P-adic Hodge theory and Topological Hochschild Homology


The picture of $A_{inf}$ by B. Bhatt. Taken from [8].
This is a joint seminar of the Center for Advanced Studies at Skoltech, Independent University of Moscow and International laboratory for Mirror Symmetry and Automorphic Forms offered by myself under the supervision of prof. Vadim Vologodsky in the 2017-2018 academic year.
The seminar is sheduled on Tuesday, room 335 (Skoltech) at 18:00.

Description

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

  • Etale cohomology $H^*_{et}({\frak X}_C, \mathbb Z_p)$ of (rigid-analytic) generic fiber ${\frak X}_C$ with $\mathbb Z_p$-coefficients,
  • Crystalline cohomology $H_{crys}^*({\frak X}_k, W(k))$ of the special fiber ${\frak X}_k$,
  • Algebraic de Rham comology $H_{dR}^*({\frak X}, \mathcal O_K)$ of the integral model ${\frak X}$

    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:

    Plan

    1. Adic spaces, perfectoid spaces, diamonds.
    2. A_inf, tilting equivalence. Fargues-Fontaine curve.
    3. Pro-etale site, some almost mathematics.
    4. Elements of rational p-adic Hodge theory. Breuil-Kisin-Fargues modules.
    5. Reminder of crystalline cohomology theory, de Rham-Witt complex.
    6. Construction of A_inf-cohomology theory via nearby cycles. Comparison with etale and crystalline cohomology. Applications.
    7. Cyclotomic spectra. Topological Hochschild and Cyclic Homology. Hopkins-Mahowald theorem. THH of group algebras and Thom spectra.
    8. A\Omega and THH. Hesselholt-Madsen theorem. Motivic filtration.
    9. Reminder of algebraic K-theory: Waldhausen's S-construction, K-theory and THH as additivizations of core groupoid and endomorphism functors respectively. Cyclotomic trace map.
    10. Goodwillie's calculus of functors, Goodwillie-Dundas-McCarthy theorem.

    Talks

    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_infArtem Kanaev September, 27; Skoltech
    Perfectoid spaces, tilting equivalence IIvan Perunov October, 4; IUM
    Perfectoid spaces, tilting equivalence II
    Pro-etale site I
    Ivan Perunov
    Dmitrii Krekov
    October, 11; Skoltech
    Pro-etale site IIDmitrii 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 IArtem Prikhodko December, 23; Skoltech
    Breuil-Kisin modules IIArtem Prikhodko December, 30; Skoltech
    Overview of the second part of the courseArtem Prikhodko January, 20; Skoltech
    Reminder on stable categories and spectraArtem Prikhodko January, 27; Skoltech
    Some higher algebraArtem Prikhodko February, 3; Skoltech
    Eventually connective cyclotomic spectra, cyclotomic structure on THHArtem Prikhodko February, 10; Skoltech
    THH of group algebrasArtem Prikhodko February, 17; Skoltech
    Aside: Non-commutative Hodge structures Ivan Yakovlev February, 24; Skoltech
    Thom spectra, THH of Thom spectraArtem Prikhodko March, 6; Skoltech
    Mahowald's theorem
    THH of F_p
    Vladimir Shajdurov
    Artem Prikhodko
    March, 13; Skoltech
    THH of perfectoid algebrasArtem Prikhodko March, 20; Skoltech
    THH of smooth algebrasArtem Prikhodko March, 27; Skoltech
    CanceledArtem Prikhodko April, 3; Skoltech
    Prismatic cohomology via THH and Nygaard filtration IArtem Prikhodko April, 10; Skoltech
    Prismatic cohomology via THH and Nygaard filtration IIArtem Prikhodko April, 17; Skoltech
    Prismatic cohomology via prismatic siteVadim Vologodsky April, 24; Skoltech
    Construction of Breuil-Kisin modules via THH Artem Prikhodko May, 5; Skoltech
    Degeneration of noncommutative Hodge to de Rham spectral sequenceIvan Perunov May, 8; Skoltech
    Goodwillie's calculusArtem Prikhodko May, 15; Skoltech
    THH and and Hasse-Weil zeta functionsArtem 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 IArtem Prikhodko June, 12; Skoltech
    Goodwillie-Dundas-McCarty theorem IIArtem Prikhodko June, 19; Skoltech
    Crystalline comparison via algebraic K-theory Nikolai Konovalov June, 20; IUM

    References

      Backgroud material on adic and perfectoid spaces

    1. Arizona Winter School 2017: Perfectoid Spaces.
    2. Peter Scholze, Peter Scholze's lectures on p-adic geometry.
    3. Peter Scholze, Perfectoid spaces.
    4. Bhargav Bhatt, Peter Scholze, The pro-etale topology for schemes.
    5. Roland Huber, Etale Cohomology of Rigid Analytic Varieties and Adic Spaces.

      P-adic Hodge Theory

    6. Jean-Marc Fontain, Representations p-adiques des corps locaux.
    7. Mark Kisn, Crystalline representations and F-crystalls.
    8. Olivier Brinon, Brian Conrad, CMI summer school notes on p-adic Hodge theory.
    9. Bhargav Bhatt, Matthew Morrow, Peter Scholze, Integral p-adic Hodge theory.
    10. Bhargav Bhatt, Specializing varieties and their cohomology from characteristic 0 to characteristic p.
    11. Peter Scholze, p-adic Hodge theory for rigid-analytic varieties.

      Hochschild Homology and K-theory

    12. Clark Barwick, On the algebraic K-theory of higher categories.
    13. Ib Madsen, Algebraic K-theory and traces.
    14. Bjorn Ian Dundas, Thomas G. Goodwillie, Randy McCarthy, The local structure of algebraic K-theory.
    15. Thomas Nikolaus, Peter Scholze, On topological cyclic homology.
    16. Lars Hesselholt, Ib Madsen, On the K-theory of finite algebras over Witt vectors of perfect fields.
    17. Bhargav Bhatt, Matthew Morrow, Peter Scholze, Topological Hochschild homology and integral p-adic Hodge theory.

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