A note on linear stacks

TL;DR

This paper explores graded functions on linear stacks in derived geometry, demonstrating how to recover quasi-coherent sheaves from graded linear stacks and generalizing the results in null characteristic.

Contribution

It introduces a method to recover sheaves from graded linear stacks and extends the understanding of graded functions in derived geometry, especially in null characteristic.

Findings

Recovered quasi-coherent sheaves from graded linear stacks.

Generalized the description of graded functions on linear stacks in null characteristic.

Provided conditions under which the recovery is possible.

Abstract

In this brief note, we investigate graded functions of linear stacks in derived geometry. In particular, we show that under mild assumptions, we can recover a quasi-coherent sheaf on a derived stack from the data of the $\mathbb{G}_m$-graded linear stack associated to it. Then we generalize this result in null caracteristic by describing graded functions on linear stacks.