For every nuclear <InlineEquation ID="IEq7"> <EquationSource Format="TEX">${\mathbb {Z}}_\ell $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>ℓ</mi> </msub> </math></EquationSource> </InlineEquation>-algebra <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation> and every small v-stack X on perfectoid spaces, we construct an <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mi>∞</mi> </math></EquationSource> </InlineEquation>-category <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {D}_{\textrm{nuc}}(X,\Lambda )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">D</mi> <mtext>nuc</mtext> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of nuclear (i.e., “ind-Banach”) <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation>-modules on X. We then construct a full 6-functor formalism for these sheaves, generalizing the étale 6-functor formalism for <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\Lambda = \mathbb {F}_\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Λ</mi> <mo>=</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>ℓ</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. Prominent choices for <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation> are <InlineEquation ID="IEq14"> <EquationSource Format="TEX">${\mathbb {Z}}_\ell $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>ℓ</mi> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\mathbb {Q}_\ell $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Q</mi> <mi>ℓ</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\overline{\mathbb {Q}_\ell }$</EquationSource> <EquationSource Format="MATHML"><math> <mover> <msub> <mi mathvariant="double-struck">Q</mi> <mi>ℓ</mi> </msub> <mo>¯</mo> </mover> </math></EquationSource> </InlineEquation>. 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.

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

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2026年2月6日

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Lucas Mann

摘要

For every nuclear ${\mathbb {Z}}_\ell $ $Z_{ℓ}$ -algebra $\Lambda $ $Λ$ and every small v-stack X on perfectoid spaces, we construct an $\infty $ $\infty$ -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 $\overline{\mathbb {Q}_\ell }$ $\bar{Q_{ℓ}}$ . 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.

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

摘要