Relative algebraic K-theory and algebraic cyclic homology

Abstract

Since its introduction over 40 years ago algebraic K-theory, which provides powerful invariants, still remains hard to compute. The subject of this work is the construction of an isomorphism between relative algebraic K-groups and relative algebraic cyclic homology in low dimensions, for certain nilpotent ideals. This isomorphism generalizes the Theorem of Goodwillie concerning rational algebras and provides a more accessible alternative to topological cyclic homology for the computation of algebraic K-groups. Following roughly the strategy of Goodwillie, the proof is structured into several parts of varying interdependencies. First, we construct a natural isomorphism between group homology and Lie ring homology of certain associated groups and Lie rings. This represents an integral generalization of a Theorem of Pickel concerning nilpotent groups and also provides a strategy for an integral version of the Theorem of Lazard concerning p-valued groups, which both considered homology with rational coefficients. The theory provides a bridge in form of a natural logarithm map from the homology of the multiplicative to that of the additive K-theory. Second, we prove that the low-dimensional homotopy groups of an infinite loop space can be identified with the primitive part of its homology by using an improved version of the Hurewicz map. This represents a variant of a Theorem of Beilinson linking both objects up to isogeny. We apply this to the infinite loop space of relative K-theory. Similarly by using an additive analogue we compute the primitive part of the homology of the Lie algebra homology of matrices as cyclic homology. This can be considered as an integral generalization of the Theorem of Loday, Quillen and Tsygan. Combining the single steps we are constructing the desired isomorphism between K-theory and cyclic homology and also compare it with the negative Chern character. Alongside the proofs we provide a comprehensive collection of required abstract tools of simplicial homotopy theory. As an application of the main theorem we compute the lower relative K-groups of truncated polynomial rings over a subring of the rationals. This shows that our Theorem can be used to obtain new results in the computation of K-groups.