We show that the relative algebraic K-theory functor fully determines the absolute cyclic homology over any field k of characteristic 0. More precisely, we prove that the tangent of K-theory, in terms of (abelian) deformation problems over k, is cyclic homology. As a consequence, any structure on K-theory is inherited by cyclic homology.
We also show that the Loday-Quillen-Tsygan generalized trace comes as the tangent morphism of the canonical map $BGL_\infty \to K$ mapping a vector bundle to… Expand

(for a precise definition, see ?1 below). By replacing everywhere K*( ) by K*( ) ? Q, one obtains the corresponding notion in rational algebraic K-theory. The above definition has an obvious… Expand

The algebraic K–theory of spaces is a variant, invented by F. Waldhausen in the late 1970’s, of the standard algebraic K–theory of rings. Until that time, applications of algebraic K–theory to… Expand

In this paper we use the theory of formal moduli problems developed by Lurie in order to study the space of formal deformations of a $k$ -linear $\infty$ -category for a field $k$ . Our main result… Expand

When A is a unital ring, the absolute Chern character is a group homomorphism Ch* : K*(A) ? HN*(A), going from algebraic K-theory to negative cyclic homology. There is also a relative version,… Expand

There is a Chern character from K-theory to negative cyclic homology. We show that it preserves the decomposition coming from Adams operations, at least in characteristic zero.

AbstractTHIS new “Higher Algebra” will be examined with great interest by all teachers and serious students of mathematics. A book of this type is certainly needed at the present time, and the one… Expand