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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

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