Theorem of the heart for Weibel's homotopy $K$-theory

Abstract

In this paper we prove the theorem of the heart for Weibel's homotopy $K$-theory $KH.$ Namely, if $\mathcal{C}$ is a small stable $\infty$-category with a bounded $t$-structure, then the realization functor $D^b(\mathcal{C}^{\heartsuit})\to \mathcal{C}$ induces an equivalence of spectra $KH(\mathcal{C}^{\heartsuit})\xrightarrow{\sim}KH(\mathcal{C}).$ In a certain sense this result is dual to the Dundas-Goodwillie-McCarthy theorem. We deduce the dévissage theorem for $KH$ of abelian categories, also on the level of spectra (in all degrees). More generally, we prove these results for dualizable categories with nice $t$-structures and for the so-called coherently assembled abelian categories. The proof is heavily based on another new result, which is a much stronger version of Barwick's theorem of the heart. Its special case states the following: if $\mathcal{C}$ is a small stable category with a bounded $t$-structure, such that for some $n\geq 1$ the realization functor induces isomorphisms on $\operatorname{Ext}^{\leq n}$ between the objects of $\mathcal{C}^{\heartsuit},$ then the map $K_j(\mathcal{C}^{\heartsuit})\to K_j(\mathcal{C})$ is an isomorphism for $j\geq -n-1,$ and a monomorphism for $j = -n-2.$ Moreover, we prove that these estimates are sharp, even for dg categories over a field. In particular the naive $K$-theoretic theorem of the heart fails for $K_{-3}.$