🔢Arithmetic Geometry Unit 11 – Étale cohomology

Foundations of Étale Cohomology

• Étale cohomology provides a powerful tool for studying algebraic varieties over arbitrary fields, extending the techniques of classical algebraic topology
• Built upon the notion of étale morphisms, which are a generalization of local isomorphisms in the context of schemes
• Allows for the development of a cohomology theory that captures arithmetic and geometric information simultaneously
• Overcomes limitations of classical cohomology theories (singular, de Rham) when working with varieties over fields of positive characteristic
• Étale cohomology groups are equipped with additional structures, such as Galois actions and Frobenius endomorphisms, which encode arithmetic data
  • These additional structures provide a bridge between algebraic geometry and number theory
• Serves as a foundation for the study of $\ell$-adic cohomology and the étale fundamental group, which have numerous applications in modern algebraic geometry and number theory

Étale Topology and Morphisms

• The étale topology is a Grothendieck topology on the category of schemes, which refines the Zariski topology
• Étale morphisms are a central notion in the étale topology, generalizing local isomorphisms
  • A morphism $f: X \to Y$ is étale if it is flat, unramified, and locally of finite presentation
• Étale morphisms satisfy important properties, such as being open, stable under base change, and composable
• The étale site of a scheme $X$, denoted $X_{\text{ét}}$, consists of étale morphisms $U \to X$ as objects and morphisms over $X$ as arrows
• Étale coverings play a crucial role in the étale topology, analogous to open coverings in classical topology
  • A family of étale morphisms $\{U_i \to X\}$ is an étale covering if $\coprod U_i \to X$ is surjective
• The étale topology allows for the development of a cohomology theory that is better suited for studying algebraic varieties over arbitrary fields compared to the Zariski topology

Sheaf Theory in the Étale Context

• Sheaves on the étale site of a scheme $X$ are central objects of study in étale cohomology
• An étale sheaf $\mathcal{F}$ on $X$ assigns an abelian group $\mathcal{F}(U)$ to each étale morphism $U \to X$, satisfying sheaf axioms with respect to the étale topology
• Étale sheaves form an abelian category, denoted $\text{Sh}(X_{\text{ét}})$, which is equipped with a natural tensor product and internal Hom
• Important examples of étale sheaves include the constant sheaf $\underline{\mathbb{Z}/n\mathbb{Z}}$, the sheaf of $\mathcal{O}_X$-modules, and the sheaf of differential forms
• The category of étale sheaves is a Grothendieck topos, allowing for the application of powerful techniques from topos theory
• Constructible sheaves, which are étale sheaves that admit a finite stratification by locally constant sheaves, play a significant role in étale cohomology
  • Constructible sheaves form a thick subcategory of $\text{Sh}(X_{\text{ét}})$ and are stable under the six operations (direct/inverse image, tensor product, internal Hom, exceptional direct/inverse image)

Cohomology Groups and Their Properties

• Étale cohomology groups $H^i(X_{\text{ét}}, \mathcal{F})$ are defined as the right derived functors of the global sections functor $\Gamma(X_{\text{ét}}, -)$ applied to an étale sheaf $\mathcal{F}$
• The étale cohomology groups satisfy the usual long exact sequence associated with a short exact sequence of étale sheaves
• For a constant sheaf $\underline{A}$ on a scheme $X$, the étale cohomology groups $H^i(X_{\text{ét}}, \underline{A})$ coincide with the classical cohomology groups of the topological space $X(\mathbb{C})$ with coefficients in $A$ when $X$ is a smooth complex variety
• Étale cohomology groups come equipped with additional structures, such as Galois actions and Frobenius endomorphisms, which encode arithmetic information
  • For a variety $X$ over a field $k$ with separable closure $k_s$, the étale cohomology groups $H^i(X_{\overline{k}}, \mathbb{Q}_\ell)$ are $\mathbb{Q}_\ell$-vector spaces with a continuous action of the absolute Galois group $\text{Gal}(k_s/k)$
• The étale cohomology groups satisfy a Künneth formula, which relates the cohomology of a product of schemes to the cohomology of the factors
• Poincaré duality holds for étale cohomology of smooth, proper varieties over separably closed fields, providing a perfect pairing between cohomology groups in complementary degrees

Comparison Theorems and Applications

• Comparison theorems relate étale cohomology to other cohomology theories, allowing for the transfer of results and techniques between different settings
• For a smooth, proper variety $X$ over the complex numbers, the comparison theorem establishes an isomorphism between the étale cohomology groups $H^i(X_{\text{ét}}, \mathbb{Z}/n\mathbb{Z})$ and the singular cohomology groups $H^i(X(\mathbb{C}), \mathbb{Z}/n\mathbb{Z})$
  • This isomorphism is compatible with the additional structures on both sides, such as the Hodge decomposition and the Galois action
• The proper base change theorem states that for a proper morphism $f: X \to Y$ and an étale sheaf $\mathcal{F}$ on $X$, the formation of the higher direct images $R^if_*\mathcal{F}$ commutes with arbitrary base change $Y' \to Y$
• The smooth base change theorem asserts that for a smooth morphism $f: X \to Y$ and an étale sheaf $\mathcal{F}$ on $X$, the formation of the higher direct images $R^if_*\mathcal{F}$ commutes with arbitrary base change $Y' \to Y$
• The Lefschetz fixed-point formula expresses the number of fixed points of a morphism $f: X \to X$ in terms of the trace of the induced endomorphism on the étale cohomology groups
• The Weil conjectures, which describe the zeta function of a variety over a finite field, can be reformulated and proven using étale cohomology (Deligne's proof)

