How Étale Cohomology Provides a Weil Cohomology Theory

This article explains how étale cohomology groups serve as a Weil cohomology theory for algebraic varieties defined over fields with positive characteristic. It outlines the limitations of classical cohomology in this setting, introduces the core concepts of étale topology, and demonstrates how these groups satisfy the specific axioms required by Weil. Finally, it highlights the significance of this theory in proving the famous Weil conjectures.

In algebraic geometry, mathematicians study shapes defined by polynomial equations. When these shapes are defined over complex numbers, standard tools like singular cohomology work well. However, problems arise when working over finite fields, which have positive characteristic. Classical topological methods fail here because the natural topology is too coarse to capture meaningful geometric information. To solve this, mathematicians needed a new way to define cohomology groups that would behave like the familiar ones from topology but work in this arithmetic setting.

André Grothendieck developed étale cohomology to bridge this gap. Instead of using open sets from standard topology, étale cohomology uses étale maps. These are algebraic maps that behave like local isomorphisms or covering spaces. By building a topology based on these maps, mathematicians can define cohomology groups for varieties over any field. To ensure these groups have the right properties, such as being finite-dimensional vector spaces, the theory uses coefficients from l-adic numbers rather than integers or real numbers.

A Weil cohomology theory must satisfy a specific set of axioms. These include having finite dimensionality, obeying Poincaré duality, and supporting a Kunneth formula. Étale cohomology meets all these requirements. The use of l-adic coefficients ensures the groups are well-behaved vector spaces. Furthermore, the theory allows for a cycle map that connects algebraic cycles to cohomology classes. This connection is crucial for counting points on varieties over finite fields.

The success of étale cohomology is best seen in its application to the Weil conjectures. These conjectures predict patterns in the number of solutions to polynomial equations over finite fields. Pierre Deligne used étale cohomology to prove the hardest part of these conjectures. By showing that étale cohomology provides a valid Weil cohomology theory, he confirmed that the eigenvalues of certain operators have specific absolute values. This achievement solidified étale cohomology as the standard tool for studying varieties in positive characteristic.

In summary, étale cohomology replaces classical topology with an algebraic equivalent suited for finite fields. By satisfying the Weil axioms through l-adic coefficients and étale maps, it provides the necessary structure to analyze arithmetic varieties. This framework not only solved longstanding mathematical problems but also unified algebraic geometry with number theory.