Let $K$ be a finite extension of $\mathbb Q_p$ with the ring of integers $\mathcal O_K$ and residue field $k$ and let $C$ be a completion of algebraic closure of $K$. To a smooth proper (formal) scheme $\frak X$ over $\mathcal O_K$ one can attach various cohomological invariants such as
and various comparison isomorphisms relating these after scalar extension by huge period rings. In the paper [7] Bhatt-Morrow-Scholze proposed a new approach to the subject. Namely they constructed a functorial (in the higher categorical sense) complex $C^*({\frak X}, A_{inf})$ of perfect $A_{inf}$-modules, where $A_{inf}$ is the Fontaine's infinitesimal period ring (you can see the picture of $A_{inf}$ on the right) such that all other cohomology theories (up to a scalar extension) can be obtained by restricting $C^*({\frak X}, A_{inf})$ to various subsets of $\mathrm{Spec}\ A_{inf}$. Using that these subsets have non-empty intersections one obtains comparison isomorphism mentioned above. Moreover under torsion-freeness assumptions and using some additional input from abstract $p$-adic Hodge theory (Breuil-Kisin and Breuil-Kisin-Fargues modules) one can even recover crystalline cohomologies from etale ones without any scalar extension.
During our seminar we will try to understand construction of this new cohomology theory and various comparison results. More concretely:
Talk | Speaker | Date, place | Notes |
---|---|---|---|
Overview of the first part of the course | Artem Prikhodko | September, 13; IUM | |
Adic spaces | Artem Prikhodko | September, 20; IUM | |
A_inf | Artem Kanaev | September, 27; Skoltech | |
Perfectoid spaces, tilting equivalence I | Ivan Perunov | October, 4; IUM | |
Perfectoid spaces, tilting equivalence II Pro-etale site I | Ivan Perunov Dmitrii Krekov | October, 11; Skoltech | |
Pro-etale site II | Dmitrii Krekov | October, 21; IUM | |
Crystalline cohomology I Crystalline cohomology II | Georgiy Shuklin Artem Prikhodko | October, 28; Skoltech | |
Crystalline cohomology III de Rham-Witt complex | Artem Prikhodko Artem Kanaev | November, 4; Skoltech | |
Rational p-adic Hodge theory | Vadim Vologodsky | November, 11; IUM | |
C*(-, A_inf): definition and basic properties | Artem Prikhodko | November, 18; IUM | |
q-de Rham complex Crystalline comparison I | Yulia Kotelnikova Artem Prikhodko | November, 25; Skoltech | |
Canceled | December, 2; Skoltech | ||
Crystalline comparison II Rational p-adic Hodge theory for rigid-analytic varieties I | Artem Prikhodko Ivan Perunov | December, 9; Skoltech | |
Rational p-adic Hodge theory for rigid-analytic varieties II | Ivan Perunov | December, 16; Skoltech | |
Breuil-Kisin modules I | Artem Prikhodko | December, 23; Skoltech | |
Breuil-Kisin modules II | Artem Prikhodko | December, 30; Skoltech | |
Overview of the second part of the course | Artem Prikhodko | January, 20; Skoltech | |
Reminder on stable categories and spectra | Artem Prikhodko | January, 27; Skoltech | |
Some higher algebra | Artem Prikhodko | February, 3; Skoltech | |
Eventually connective cyclotomic spectra, cyclotomic structure on THH | Artem Prikhodko | February, 10; Skoltech | |
THH of group algebras | Artem Prikhodko | February, 17; Skoltech | |
Aside: Non-commutative Hodge structures | Ivan Yakovlev | February, 24; Skoltech | |
Thom spectra, THH of Thom spectra | Artem Prikhodko | March, 6; Skoltech | |
Mahowald's theorem THH of F_p | Vladimir Shajdurov Artem Prikhodko | March, 13; Skoltech | |
THH of perfectoid algebras | Artem Prikhodko | March, 20; Skoltech | |
THH of smooth algebras | Artem Prikhodko | March, 27; Skoltech | |
Canceled | Artem Prikhodko | April, 3; Skoltech | |
Prismatic cohomology via THH and Nygaard filtration I | Artem Prikhodko | April, 10; Skoltech | |
Prismatic cohomology via THH and Nygaard filtration II | Artem Prikhodko | April, 17; Skoltech | |
Prismatic cohomology via prismatic site | Vadim Vologodsky | April, 24; Skoltech | |
Construction of Breuil-Kisin modules via THH | Artem Prikhodko | May, 5; Skoltech | |
Degeneration of noncommutative Hodge to de Rham spectral sequence | Ivan Perunov | May, 8; Skoltech | |
Goodwillie's calculus | Artem Prikhodko | May, 15; Skoltech | |
THH and and Hasse-Weil zeta functions | Artem Kanaev | May, 22; IUM | |
Fontaine-Fargues curve | Roman Kositsin | May, 29; Skoltech | |
Universal property of Waldhausen's S-construction | Artem Prikhodko | June, 5; Skoltech | |
Goodwillie-Dundas-McCarty theorem I | Artem Prikhodko | June, 12; Skoltech | |
Goodwillie-Dundas-McCarty theorem II | Artem Prikhodko | June, 19; Skoltech | |
Crystalline comparison via algebraic K-theory | Nikolai Konovalov | June, 20; IUM |