The main technical theorem is an extension of known devissage theorems in algebraic \(K\)-theory: let \(F \colon \mathcal{C} \to \mathcal{D}\) be an exact functor of stable \(\infty\)-categories such that the image of \(F\) generates \(\mathcal{D}\) under finite colimits and retracts, and assume moreover that \(\mathcal{C}\) carries a bounded t-structure \((\mathcal{C}_{\geq 0}, \mathcal{C}_{\leq 0})\) such that \(F\) is fully faithful on the heart \(\mathcal{C}^\heartsuit\). The authors show that \(\mathcal{D}\) then also carries a bounded t-structure, and \(F\) restricts to a functor \(\mathcal{C}^\heartsuit \to \mathcal{D}^\heartsuit\) satisfying the assumptions of Quillen’s devissage theorem. Combining this with Barwick’s theorem of the heart, it follows that \(F\) induces an equivalence on connective algebraic \(K\)-theory. By a vanishing result of Antieau-Gepner-Heller, the induced map \(K_{-1}(F)\) is also an equivalence since both domain and codomain are trivial.
As a first application, the authors show that, if \(R\) is a coconnective ring spectrum such that \(\pi_0(R)\) is left regular coherent and the \(-1\)-truncation \(\tau_{\leq -1}(R)\) has Tor-amplitude in \([-\infty,-1]\) as a right \(\pi_0(R)\)-module, then the induced map \(K(\pi_0(R)) \to K(R)\) is an equivalence.
Using work of Land and Tamme, they also formulate a criterion when connective algebraic \(K\)-theory preserves pushouts of idempotent complete stable \(\infty\)-categories, and conclude that \(K\)-theory preserves pushouts of discrete rings \(B \sqcup_A C\) along right faithfully flat maps over a left regular coherent ring \(A\).
Further applications include \(\mathbb{A}^1\)-invariance of the algebraic \(K\)-theory of stable \(\infty\)-categories with bounded t-structure and the existence of bounded t-structures on categories of representations.
The article contains many examples illustrating both applications and limitations of the main results.