Noncommutative rigidity

Abstract

Abstract. In this article we prove that the numerical Grothendieck group of every smooth proper dg category is invariant under primary field extensions, and also that the mod-n algebraic K-theory of every dg category is invariant under extensions of separably closed fields. As a byproduct, we obtain an extension of Suslin's rigidity theorem, as well as of Yagunov-Østvaer's equivariant rigidity theorem, to singular varieties. Among other applications, we show that base-change along primary field extensions yields a faithfully flat morphism between noncommutative motivic Galois groups. Finally, along the way, we introduce the category of n-adic noncommutative mixed motives. IntroductionLet l/k be a field extension and X an algebraic k-variety. On the one hand, it is (well-)known that when the field extension l/k is primary 1 and the algebraic k-variety X is smooth and proper, base-change induces an isomorphism between the Q-vector spaces of algebraic cycles up to numerical equivalence:On the other hand, when l/k is an extension of algebraically closed fields, a remarkable result of Suslin [12] asserts that, for every integer n ≥ 2 coprime to char(k), base-change induces an isomorphism in mod-n G-theory:Among other applications, the isomorphism (1.2) (with X = Spec(k)) enabled Suslin to describe the torsion of the algebraic K-theory of every algebraically closed field of positive characteristic, thus solving a longstanding conjecture of QuillenLichtenbaum; consult Suslin's ICM address [13] for further applications. The main goal of this article is to establish far-reaching noncommutative generalizations of the above rigidity isomorphisms (1.1)-(1.2); consult §2 for applications. Statement of results.A differential graded (=dg) category A, over a base field k, is a category enriched over complexes of k-vector spaces; see §3.1. Every (dg) kalgebra A gives naturally rise to a dg category with a single object. Another source of examples is proved by algebraic varieties (or more generally by algebraic stacks) since the category of perfect complexes perf(X), resp. the bounded derived category of coherent O X -modules D b (coh(X)), of every algebraic k-variety X admits a canonical dg enhancement perf dg (X), resp. D b dg (coh(X)); see [5, §4.6]. Following