The Formalism of ℓ-adic Sheaves

Gaitsgory, Dennis; Lurie, Jacob

doi:10.23943/princeton/9780691182148.003.0002

Weil's Conjecture for Function Fields: Volume I (AMS-199)

Dennis Gaitsgory and Jacob Lurie

https://doi.org/10.23943/princeton/9780691182148.001.0001

Published online:

19 September 2019

Published in print:

19 February 2019

Online ISBN:

9780691184432

Print ISBN:

9780691182148

One Introduction
- Notes
- Notes
- 2.1 Higher Category Theory 2.1 Higher Category Theory
  - 2.1.1 Motivation: Deficiencies of the Derived Category 2.1.1 Motivation: Deficiencies of the Derived Category
  - 2.1.2 The Differential Graded Nerve 2.1.2 The Differential Graded Nerve
  - 2.1.3 The Weak Kan Condition 2.1.3 The Weak Kan Condition
  - 2.1.4 The Language of Higher Category Theory 2.1.4 The Language of Higher Category Theory
  - 2.1.5 Example: Limits and Colimits 2.1.5 Example: Limits and Colimits
  - 2.1.6 Stable ∞-Categories 2.1.6 Stable ∞-Categories
- 2.2 Étale Sheaves 2.2 Étale Sheaves
  - 2.2.1 Sheaves of Λ-Modules 2.2.1 Sheaves of Λ-Modules
  - 2.2.2 The t-Structure on Shv(X; Λ) 2.2.2 The t-Structure on Shv(X; Λ)
  - 2.2.3 Functoriality 2.2.3 Functoriality
  - 2.2.4 Compact Generation of Shv(X; Λ) 2.2.4 Compact Generation of Shv(X; Λ)
  - 2.2.5 The Exceptional Inverse Image 2.2.5 The Exceptional Inverse Image
  - 2.2.6 Constructible Sheaves 2.2.6 Constructible Sheaves
  - 2.2.7 Sheaves of Vector Spaces 2.2.7 Sheaves of Vector Spaces
  - 2.2.8 Extension of Scalars 2.2.8 Extension of Scalars
  - 2.2.9 Stability Properties of Constructible Sheaves 2.2.9 Stability Properties of Constructible Sheaves
- 2.3 ℓ-Adic Sheaves 2.3 ℓ-Adic Sheaves
  - 2.3.1 ℓ-Completeness 2.3.1 ℓ-Completeness
  - 2.3.2 Constructible ℓ-adic Sheaves 2.3.2 Constructible ℓ-adic Sheaves
  - 2.3.3 Direct and Inverse Images 2.3.3 Direct and Inverse Images
  - 2.3.4 General ℓ-adic Sheaves 2.3.4 General ℓ-adic Sheaves
  - 2.3.5 Cohomological Descent 2.3.5 Cohomological Descent
  - 2.3.6 The t-Structure on l-Constructible Sheaves 2.3.6 The t-Structure on l-Constructible Sheaves
  - 2.3.7 Exactness of Direct and Inverse Images 2.3.7 Exactness of Direct and Inverse Images
  - 2.3.8 The Heart of ${Shv}_{l}^{c} (X)$ 2.3.8 The Heart of ${Shv}_{l}^{c} (X)$
  - 2.3.9 The t-Structure on Shvl(X) 2.3.9 The t-Structure on Shvl(X)
- 2.4 Base Change Theorems 2.4 Base Change Theorems
  - 2.4.1 Digression: The Beck-Chevalley Property 2.4.1 Digression: The Beck-Chevalley Property
  - 2.4.2 Smooth and Proper Base Change 2.4.2 Smooth and Proper Base Change
  - 2.4.3 Direct Images and Extension by Zero 2.4.3 Direct Images and Extension by Zero
  - 2.4.4 Base Change for Exceptional Inverse Images 2.4.4 Base Change for Exceptional Inverse Images
- Notes
- Notes
Three E∞-Structures on l-Adic Cohomology
- Notes
- Notes
Four Computing the Trace of Frobenius
- Notes
- Notes
Five The Trace Formula for BunG(X)
- Notes
- Notes
- Bibliography
  - Notes
  - Notes

Chapter

Two The Formalism of ℓ-adic Sheaves

Get access

https://doi.org/10.23943/princeton/9780691182148.003.0002

Pages

62–128

Published:

February 2019

Abstract

The ℓ-adic product formula discussed in Chapter 4 will need to make use of analogous structures, which are simply not visible at the level of the triangulated category D_ℓ(X). This chapter attempts to remedy the situation by introducing a mathematical object Shv_ℓ (X), which refines the triangulated category D_ℓ (X). This object is not itself a category but instead is an example of an ∞-category, which is referred to as the ∞-category of ℓ-adic sheaves on X. The triangulated category D_ℓ (X) can be identified with the homotopy category of Shv_ℓ (X); in particular, the objects of D_ℓ (X) and Shv_ℓ (X) are the same. However, there is a large difference between commutative algebra objects of D_ℓ (X) and commutative algebra objects of the ∞-category Shv_ℓ (X). We can achieve (b') by viewing the complex B as a commutative algebra of the latter sort.

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Contents

Two The Formalism of ℓ-adic Sheaves

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