On the derived category of quasi-coherent sheaves on an Adams geometric stack

TL;DR

This paper demonstrates that the derived category of quasi-coherent sheaves on an Adams geometric stack forms a stable homotopy category and explores its relation to comodules over a Hopf algebroid.

Contribution

It establishes the stable homotopy structure of the derived category of quasi-coherent sheaves on Adams stacks and links it to comodules over associated Hopf algebroids.

Findings

Derived category satisfies stable homotopy axioms

Connection between sheaf categories and Hopf algebroid comodules

Framework for applying homotopy-theoretic methods to algebraic stacks

Abstract

Let $\mathbf{X}$ be an Adams geometric stack. We show that $D(A_{qc}(\mathbf{X}))$, its derived category of quasi-coherent sheaves, satisfies the axioms of a stable homotopy category defined by Hovey, Palmieri and Strickland. Moreover we show how this structure relates to the derived category of comodules over a Hopf algebroid that determines $\mathbf{X}$.