A Jacobian Criterion for Artin $v$-stacks

TL;DR

This paper extends the Jacobian criterion to Artin v-stacks over the Fargues-Fontaine curve, enabling the analysis of their smoothness and dimension, with applications to the geometric Langlands program.

Contribution

It generalizes Fargues-Scholze's Jacobian criterion to Artin stacks, facilitating the study of their smoothness and dimensions in the Fargues-Fontaine setting.

Findings

Moduli stacks in the Fargues-Scholze program are cohomologically smooth Artin v-stacks.

The paper computes the $ ext{ell}$-dimensions of these stacks.

The generalized criterion applies to stacks quotient of smooth varieties by algebraic groups.

Abstract

We prove a generalization of the Jacobian criterion of Fargues-Scholze for spaces of sections of a smooth quasi-projective variety over the Fargues-Fontaine curve. Namely, we show how to use their criterion to deduce an analogue for spaces of sections of a smooth Artin stack over the (schematic) Fargues-Fontaine curve obtained by taking the stack quotient of a smooth quasi-projective variety by the action of a linear algebraic group. As an application, we show various moduli stacks appearing in the Fargues-Scholze geometric Langlands program are cohomologically smooth Artin $v$-stacks and compute their $\ell$-dimensions.