In the fall we will largely focus on understanding key examples and calculations as well as proofs of serious theorems concerning etale sheaf theory, aiming to get through much of Chapter 1 of the book of Freitag and Kiehl. That will get us through the important smooth and proper base change theorems, as well as the basic formalism of l-adic cohomology. In the winter we will delve further into the cohomology theory (especially to duality theorems and Kunneth formulas), and then move on to Laumon's technique of l-adic Fourier transforms in the sheaf setting.
Here are some references relevant to this year's seminar (in approximate order of appearance):
Some notes that Conrad wrote long ago that we will be following as the template
for the fall and winter (supplemented by other references for omitted details indicated therein); this is an edited .pdf
file, explaining some occasional (irrelevant) blank spaces in the middle of text
[FK] "Etale Cohomology and the Weil Conjectures" by Freitag and Kiehl
[Mi] "Etale Cohomology" by Milne
[KW] "Weil Conjectures, Perverse Sheaves, and l-adic Fourer transform" by Kiehl and Weissauer
[M1] "Analytic etale duality" (preprint) and [M2] "q-crystalline cohomologies" (in preparation)
by Masullo
Fall quarter | ||||
1 | Oct. 5 | Conrad | Overview (main goals and etale morphisms) | |
2 | Oct. 12 | Conrad | Smooth maps, etale topology/sheaves, sheaf operations, stalks | |
3 | Oct. 19 | Rosengarten | Constructibility, fundamental group, henselian rings, and applications (1.1.7, 1.2.1-1.2.5) | |
4 | Oct 26 | Sherman | First calculations: Zariski comparison, Kummer/Artin-Schreier sequences, cohomology of curves (1.2.6-1.2.7, [9]) | |
5 | Nov. 2 | Warner | Cohomology and limits, and reduction of proper base change to constant coefficients (1.3.1-1.3.4.2, [9]) | |
6 | Nov. 9 | Venkatesh | Artin approximation and proof of proper base change (1.3.4.2, [3]) | |
7, 8 | Nov. 16, 30 | Landesman | Smooth base change, local acyclicity, and vanishing cycles (1.3.5, [6]) | |
9 | Dec. 14 | Masullo | Formal GAGA (EGA III) | |
Winter quarter | ||||
10 | Jan. 18 | Tam | Cohomology with proper supports and Ehresmann's theorem (1.3.6-1.3.7, omit proof of 1.3.6.4) | |
11 | Jan. 25 | Zavyalov | Relative purity (section 10 of [9], 7.4.5 of [2] for slicker method) | |
12, 13 | Feb. 1, 8 | Feng | Poincare duality (1.3.8, [26]) | |
14 | Feb. 15 | Silliman | Kunneth formula and Artin comparison (1.3.9-1.3.10, [9], [2]) | |
15 | Feb. 22 | Devadas | Basic adic formalism (1.4.1-1.4.4.6) | |
16 | March 1 | Lawrence | Advanced adic formalism (1.4.4.7-1.4.6) | |
17 | March 8 | Raksit | Adic Artin comparison (1.4.7-1.4.8) | |
March 15 | Cancelled | Arizona Winter School | ||
18 | March 22 | Ronchetti | Sheaf Frobenius, Lefschetz trace formula, and purity (1.5) | |
Spring quarter | ||||
19 | April 12 | Lim, Dore | Weil sheaves and weights [KW, I.1-I.2.11] | |
20 | April 19 | Feng | Proof of Lefschetz trace formula [FK, II, 2-4] | |
21 | April 26 | Kemeny | Convergence radius and determinant weights [KW, I.2.12-I.3.2] | |
22 | May 3, 10 | Rosengarten | Monodromy and real sheaves [KW, I.3.3-I.4] | |
May 17 | Cancelled | Scheduling conflict | ||
23 | May 24 | Venkatesh | l-adic Fourier transform [KW, I.5] with examples | |
May 31 | Masullo | Analytic duality and de Rham cohomologies [M1], [M2] | ||
24 | June 7 | Sherman, Tam | Weil conjectures [KW, I.6-I.7] |