A 6-functor formalism for {Z}_ - and Q_ -sheaves on diamonds

Abstract For every nuclear {Z}_ Z ℓ -algebra Λ and every small v-stack X on perfectoid spaces, we construct an ∞ -category D₍ₔ₂ (X, ) D nuc (X, Λ) of nuclear (i. e. , “ind-Banach”) Λ -modules on X. We then construct a full 6-functor formalism for these sheaves, generalizing the étale 6-functor formalism for = F_ Λ = F ℓ. Prominent choices for Λ are {Z}_ Z ℓ, Q_ Q ℓ and Q_ 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.