FSF3607 Étale Cohomology 7.5 credits

• Étale morphisms and the étale topology

• Étale cohomology

• The étale fundamental group

• Examples: curves, Pic and Brauer groups

• Torsors

• Comparison theorems

• Constructible sheaves

• Proper and smooth base change theorems

• Finiteness theorems

• l-adic sheaves

• Lefschetz trace formula

• Weil conjectures

After the course, the student should have sufficient knowledge of étale cohomology to be able to read more advanced topics such as the proof of the Weil conjectures (Deligne), perverse sheaves and the decomposition theorem (Beilinson, Bernstein, Deligne and Gabber). In particular, the student should

• have a thorough understanding of étale morphisms and the étale topology,

• have a working knowledge of the machinery behind étale cohomology,

• be able to do computations with torsors and the étale fundamental group,

• be able to use the fundamental theorems (finiteness, base change),

• have a cursory knowledge of constructible sheaves and l-adic cohomology.

Knowledge of basic algebraic geometry (schemes, sheaves, etc.) on the level of Algebraic Geometry II (FSF3605). This implies that a solid ba-sic knowledge of topology and commutative algebra is needed, for instance SF2735 Homological Algebra and Algebraic Topology and SF2737 Commutative Algebra and Algebraic Geometry. A second course in commutative algebra (FSF3603) is also desirable as is a course on sheaf cohomology (FSF3606).

Main text book:  J.S. Milne, Étale cohomology, Princeton University Press, 1980

Other references:

• S. Milne, Lectures on Étale Cohomology, http://www.jmilne.org/math/CourseNotes/lec.html.

• Stacks project, Chapter on Etale cohomology, http://stacks.math.columbia. edu/download/etale-cohomology.pdf.

• Lei Fu, Étale cohomology theory, World Scientific, 2015.

• SGA 4, SGA 41⁄2, SGA 5, Springer Lecture Notes in mathematics 269, 270, 305, 569, 589.

• G. Tamme, Introduction to Étale Cohomology, Springer-Verlag, 1994.

• E. Freitag and R. Kiehl, Etale Cohomology and the Weil Conjecture, Springer-Verlag, 1988.

If the course is discontinued, students may request to be examined during the following two academic years.

• INL1 - Assignment, 7.5 credits, grading scale: P, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

Homework assignments.

Homework assignments completed.

David Rydh

• All members of a group are responsible for the group's work.
• In any assessment, every student shall honestly disclose any help received and sources used.
• In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.