Classification of localizing subcategories along t-structures

Abstract

We study the interplay between localizing subcategories in a stable $\infty$-category $\mathcal{C}$ with $t$-structure $({\mathcal{C}}_{\ge 0},{\mathcal{C}}_{\le 0})$, the prestable $\infty$-category ${\mathcal{C}}_{\ge 0}$ and the abelian category ${\mathcal{C}}^{♡}$. We prove that weak localizing subcategories of ${\mathcal{C}}^{♡}$ are in bijection with the localizing subcategories of $\mathcal{C}$ where object-containment can be checked on the heart. This generalizes similar known correspondences for noetherian rings and bounded $t$-structures. We also prove that this restricts to a bijection between localizing subcategories of ${\mathcal{C}}^{♡}$, and localizing subcategories of $\mathcal{C}$ that are kernels of $t$-exact functors -- lifting Lurie's correspondence between localizing subcategories in ${\mathcal{C}}_{\ge 0}$ and ${\mathcal{C}}^{♡}$ to the stable category $\mathcal{C}$.

26 pages. Comments very welcome!