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A Nomizu-van Est theorem in Ekedahl's derived $\ell$-adic setting

Olivier Taïbi

TL;DR

The paper develops a derived $\ell$-adic Nomizu–van Est framework by embedding continuous $\ell$-adic representations into Ekedahl’s triangulated categories and constructing a finite-type, canonical polynomial-cochain model that replaces the intractable infinite-dimensional polynomial complexes. It provides explicit Hochschild–Serre formulas for semi-direct products, proves polynomial-cochain and filtrations results for torsion and torsion-free pro-$\ell$ nilpotent groups, and then extends the theory to characteristic zero via unipotent algebraic groups and their Lie algebras. The central achievement is a derived isomorphism $\nu_{\mathbf N,\mathbf H,K}(V^\bullet)$ linking $R\Gamma(N_K,F(V^\bullet))$ with $F(R\Gamma_{\mathrm{Lie}}(\mathfrak n,V^\bullet))$ in $D^+(H_K,E)$, compatible with restriction and base change, enabling application to Pink’s $\ell$-adic Shimura-variety framework and related automorphic/derived contexts. The results provide a robust, triangulated-category analogue of Pink–Nomizu–van Est in the $\ell$-adic setting, with potential impact on the analysis of $\ell$-adic local systems on Shimura varieties and their minimal compactifications.

Abstract

A theorem of Nomizu and van Est computes the cohomology of a compact nilmanifold, or equivalently the group cohomology of an arithmetic subgroup of a unipotent linear algebraic group over $\mathbb{Q}$. We prove a similar result for the cohomology of a compact open subgroup of a unipotent linear algebraic group over $\mathbb{Q}_{\ell}$ with coefficients in a complex of continuous $\ell$-adic representations. We work with the triangulated categories defined by Ekedahl which play the role of ``derived categories of continuous $\ell$-adic representations''. This is motivated by Pink's formula computing the derived direct image of an $\ell$-adic local system on a Shimura variety in its minimal compactification, and its application to automorphic perverse sheaves on Shimura varieties. The key technical result is the computation of the cohomology with coefficients in a unipotent representation with torsion coefficients by an explicit complex of polynomial cochains which is of finite type.

A Nomizu-van Est theorem in Ekedahl's derived $\ell$-adic setting

TL;DR

The paper develops a derived -adic Nomizu–van Est framework by embedding continuous -adic representations into Ekedahl’s triangulated categories and constructing a finite-type, canonical polynomial-cochain model that replaces the intractable infinite-dimensional polynomial complexes. It provides explicit Hochschild–Serre formulas for semi-direct products, proves polynomial-cochain and filtrations results for torsion and torsion-free pro- nilpotent groups, and then extends the theory to characteristic zero via unipotent algebraic groups and their Lie algebras. The central achievement is a derived isomorphism linking with in , compatible with restriction and base change, enabling application to Pink’s -adic Shimura-variety framework and related automorphic/derived contexts. The results provide a robust, triangulated-category analogue of Pink–Nomizu–van Est in the -adic setting, with potential impact on the analysis of -adic local systems on Shimura varieties and their minimal compactifications.

Abstract

A theorem of Nomizu and van Est computes the cohomology of a compact nilmanifold, or equivalently the group cohomology of an arithmetic subgroup of a unipotent linear algebraic group over . We prove a similar result for the cohomology of a compact open subgroup of a unipotent linear algebraic group over with coefficients in a complex of continuous -adic representations. We work with the triangulated categories defined by Ekedahl which play the role of ``derived categories of continuous -adic representations''. This is motivated by Pink's formula computing the derived direct image of an -adic local system on a Shimura variety in its minimal compactification, and its application to automorphic perverse sheaves on Shimura varieties. The key technical result is the computation of the cohomology with coefficients in a unipotent representation with torsion coefficients by an explicit complex of polynomial cochains which is of finite type.
Paper Structure (11 sections, 28 theorems, 161 equations)

This paper contains 11 sections, 28 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. "Derived categories" of $\ell$-adic representations
  3. Group cohomology as a derived functor
  4. Group cohomology using continuous cochains
  5. Explicit Hochschild-Serre for semi-direct products
  6. Polynomial cochains for $\mathbb{Z}_{\ell}$
  7. Polynomial cochains for torsion-free pro-$\ell$ nilpotent groups
  8. Characteristic zero coefficients
  9. Lie algebra cohomology
  10. Algebraic group cohomology and Lie algebra cohomology
  11. A Nomizu-van Est theorem

Key Result

Proposition 2.2

Let $\operatorname{Rep}_{\mathrm{fg}, \mathrm{cont}}(K, \mathcal{O}_E)$ be the abelian category of finitely generated $\mathcal{O}_E$-modules with a continuous linear action of $K$. The functor induces an exact functor $F: D^b(\operatorname{Rep}_{\mathrm{fg}, \mathrm{cont}}(K, \mathcal{O}_E)) \to D^b(K, \mathcal{O}_E)$.

Theorems & Definitions (63)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • Lemma 3.1
  • ...and 53 more