The moduli stack and motivic Hall algebra for the bounded derived category

We give an alternate formulation of pseudo-coherence over an arbitrary derived stack X. The full subcategory of pseudo-coherent objects forms a stable sub-infinity-category of the derived category associated to X. Using relative Tor-amplitude we define a derived stack classifying pseudo-coherent objects. For reasonable base schemes, this classifies the bounded derived category. In the case that X is a projective derived scheme flat over the base, we show the moduli is locally geometric and locally of almost finite type. Using this result, we prove the existence of a derived motivic Hall algebra associated to X.

💡 Research Summary

The paper develops a new framework for studying pseudo‑coherent objects on arbitrary derived stacks and uses this to construct a derived motivic Hall algebra. It begins by redefining pseudo‑coherence in the language of ∞‑categories: instead of relying on classical notions that work only for ordinary schemes, the authors describe pseudo‑coherent complexes on a derived stack X via stable ∞‑categorical data, namely the existence of coCartesian morphisms and coCartesian push‑outs that encode the homotopical gluing of complexes. They prove that the full sub‑∞‑category of pseudo‑coherent objects is stable, closed under shifts and cones, and therefore forms a triangulated subcategory of the derived ∞‑category D(X).

Next, the authors introduce relative Tor‑amplitude. For a morphism X→S, a complex E has Tor‑amplitude in