Products, Homotopy Limits and Applications

TL;DR

This paper investigates the behavior of derived functors of infinite products and homotopy limits in the context of quasi-coherent sheaves on Deligne-Mumford stacks, revealing convergence properties and category comparisons.

Contribution

It introduces new insights into the convergence of homotopy limits and compares derived categories of quasi-coherent sheaves with related triangulated categories.

Findings

Derived functors of product vanish after finite stages in QC(X)

Convergence properties of certain homotopy limits established

Comparison results between derived categories of QC(X) and other categories

Abstract

In this note, we discuss the derived functors of infinite products and homotopy limits. $QC(X)$, the category of quasi-coherent sheaves on a Deligne-Mumford stack $X$, usually has the property that the derived functors of product vanish after a finite stage. We use this fact to study the convergence of certain homotopy limits and apply it compare the derived category of $QC(X)$ with certain other closely related triangulated categories.