The 6-Functor Formalism for $\mathbb Z_\ell$- and $\mathbb Q_\ell$-Sheaves on Diamonds

TL;DR

This paper develops a comprehensive 6-functor formalism for nuclear sheaves over v-stacks with $bZ_l$- and $bQ_l$-coefficients, extending previous étale formalism and enabling new nuclear representation theories.

Contribution

It constructs a 6-functor formalism for nuclear $bZ_l$- and $bQ_l$-sheaves on v-stacks, a significant generalization of existing étale formalisms.

Findings

Established a 6-functor formalism for nuclear sheaves on v-stacks.

Created a theory of nuclear representations on classifying stacks.

Extended the formalism to $bQ_l$ and $ar{bQ}_l$ coefficients.

Abstract

For every nuclear $\mathbb Z_\ell$-algebra $\Lambda$ and every small v-stack $X$ we construct an $\infty$-category $\mathcal D_{\mathrm{nuc}}(X,\Lambda)$ of nuclear $\Lambda$-modules on $X$. We then construct a full 6-functor formalism for these sheaves, generalizing the \'etale 6-functor formalism for $\Lambda = \mathbb F_\ell$. Prominent choices for $\Lambda$ are $\mathbb Z_\ell$, $\mathbb Q_\ell$ and $\bar{\mathbb Q_\ell}$ and especially in the latter two cases, no satisfying 6-functor formalism has been found before. Applied to classifying stacks we obtain a theory of nuclear representations, i.e. continuous representations on filtered colimits of Banach spaces.