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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
##### Keywords:
relative $$K_ 2$$; formal group; K-groups; $$\lambda$$-rings
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##### References:
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