Categorical K\"unneth formulas for analytic stacks

TL;DR

This paper extends categorical K"unneth formulas to analytic stacks, enabling new results in Tannakian reconstruction and a $p$-adic Drinfeld's lemma, advancing the understanding of derived and analytic stacks.

Contribution

It generalizes categorical K"unneth formulas to analytic stacks and applies these to Tannakian reconstruction and $p$-adic Langlands conjectures.

Findings

Established categorical K"unneth formulas for analytic stacks

Derived a Tannakian reconstruction theorem for analytic stacks

Proved a $p$-adic Drinfeld's lemma for specific stacks

Abstract

In arXiv:0805.0157v5, the authors define a class of derived stacks, called "perfect stacks" and show that for this class the categories of quasi-coherent sheaves satisfy a categorical K\"unneth formula. Motivated to extend their results to the theory of analytic stacks as developed by Clausen-Scholze, we investigate categorical K\"unneth formulas for general $6$-functor formalisms. As applications we show a general Tannakian reconstruction result for analytic stacks and, following recent work of Ansch\"utz, Le Bras and Mann arXiv:2412.20968v1, show a $p$-adic version of Drinfeld's lemma for certain stacks that appear conjecturally in a categorical $p$-adic Langlands program.