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

Abstract For every nuclear $${\mathbb {Z}}_\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>ℓ</mml:mi> </mml:msub> </mml:math> -algebra $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> and every small v-stack X on perfectoid spaces, we construct an $$\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∞</mml:mi> </mml:math> -category $$\mathcal {D}_{\textrm{nuc}}(X,\Lambda )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>D</mml:mi> <mml:mtext>nuc</mml:mtext> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of nuclear (i.e., “ind-Banach”) $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> -modules on X . We then construct a full 6-functor formalism for these sheaves, generalizing the étale 6-functor formalism for $$\Lambda = \mathbb {F}_\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Λ</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>ℓ</mml:mi> </mml:msub> </mml:mrow> </mml:math> . Prominent choices for $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> are $${\mathbb {Z}}_\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>ℓ</mml:mi> </mml:msub> </mml:math> , $$\mathbb {Q}_\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>ℓ</mml:mi> </mml:msub> </mml:math> and $$\overline{\mathbb {Q}_\ell }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>ℓ</mml:mi> </mml:msub> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> . 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.