K-theory and the bridge from motives to noncommutative motives

Abstract

In this work we present a new approach to the theory of noncommutative motives and use it to explain the different flavors of algebraic K-theory of schemes and dg-categories. The work is divided into three main parts. In the first part we use the techniques of higher algebra developed in [63] to provide a universal characterization for the symmetric monoidal (∞,1)-category underlying the motivic stable A1- homotopy theory of Morel–Voevodsky [107,67]. More precisely, given a symmetric monoidal model category V together with an object X∈V, we characterize the underlying symmetric monoidal (∞,1)-category of the symmetric monoidal model category SpΣ(V,X) introduced by Hovey in [43], by means of a universal property amongst symmetric monoidal (∞,1)- categories. This characterization trivializes the problem of finding motivic monoidal realizations.In the second part we introduce a new approach to the theory of noncommutative motives by constructing a stable motivic homotopy theory for the noncommutative spaces of Kontsevich [56,55,54]. The key ingredient is a notion of Nisnevich topology in the noncommutative setting, compatible with the classical notion. This compatibility, together with the universal property proved in the first part, ensures the existence of a canonical monoidal map from the stable motivic theory of Morel–Voevodsky towards these new noncommutative motives that allow us to compare the two theories.In the last part of this paper we explain how this bridge can be used to explain the various flavors of algebraic K-theory of dg-categories. More precisely, we prove that the non-connective K-theory of dg-categories introduced by Schlichting [82] is the (non-commutative) Nisnevich sheafification of connective algebraic K-theory. Then we prove that its further (non-commutative) A1-localization is a tensor unit in our noncommutative motives. As a corollary we obtain a precise proof for an original conjecture of Kontsevich claiming that K-theory gives the correct mapping spaces in noncommutative motives. Our major application is the discovery of a canonical factorization of our motivic bridge through the (∞,1)-category of modules over the commutative algebra object representing homotopy invariant algebraic K-theory of schemes. The results in [77] imply that this bridge is fully faithful over a field k with resolutions of singularities, so that, at the motivic level, no information (below K-theory) is lost by passing to the noncommutative world.