The goal of this study group is to read through Bhargav Bhatt's course notes on prismatic F-gauges. Prismatic F-gauges are the analogue, when the prime p is no longer invertible on your scheme, of lisse ℤp sheaves in étale cohomology. They are defined using the prismatisation and syntomification of p-adic formal schemes, so we will aim to study these constructions in some detail.
We meet on Friday mornings, 10am-12pm, Orsay Mathematics Institute, usually room 3L8 (but see the schedule below for exceptions). Everyone is welcome, even if you have not attended previously.
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The following might be useful (to be updated as the study group progresses):
Dates are approximate, as talks may run over to the following week. Details of further talks will be added in due time.
| Date | Room | Speaker | Material |
|---|---|---|---|
| 26/9/25 | 2L8 (not 3L8!) | MM | I will quickly introduce the study group by running over the introduction of Bhatt's notes, then spend some time setting up the necessary machinary of stacks, presheaves of spaces, groupoids, etc. Examples should include at least B𝔾m and 𝔸1/𝔾m and their relation to linear algebra. In terms of the F-gauges notes, this should cover most of section 2.2. |
| 3/10/25, 10h15-12h15 | 0E1 (not 3L8!) | MM | Continuation of previous talk. NOTE CHANGE OF ROOM AND TIME! |
| 10/10/25 | 3L8 | Tasos Moulinos | First MM finishes 𝔸1/𝔾m. Then Tasos does De Rham cohomology: review the standard theory (section 2.1) and explain the stacky approach in characteristic 0 (section 2.3). |
| 17/10/25 | 3L8 | Tasos Moulinos | Tasos continues from last time. |
| 24/10/25 | 3L8 | Drimik Roy Chowdhury | Stacky approach to de Rham cohomology over 𝔽p and in mixed characteristic (sections 2.4 and 2.5); probably not so many proofs, because they are mostly variants of what we already saw in characteristic zero. |
| 31/10/25 | - | - | There will be no talk on 31 October 2025, because it is both university vacation and the IHES conference New Structures and Techniques in p-adic Geometry. |
| 7/11/25 | 3L8 | Lucas Gerth | Stacky approach to the conjugate filtration in characteristic p (section 2.6 and 2.7.1) |
| 14/11/25 | 3L8 | Jason Kountouridis | Transmutation approach to the conjugate filtration in characteristic p (section 2.7.2) |
| 21/11/25 | 3L8 | Yuanyang Jiang | Sections 3.1 and 3.2. Introduce the prismatisation of smooth k-schemes where k is a perfect field of characteristic p. This might require quickly repeating some constructions from previous talks. Then review the Nygaard filtration on crystalline cohomology (also mention the approach via the de Rham--Witt complex, which Bhatt omits -- see Definition 8.1 of THH and integral p-adic Hodge theory) |
| 28/11/25 | 3L8 | Yicheng Zhou | Sections 3.3 and 3.4: for smooth k-schemes, define the Nygaard filtered prismatision and gauges. In particular, state and sketch Theorem 3.3.5, which explains how cohomology on the syntomification encodes crystalline cohomology, its Frobenius twist, the Hodge, conjugate, and Nygaard filtrations, and Hodge cohomology. Be sure to describe gauges following Fontaine--Jannsen (Remark 2.4.6). Most of the material from Lemma 3.4.11 til the end of section 3.4 can probably be omitted for the moment (it is mainly used for Mazur's theorem in section 3.5, which I suggest we skip for the sake of time), but mention the structure of vector bundles (Proposition 3.4.12) since we will need it later. | 5/12/25 | 3L8 | Hao Fu | Sections 4.1 and 4.2: for smooth k-schemes, define the syntomification and F-gauges. | 12/12/25 | 3L8 | Matthias Alizon | Sections 4.3 and 4.4: vector bundles on the syntomification and syntomic cohomology via the syntomification. | 19/12/25 | 3L8 | Niklas Kipp | Sections 4.5, i.e., duality. Explain Serre duality for the syntomification of 𝔽p | and use it to obtain both Poincaré duality for F-gauges and Milne's duality over finite fields.