The 6-Functor Formalism for ${\mathbb{Z}}_{\ell }$- and ${\mathbb{Q}}_{\ell }$-Sheaves on Diamonds

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 étale 6-functor formalism for $\Lambda ={\mathbb{F}}_{\ell }$. Prominent choices for $\Lambda$ are ${\mathbb{Z}}_{\ell }$, ${\mathbb{Q}}_{\ell }$ and $\overline{{\mathbb{Q}}_{\ell }}$ 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.

65 pages, comments welcome!