Geometric derived Hall algebra

TL;DR

This paper provides a geometric formulation of To"en's derived Hall algebra using Grothendieck's six operations on derived categories of sheaves, and offers an $alculus-based explanation of derived stacks.

Contribution

It introduces a geometric approach to derived Hall algebras and extends the theory with an $alculus perspective on derived stacks.

Findings

Constructs Grothendieck's six operations for derived categories on stacks

Provides an $alculus-based explanation of derived stacks

Reformulates To"en's derived Hall algebra geometrically

Abstract

We give a geometric formulation of To\"en's derived Hall algebra by constructing Grothendieck's six operations for the derived category of lisse-\'etale constructible sheaves on the derived stacks of complexes. Our formulation is based on an variant of Laszlo and Olsson's theory of derived categories and six operations for algebraic stacks. We also give an $\infty$-theoretic explanation of the theory of derived stacks, which was originally constructed by To\"en and Vezzosi in terms of model theoretical language.