Time: Monday 2:30- 3:50 pm,  Zoom link (please contact the organizers for the link)

Room:  SCHM 103 (Purdue)

Schedule and Plan:  

Here is the Introduction and  Plan 

1. Jan 12:  Overview: Explain the main goal of this semester. Speaker: Shubhodip Mondal (Purdue)

            Title: Overview of prismatic F-gauges.

           Abstract: This talk will be an introduction and a brief overview of prismatic F-Gauges and their connection to p-adic cohomology theories.

2. January 22 (Note the changed time: 4:00-5:20 pm)   Prismatic cohomology. Speaker: George Nicolas Diaz-Wahl (Purdue) 

Title: Overview of Prismatic Cohomology

Abstract: We will define the notion of prisms and prismatic cohomology, emphasizing the examples of crystalline, perfect, and Breuil-Kisin prisms. We will also review the Nygaard filtration on prismatic cohomology in the quasi-syntomic setting

3. Jan 26. Review Stacks and anything needed for Chapter 2. Speaker Mansimar Singh (Purdue)

Title: Stacks

Abstract : The goal of this talk is to introduce stacks with a focus on quotient stacks, especially the classifying stack BG. We begin with prestacks, or categories fibered in groupoids, and work through concrete examples. To understand quotient prestacks, we will introduce G-torsors and discuss their local triviality. We then explain how to pass from prestacks to stacks via descent theory, recalling Grothendieck topologies and focusing on the fpqc topology as the correct notion for descent. Throughout, the emphasis will be on motivation and examples, leading to a conceptual understanding of classifying stack BG.

4. Feb 2. Review the main results from Chapter 2. Speaker: Zichuan Wang (IU)

Title: Stacky approach to filtered de Rham cohomology

Abstract: Following chapter 2, we review quasicoherent sheaves on quotient stacks and representations of formally completed vector bundles, with examples for A^1/G_m and the de Rham stack A^{1,dR}. We then review the realization of (filtered) de Rham cohomology via stacks obtained by "transmutation". Over a perfect field k of char. p, we discuss combining the Hodge and conjugate filtered stacks by gluing their Witt vector models, which serves as a “mod-p prelude” to the Nygaard/Syntomic constructions in later chapters.

5.  Feb 9. §3.1: The prismatization over k. Speaker: Matthias Strauch (IU)

Title: The prismatization over k

Abstract: We will cover some of the material in sections 3.1 and 3.2 in Bhatt's paper on Prismatic F-gauges. We start with the prismatization construction in 3.1.1. Then we discuss the definition of the Nygaard filtration following Theorem 3.2.1 (without proof). Finally, our goal is to explain the actual construction of the Nygaard filtration using the de Rham Witt complex following Def. 8.1 in the paper "Topological Hochschild Homology and integral p-adic Hodge Theory" by Bhatt, Morrow, and Scholze (https://arxiv.org/pdf/1802.03261).

6. Feb 16. §3.3 The (Nygaard) filtered prismatization. Speaker: Zhilin Luo (Purdue)

Title: Nygaard filtered prismatization.

Abstract: Let k be a perfect field of characteristic p and W=W(k). For a smooth k-scheme X, crystalline (identified as prismatic via Frobenius pull-back in this setting) cohomology comes with Frobenius, which becomes isomorphism after inverting p. The main theme of this talk is the Nygaard filtration, which is an integral refinement living on the Frobenius pull-back of crystalline cohomology characterized by the p-divisibility of the Frobenius.

Following section 3.3 of Bhatt's note, we will briefly explain how the Nygaard filtration can be geometrized via the Nygaard filtered prismatization: We will introduce a (p-adic formal) stack k^N = Spf(W[u,t]/(ut-p))/G_m, called the filtered prismatization of Spec(k). Via transmutation we introduce a stack X^N\to k^N, such that the cohomology of its structure sheaf simultaneously encodes 1) The Nygaard filtration as p^\bullet W filtered structure 2) two open charts u\neq 0 and t\neq 0 recovering prismatic = crystalline cohomology glued by Frobenius twist 3) two boundary locus u=0 and t=0 recovering the Hodge and the conjugate filtration on de-Rham cohomology, with the closed stratum u=t=0 yielding the graded Hodge pieces. 

