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Relative algebraic K-theory and cyclic homology. (English) Zbl 0627.18004
The paper establishes the rational equivalence of relative K-theory and relative cyclic homology for simplicial rings. More precisely: Given a homomorphism $$f: R\to S$$ of simplicial rings such that $$f_*: \pi_ 0R\to \pi_ 0S$$ is surjective with nilpotent kernel, then $$K_ n(f)\otimes {\mathbb{Q}}\cong HC_{n-1}(f)\otimes {\mathbb{Q}}$$. For 2-connected maps this result implies the rational equivalence of Waldhausen’s relative K-theory of spaces with the relative $$S^ 1$$-equivariant Borel homology of the free loop space functor modulo a dimension shift.
A cyclic object in the sense of Connes of abelian groups gives rise to a 2-periodic double complex whose homology is known as the “periodic homology” $$PH_*$$. The homology of its “positive half” is Connes’ cyclic homology $$HC_*$$, the homology $$HC^-_*$$ of its “negative half” has become an object of interest since the publication of the present paper. Both “halves” meet in the Hochschild complex inducing Hochschild homology $$H_*$$. By construction, there is an exact sequence $...\to \quad HC_{*-1}\quad \to^{\beta}\quad HC^-_*\quad \to \quad HP_*\quad \to \quad HC_{*-2}\quad \to \quad...$ and an obvious map p: HC$${}^-_*\to H_*$$. The important property of $$HC^- _*$$ is that the Dennis trace map $$K_*\to H_*$$ has a natural lift $$\alpha$$ : $$K_*\to HC^-_*$$. The proof of the theorem is now established in two steps: Applying the main result of an earlier paper the author shows that $$\beta$$ is a rational equivalence under the given assumptions. The main body of the paper then consists of the proof that $$\alpha$$ is a rational equivalence (the relative versions are obtained by using the algebraic mapping cone). This again is done in two steps: First the problem is reduced to the case where $$f: R\to S$$ is a split extension of discrete rings with kernel a square-zero ideal I such that I is a free S-bimodule. Then the rational K-theory $$K_*(f)\otimes {\mathbb{Q}}$$ and the rational cyclic homology $$HC_{*-1}(f)\otimes {\mathbb{Q}}$$ are explicitly computed in this special case. Here the K-theory calculation is the harder part: It relies on the computation of the homology of tensor products of adjoint representations and derived from those on computations of $$H_*(Gl(S)$$; $$\bigwedge^ rM(I\otimes {\mathbb{Q}}))$$, where M( ) stands for matrices.
The paper is very carefully written and a pleasure to read. It starts with a recollection of the algebraic K-theory and cyclic homology of simplicial rings including a section on simplicial tools, then the lift of $$\alpha$$ of the Dennis trace map is constructed, the reduction to the special case follows, and the paper ends with the involved explicit calculations.
Reviewer: R.Vogt

MSC:
 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects) 18G30 Simplicial sets; simplicial objects in a category (MSC2010) 55N99 Homology and cohomology theories in algebraic topology
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