The main topics of this course are the concepts and properties of étale cohomology, often used for the geometry of numbers. After reviewing Galois theory for fields, students will study the definition and properties of Galois cohomology, and specific examples for low orders. The étale cohomology of schemes for fields is Galois cohomology. Étale cohomology is a natural extension of Galois cohomology. Students will then review the basics of schemes and sheaves, studying étale morphism and Grothendieck topology. Using that, the instructor will define étale cohomology, and cover its properties. Specific examples for low orders will be investigated. This course is followed by Advanced Topics in Algebra H.
Étale cohomology is a basic, commonly used tool in the geometry of numbers. Students in this course will gain an understanding of étale cohomology, and accurately describe low-order étale cohomology for algebraic curves over number fields in particular.
By the end of this course, students will be able to:
1) Understand the definition and some of the basic properties of étale cohomologies,
2) Understand the relation between étale cohomologies, Galois cohomologies and Zariski cohomologies,
3) Calculation of low-dimensional Galois cohomologies,
4) Calculation of low-dimensional étale cohomologies.
Galois cohomology, scheme, Zariski cohomology, étale morphism, Grothendieck topology, étale cohomology,
Intercultural skills | Communication skills | Specialist skills | Critical thinking skills | Practical and/or problem-solving skills |
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Standard lecture course.
Course schedule | Required learning | |
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Class 1 | review of the Galois theory | Details will be provided during each class session. |
Class 2 | the definition and some properties of Galois cohomologies | Details will be provided during each class session. |
Class 3 | the finite-dimensional Galois cohomologies | Details will be provided during each class session. |
Class 4 | review of the commutative rings, schemes and sheaves | Details will be provided during each class session. |
Class 5 | étale morphisms, Grothendieck topologies | Details will be provided during each class session. |
Class 6 | the definition and some of the basic properties of Étale cohomologies | Details will be provided during each class session. |
Class 7 | the finite-dimensional étale cohomologies | Details will be provided during each class session. |
Class 8 | some cohomology theories on arithmetic geometry | Details will be provided during each class session. |
Unspecified.
Course materials are provided during class.
Learning achievement is evaluated by reports.
Students must have successfully completed Algebra I, Algebra II and Algebra III;
or, they must have equivalent knowledge.