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 $D^b({\mathcal{C}}^{♡})\to \mathcal{C}$ induces an equivalence of spectra $KH({\mathcal{C}}^{♡})\stackrel{\sim }{\to }KH(\mathcal{C}).$ 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 $n\ge 1$ the realization functor induces isomorphisms on ${\mathrm{Ext}}^{\le n}$ between the objects of ${\mathcal{C}}^{♡},$ then the map $K_j({\mathcal{C}}^{♡})\to K_j(\mathcal{C})$ is an isomorphism for $j\ge -n-1,$ 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 $K_{-3}.$

93 pages; v2: minor changes, typos fixed, references added