Number Theory Learning Seminar on Étale Cohomology
Spring-Summer 2023
Format: Each meeting will last for 90 minutes. The speaker will present the material during the first 60 minutes, and the remaining time will be devoted to going over exercises from the previous talk.
There is an Overleaf page containing exercises related to each talk and collaborated solutions for the exercises. Please email me if you would like to access it.
References:
[LM] D. Lombardo, A. Maffei. Introduction to étale cohomology. https://people.dm.unipi.it/tdn/CoomologiaEtale/Note.pdf
[Mil] J.S. Milne. Lectures on Étale Cohomology. https://www.jmilne.org/math/CourseNotes/LEC.pdf
Tentative schedule
Talk 1 (Date: May 30 2-3:30pm; Speaker: Steve Huang)
Topics: Review of cohomology theories; motivation; unramified and étale morphisms
References: [LM] Chapter 1, 7; [Mil] Chapter 1 sections 1-2
Talk 2 (Date: 6/6 2-3:30pm; Speaker: Xinyu Wu)
Topics: Étale fundamental group, sites
References: [Mil] Chapter 1 sections 3 & 5; [LM 11.1]
Talk 3 (Date: 6/13 2-3:30pm; Speaker: Casey Perdue)
Topics: Sheaves for Étale topology I
References: [Mil] Chapter 1 section 6; [LM] Chapter 12.3
Talk 4 (Date: 7/3 11-12:30; Speaker: Casey Perdue)
Topics: Sheaves for Étale topology II
References: [LM] Chapter 11 [Mil] Chapter 1 sections 7-8
Talk 5 (Date: 7/7 11 am-12:30 pm; Speaker: Matt Broe)
Topics: Stalks of etale sheaves
References: [LM] Chapter 13; [Mil] Chapter 1 section 4
Talk 6 (Date: 7/11 11 am-12:30 pm; Speaker: Ramanujan Srihari)
Topics: Direct and inverse image of sheaves
References: [LM] Chapter 16; [Mil] Chapter 1 section 8
Talk 7 (Date: 7/18 9-10:30 am; Speaker: Steve Huang)
Topics: Cohomology and Čech cohomology
References: [LM] Chapter 15; [Mil] Chapter 1 sections 9-10
Talk 8 (Date: 7/21 11 am-12:30 pm; Speaker: Ramanujan Srihari)
Topics: Torsors and H^1
References: [LM] Chapter 18.2; [Mil] Chapter 1 section 11
Talk 9 (Date: 7/24 11 am-12:30 pm; Speaker: Max Johnson)
Topics: Crashcourse on spectral sequences
References: [Mil] Chapter 1 section 12; [LM] Chapter 14
Talk 10 (Date: 7/28 11 am-12:30 pm; Speaker: Hargun Bhatia)
Topics: Cohomology of G_m
References: [Mil] Chapter 1 section 13; [LM] 17.1 & 17.3
Talk 11 (Date: 7/31 11 am-12:30 pm; Speaker: Chris Xu)
Topics: Cohomology of curves I
References: [Mil] Chapter 1 section 14; [LM] Chapter 17
Talk 12 (Date: 8/4 11 am-12:30 pm; Speaker: Jianxiang Tan)
Topics: Cohomology of curves II
References: [Mil] Chapter 1 section 14; [LM] Chapter 17