Étale cohomology

Introduction

Étale cohomology is a sophisticated tool in algebraic geometry and number theory, providing a way to study the properties of algebraic varieties and schemes through the lens of cohomology. It extends the classical notion of cohomology to the setting of étale topology, which is a Grothendieck topology that allows for a finer analysis of the structure of algebraic varieties. Étale cohomology has been instrumental in proving several deep results, including the Weil conjectures.

Historical Context

Étale cohomology was developed by Alexander Grothendieck and his collaborators in the 1960s. The motivation behind its development was to find a cohomology theory for algebraic varieties that could handle the arithmetic properties of these varieties, particularly over fields of positive characteristic. The classical cohomology theories, such as singular cohomology, were inadequate for this purpose because they relied heavily on the topology of complex numbers, which does not generalize well to other fields.

Étale Topology

Étale topology is a Grothendieck topology on the category of schemes. It is defined using étale morphisms, which are a type of morphism between schemes that generalize the notion of a local isomorphism. An étale morphism is flat, unramified, and locally of finite presentation. The étale topology allows for the definition of sheaves and cohomology in a way that is more flexible than the classical Zariski topology.

An image of an algebraic variety, showing its complex structure and geometric properties.

Sheaves in Étale Topology

A sheaf on a scheme in the étale topology is a functor that assigns to each étale morphism a set, group, or module, satisfying certain gluing conditions. The category of sheaves on a scheme with respect to the étale topology is an abelian category, which allows for the definition of derived functors and cohomology.

Étale Cohomology Groups

Étale cohomology groups are defined as the derived functors of the global sections functor on the category of sheaves in the étale topology. For a scheme \( X \) and a sheaf \( \mathcal{F} \) on \( X \) in the étale topology, the \( i \)-th étale cohomology group \( H^i_{\text{ét}}(X, \mathcal{F}) \) is the \( i \)-th right derived functor of the global sections functor \( \Gamma(X, -) \).

Properties and Applications

Étale cohomology has several important properties that make it a powerful tool in algebraic geometry and number theory. Some of these properties include:

**Finiteness**: For a proper scheme over a finite field, the étale cohomology groups are finite-dimensional vector spaces over the field of coefficients.
**Base Change**: Étale cohomology commutes with base change, making it a flexible tool for studying families of varieties.
**Comparison Theorem**: There is a comparison theorem that relates étale cohomology to singular cohomology for schemes over the complex numbers.
**Galois Action**: Étale cohomology groups come equipped with an action of the absolute Galois group of the base field, providing a link between algebraic geometry and Galois theory.

Étale cohomology has been used to prove several deep results in mathematics, including the Weil conjectures, which relate the number of points on a variety over a finite field to the eigenvalues of the Frobenius endomorphism acting on the étale cohomology groups.

Comparison with Other Cohomology Theories

Étale cohomology is one of several cohomology theories used in algebraic geometry. Other notable cohomology theories include:

**Singular cohomology**: Used primarily for topological spaces, particularly complex algebraic varieties.
**De Rham cohomology**: Used for smooth varieties over fields of characteristic zero, particularly the complex numbers.
**Crystalline cohomology**: Used for varieties over fields of positive characteristic, particularly in the study of p-adic Hodge theory.

Each of these cohomology theories has its own strengths and weaknesses, and they are often used in conjunction to study different aspects of algebraic varieties.

Advanced Topics in Étale Cohomology

1. 1. Étale Fundamental Group

The étale fundamental group is an analogue of the classical fundamental group in algebraic topology. For a connected scheme \( X \) with a geometric point \( \bar{x} \), the étale fundamental group \( \pi_1^{\text{ét}}(X, \bar{x}) \) is defined as the automorphism group of the fiber functor from the category of finite étale coverings of \( X \) to the category of finite sets. This group encodes information about the covering spaces of \( X \) and has applications in arithmetic geometry.

1. 1. L-Functions and Zeta Functions

Étale cohomology plays a crucial role in the study of L-functions and zeta functions of algebraic varieties. These functions encode deep arithmetic information about the varieties and are connected to the eigenvalues of the Frobenius endomorphism acting on the étale cohomology groups. The study of these functions is a central topic in number theory and arithmetic geometry.

1. 1. Étale Cohomology with Compact Support

Étale cohomology with compact support is a variant of étale cohomology that is used to study non-proper schemes. For a scheme \( X \) and a sheaf \( \mathcal{F} \) on \( X \), the compactly supported étale cohomology groups \( H^i_{c, \text{ét}}(X, \mathcal{F}) \) are defined using a complex of sheaves that takes into account the compactness properties of \( X \). These groups are particularly useful in the study of motivic cohomology and intersection cohomology.

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