A 6-functor formalism for $${\mathbb {Z}}_\ell$$- and $$\mathbb {Q}_\ell$$-sheaves on diamonds

Source abstract

Abstract For every nuclear $${\mathbb {Z}}_\ell$$ Z ℓ -algebra $$\Lambda$$ Λ and every small v-stack X on perfectoid spaces, we construct an $$\infty$$ ∞ -category $$\mathcal {D}_{\textrm{nuc}}(X,\Lambda )$$ D nuc ( X , Λ ) of nuclear (i.e., “ind-Banach”) $$\Lambda$$ Λ -modules on X . We then construct a full 6-functor formalism for these sheaves, generalizing the étale 6-functor formalism for $$\Lambda = \mathbb {F}_\ell$$ Λ = F ℓ . Prominent choices for $$\Lambda$$ Λ are $${\mathbb {Z}}_\ell$$ Z ℓ , $$\mathbb {Q}_\ell$$ Q ℓ and $$\overline{\mathbb {Q}_\ell }$$ Q ℓ ¯ . We also provide and study an abstract notion of ULA sheaves in this setting, whose definition and basic properties can be carried over to any 6-functor formalism. Applied to classifying stacks, we obtain a robust theory of nuclear representations, i.e., continuous representations on filtered colimits of Banach spaces.