
[7] J. Stienstra , Cartier-Dieudonné theory for Chow groups , J. Reine Angew. Math. 355 (1985) 1-66. · Zbl 0545.14013 · doi:10.1515/crll.1985.355.1
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