Connections to Number Theory and Algebraic Geometry

• Étale cohomology provides a powerful tool for studying arithmetic and geometric properties of algebraic varieties over arbitrary fields
• The étale fundamental group $\pi_1^{\text{ét}}(X, \overline{x})$ of a pointed scheme $(X, \overline{x})$ classifies finite étale covers of $X$ and encodes arithmetic information
  • For a variety $X$ over a field $k$, the étale fundamental group fits into an exact sequence $1 \to \pi_1^{\text{ét}}(X_{\overline{k}}, \overline{x}) \to \pi_1^{\text{ét}}(X, \overline{x}) \to \text{Gal}(k_s/k) \to 1$, relating the geometric and arithmetic fundamental groups
• Étale cohomology plays a central role in the study of $\ell$-adic representations of Galois groups, which are crucial in modern number theory
  • The étale cohomology groups $H^i(X_{\overline{k}}, \mathbb{Q}_\ell)$ of a variety $X$ over a field $k$ are $\mathbb{Q}_\ell$-vector spaces with a continuous action of the absolute Galois group $\text{Gal}(k_s/k)$, providing examples of $\ell$-adic representations
• The theory of étale cohomology is closely related to the study of motives, which aim to provide a unified framework for understanding cohomology theories in algebraic geometry
• Étale cohomology has applications to the study of algebraic cycles, $K$-theory, and the Hodge conjecture, among other areas of algebraic geometry and number theory

Key Examples and Computations

• The étale cohomology groups of the projective space $\mathbb{P}^n_k$ over a field $k$ are given by $H^i(\mathbb{P}^n_{k, \text{ét}}, \mathbb{Z}/n\mathbb{Z}) \cong \mathbb{Z}/n\mathbb{Z}$ for $i = 0, 2, \ldots, 2n$ and vanish otherwise
• For an elliptic curve $E$ over a field $k$, the étale cohomology groups are $H^0(E_{\text{ét}}, \mathbb{Z}/n\mathbb{Z}) \cong \mathbb{Z}/n\mathbb{Z}$, $H^1(E_{\text{ét}}, \mathbb{Z}/n\mathbb{Z}) \cong E[n](k_s)$ (the $n$-torsion points of $E$ over the separable closure $k_s$), and $H^2(E_{\text{ét}}, \mathbb{Z}/n\mathbb{Z}) \cong \mathbb{Z}/n\mathbb{Z}$
  • The Galois action on $H^1(E_{\text{ét}}, \mathbb{Z}/n\mathbb{Z})$ encodes arithmetic information about the elliptic curve $E$
• For a smooth, proper curve $X$ over a finite field $\mathbb{F}_q$, the zeta function $Z(X, t)$ can be expressed in terms of the étale cohomology groups as $Z(X, t) = \frac{P_1(t)}{(1-t)(1-qt)}$, where $P_1(t)$ is the characteristic polynomial of the Frobenius endomorphism acting on $H^1(X_{\overline{\mathbb{F}}_q}, \mathbb{Q}_\ell)$
• The étale cohomology of abelian varieties over finite fields can be used to study the Tate conjecture, which relates the $\ell$-adic Tate module of an abelian variety to the Galois action on its étale cohomology groups
• The étale cohomology of Shimura varieties, such as modular curves and Siegel modular varieties, plays a crucial role in the study of automorphic forms and Galois representations in number theory

Advanced Topics and Current Research

• The theory of perverse sheaves, which are complexes of étale sheaves satisfying certain support and dimension conditions, provides a powerful tool for studying the geometry and topology of algebraic varieties
  • The decomposition theorem for perverse sheaves, proved by Beilinson, Bernstein, Deligne, and Gabber, has numerous applications in representation theory and algebraic geometry
• The étale cohomology of $p$-adic analytic spaces, developed by Berkovich and Huber, extends the theory of étale cohomology to non-algebraic settings and has connections to $p$-adic Hodge theory
• The theory of étale homotopy, introduced by Artin and Mazur, aims to develop a homotopy theory for schemes analogous to classical homotopy theory for topological spaces
  • The étale homotopy type of a scheme encodes information about its étale cohomology groups and fundamental group
• The study of étale cohomology with coefficients in $p$-adic sheaves, such as $\mathbb{Q}_p$ or $\overline{\mathbb{Q}}_p$, has led to the development of $p$-adic Hodge theory, which relates $p$-adic Galois representations to $p$-adic differential equations
• The theory of motives, initiated by Grothendieck, aims to provide a unified framework for understanding cohomology theories in algebraic geometry, with étale cohomology playing a central role
  • The existence of a motivic $t$-structure on the triangulated category of mixed motives, conjectured by Beilinson and Vologodsky, would have significant implications for the study of algebraic cycles and the Hodge conjecture
• Current research in étale cohomology includes the study of $p$-adic and $\ell$-adic sheaves on algebraic stacks, the development of a theory of étale cohomology for schemes over $\mathbb{F}_1$ (the "field with one element"), and the application of étale cohomology techniques to the study of algebraic cycles and motives.