2026/03/09 by Alexander I. Efimov · 1 voice · 2 citations
Mathematics · #math.KT #math.AG #math.AT #math.CT #math.NT
arxiv published 2026/03/09 · arxiv updated 2026/03/16
In this paper we prove the theorem of the heart for Weibel's homotopy K-theory KH. Namely, if C is a small stable ∞-category with a bounded t-structure, then the realization functor Db(C\heartsuit)→ C induces an equivalence of spectra KH(C\heartsuit)\xrightarrow∼KH(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 C is a small stable category with a bounded t-structure, such that for some n≥ 1 the realization functor induces isomorphisms on Ext≤ n between the objects of C\heartsuit, then the map Kj(C\heartsuit)→ Kj(C) is an isomorphism for j≥ -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.