A Jacobian Criterion for Artin $v$-stacks

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.

💡 Research Summary

This paper establishes a generalized Jacobian criterion for Artin v-stacks, extending the foundational work of Fargues-Scholze on spaces of sections over the Fargues-Fontaine curve.

The core result, Theorem 1.8, states the following: Given a perfectoid space S, a linear algebraic group H over Q_p, and a scheme Z smooth and quasi-projective over the schematic Fargues-Fontaine curve X_S equipped with an H-action, the moduli space M_{