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The K-groups of \(\lambda\)-rings. (English) Zbl 0626.18008
It is well-known that Dennis-Stein symbols satisfy \((*)\quad <a,b><a,c>=<a,b+c-abc>,\) a nonlinear relation, and that the universal group D(R,I) is isomorphic to \(K_ 2(R,I)\) for I contained in the Jacobson radical of R. Suppose one replaces the nonlinear relation (*) in D(R,I) by the corresponding linear relation, i.e. \([a,b][a,c]=[a,b+c].\) Call the resulting group \(K_{2,L}(R,I)\). It is shown that in certain \(\lambda\)-rings R, there is a continuous map \(L: D^{top}(R,I)\to K^{top}_{2,L}(R,I)\) such that \(L(<a,b>)=[a,b]+ higher\) order terms where, for a functor F, \(F^{top}=\lim_{\leftarrow}F(R/J^ n,(I+J^ n)/J^ n)\) for some ideal J. In the process of proving the existence of L, logarithm and exponential maps are defined. An introduction to \(\lambda\)-rings is also included.
Reviewer: S.C.Geller

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