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

Abstract

For every nuclear $\mathbb Z_\ell$-algebra $Λ$ and every small v-stack $X$ we construct an $\infty$-category $\mathcal D_{\mathrm{nuc}}(X,Λ)$ of nuclear $Λ$-modules on $X$. We then construct a full 6-functor formalism for these sheaves, generalizing the étale 6-functor formalism for $Λ= \mathbb F_\ell$. Prominent choices for $Λ$ 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.