The categories ${\mathcal T}^c$ and ${\mathcal T}^b_c$ determine each other
Amnon Neeman
TL;DR
The article develops an intrinsic, enhancement-free framework to recover and relate key triangulated categories via metric completions. By introducing good metrics on essentially small triangulated categories and constructing Cauchy sequences, Neeman defines the triangulated category ${\mathfrak S}({\mathscr S})$ from any ${\mathscr S}$, enabling an intrinsic passage between ${\mathscr T}^c$ (e.g., ${\mathbf D}^{perf}$) and ${\mathscr T}^b_c$ (e.g., ${\mathbf D}^b_{coh}$) without relying on extra enhancements. The main results show that, under suitable approximability and coherence hypotheses, ${\mathfrak S}({\mathscr T}^c)$ recovers ${\mathscr T}^b_c$ and conversely, and that ${\mathscr T}^b_c$ determines ${\mathscr T}^c$ intrinsically. These findings yield a robust, metric-driven equivalence between fundamental derived-category subcategories, with explicit consequences for noetherian schemes such as ${\mathbf D}^{perf}(X)$ and ${\mathbf D}^b_{coh}(X)$, thereby unifying prominent invariants in birational geometry and homological algebra with minimal reliance on enhancements.
Abstract
Given an essentially small triangulated category it is possible to give a metric on it, to complete it with respect to the metric, and to look at the subcategory of objects in the completion which are compactly supported with respect to the metric. The main theorem says that this procedure produces a new triangulated category. And then we give examples: for example we learn that it is possible, for suitable choices of metrics, to produce the categories $D^b(R-\text{mod})$ and $K^b(R-\text{proj})$ out of each other.