7. Feb 23. §3.4 Gauges over k. Speaker: Ruipeng Zou (Purdue)

Title: Prismatic Gauges as filtered objects and Local Structures

Abstract: In this talk, we will define prismatic gauges using the filtered prismatization stack $X^\mathcal{N}$. We will unpack the geometry of $X^\mathcal{N}$ and its explicit local charts. We will reinterpret gauges as filtered objects and visualize them explicitly as $p$-complete complexes. If time permits, we will also talk about the example $\mathcal{H}_{\mathcal{N}}(X)$.

8. Mar 2. §4.1, §4.2. Speaker: Matthias Strauch (IU)

Title: Syntomification in characteristic p and F-gauges

Abstract: We plan to cover the essential points of sections 4.1 and 4.2 of Bhatt's lectures on Prismatic F-gauges. More in detail, we start by introducing the syntomification X^{Syn} of a smooth scheme X over a perfect field k of characteristic p. This stack is obtained by identifying the two open substacks {u \new q 0} and {t \neq 0} of the Nygaard filtered prismatization of X. It contains the prismatization of X as an open substack and the Hodge stack associated to X as a closed substack. We discuss some properties of X^{Syn} and consider the syntomification of the affine line. Then we define the category of F-gauges on X as the category of quasi-coherent sheaves on X^{Syn}. The latter half of the talk will be devoted of explicit descriptions of the cohomology of some F-gauges. We may also try to briefly mention some aspects of the paper "Frobenius gauges and a new theory of p-torsion sheaves in characteristic p" by Fontaine and Jannsen. 

9. March 9th, §4.3, §4.4. Speaker: Zichuan Wang (IU): 

Title: Syntomic cohomology of F-gauges in characteristic p 

Abstract: Following the previous talks and sections 4.3, 4.4 of Bhatt's notes, we continue studying the syntomification of a smooth scheme X over a perfect field k of char p. More precisely, we begin by describing vector bundle F-gauges on k^{Syn}, with a classification of the invertible F-gauges, in terms of semi-linear algebra data. Then we define the Breuil-Kisin twists O{i} and the weight i syntomic cohomology in char p. The latter half of the talk will be devoted to a comparison between syntomic cohomology of Z_p(1) and the étale cohomology of G_m, and a list of cohomology theories on X that can be captured by the coherent cohomology on X^{Syn}. If time permits, we may try to relate Fontaine-Jannsen’s gauge cohomology and Fontaine-Messing’s syntomic sheaves to this stacky picture.

Happy Spring Break !  

10. March 23rd, §5.1 The prismatization. Speaker: Daniel Le (Purdue University)

     Title: Prismatization

Abstract: The goal of this talk is to introduce the prismatization of a bounded p-adic formal scheme. In good cases, the prismatic cohomology of a p-adic formal scheme can be computed by the coherent cohomology of the structure sheaf of its prismatization. We will review aspects of prismatic cohomology related to the construction. 

References and useful links (also see the website of learning seminar during Fall 2025 )

Prismatic F-gauge (Bhatt's notes for a class taught at Princeton, which will be our main reference)
Prisms and Prismatic Cohomology (Bhatt and Scholze's paper on prismatic cohomology)
Prismatization (Drinfeld's article on the stacks \Sigma, \Sigma', \Sigma'', also called ``prismatization", ``Nygaard filtered prismatization" and ``syntomification" of Z_p)
Absolute prismatic cohomology (Bhatt and Lurie's article on ``prismatization", which is called Cartier-Witt stack in their paper)
Dieudonn\'e theory via cohomology of classifying stacks and prismatic F-gauges (Mondal's article containing an approach to prismatic F-gauges via quasisyntomic descent)
An algebraicity conjecture of Drinfeld and the moduli of p-divisible groups (article of Gardner-Madapusi containing an exposition of the stacks we need equipped with suitable derived structures)
Prismatic F-gauges and a result of T. Liu (containing a summary of the stacks we need)
Paper by Fontaine-Jannsen