Following up on our previous work on [1] and [4] in the winter, we will study the recent paper [5] of Bhatt and Scholze comparing prismatic $F$-crystals to crystalline Galois representations. The goal is to work through Section 6, leaving the results of Section 7 for another time.

The seminar meets at 1500 central time on Mondays in Lunt 105. For the listings on the NU Math Calendar, see here.

# Syllabus

09/27. Benjamin Antieau. Introduction. Background on $p$-adic Galois representations and comparison theorems. Definitions of the main objects and statements of the main theorems.

10/04. Benjamin Antieau. Computational practicum. Zeta functions, Weil conjectures, crystalline cohomology, and $p$-divisible groups.

10/11. Noah Riggenbach. Background. Cover $\delta$-rings, prisms, and prismatic cohomology and state the main comparison theorems from [4]. Then, following [3], explain how Nygaard-completed absolute prismatic cohomology filters $\TP$ and how the syntomic complexes $\bZ_p(n)$ filter $\TC$.

10/18. Micah Darrell. Section 2. The prismatic and quasisyntomic sites. Descent for coherent sheaves in rigid analytic geometry after Drinfeld–Mathew. The Breuil–Kisin and $A_{\mathrm{inf}}$-prisms are covers. Implications for the category of prismatic crystals. Proofs of 2.2 (at least a sketch), 2.7, 2.13, and 2.14.

10/25. Bao Le Hung. Section 3. Just $\epsilon$ on the cohomology of diamonds. Lisse $\bZ_p$-sheaves on the generic fiber. Laurent $F$-crystals and the comparison results of 3.6, 3.7, 3.8, and 3.11.

11/01. NO SEMINAR.

11/08. Carlos Cortez. Section 4. Prismatic $F$-crystals, the definition and all the examples. The étale and crystalline realizations.

11/15. Elchanan Nafcha. Section 5. Prismatic $F$-crystals on perfectoids: Fargues’ theorem, construction of the functor, and 3.4 are crucial. Proof of fully faithfulness following [2], Remark 4.29. Period rings are used seriously here. Proof of the crystalline property 5.3. Go through the proof of fully faithfulness in 5.6.

11/22. Deven Manam. Sections 6.1 and 6.2. Some period sheaves on the quasisyntomic and prismatic sites and the production of crystals from filtered $\varphi$-modules. Proof of 6.8.

11/29. Adam Holeman. Sections 6.3 and 6.4. Proofs of 6.9 and 6.10. Finally, the proof of essential surjectivity.

# COVID policy

All attendees from outside Northwestern University will be required to have been fully vaccinated or have received a negative COVID test within 48 hours of the event start, as well as comply with all other University safety protocols that are in place at the time of the event. Participants unwilling or unable to abide by these requirements should not attend.

# References

[1] Anschütz and Le Bras, Prismatic Dieudonné theory, arXiv:1907.10525.

[2] Bhatt, Morrow, and Scholze, Integral $p$-adic Hodge theory, Publ. Math. Inst. Hautes Études Sci. 128 (2018), 219-397.

[3] Bhatt, Morrow, and Scholze, Topological Hochschild homology and integral p-adic Hodge theory, Publ. Math. Inst. Hautes Études Sci. 129 (2019), 199–310.

[4] Bhatt and Scholze, Prisms and prismatic cohomology, arXiv:1905.08229.

[5] Bhatt and Scholze, Prismatic $F$-crystals and crystalline Galois representations, arXiv:2106.14735.