Relative algebraic K-theory and cyclic homology.

*(English)*Zbl 0627.18004The 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.

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 |