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Outline

This is a small seminar held in 2025 winter. The goal of this seminar is to understand how to construct the so called Bhatt-Morrow-Scholze filtration on topological cyclic homology (and its related variant) to extract the information of some usual p-adic cohomology theories.

The seminar is divided into the introductory part and three main parts. The introductory part introduce some backgrounds in p-adic Hodge theory and state some of the main results of Bhatt-Morrow-Scholze.

The first part mainly follows [BMS2], we briefly go through its main contents: constructing a so called $A_{\mathrm{inf}}$-cohomology theory by constructing a filtration on $\mathrm{THH},\mathrm{TC}^-$ and using some quasi-syntomic descent techniques. We also give the proof of all comparison theorems except for the $A_{\mathrm{inf}}$-etale comparison theorem.

The second part mainly follows [BS19], we study the complete version of $A_{\mathrm{inf}}$-cohomology, which is called prismatic cohomology. We introduce some basic properties and computational techniques such as Cech-Alexander complexes, prismatic coperfection etc. In this part we prove the crystalline comparison and Hodge-Tate comparison theorem. Moreover we use prisms to complete the proof of etale comparison theorem.

The third part mainly follows [Bha23], in this part we encode the cohomology theories we constructed before in a stack. To be precise, for any geometric object $X$, we associate a stack $X^C$ such that the cohomology of this stack encodes the $C$-cohomology theory of $X$. (For example one can take $C=HT,dR,$prismatic etc.) Moreover one can encode the additional structure of the cohomology theories, such as Hodge filtration, conjugate filtration (in pure characteeristic $p$) and Nygaard filtration. This is part is not discussed in the seminar but only appears in the seminar notes.

Main references (please check the seminar notes for detailed references):

• BMS2: Topological Hochschild homology and p-adic Hodge theory.
• NS18: On topological cyclic homology
• BS19: Prism and prismatic cohomology
• Kedlaya prismatic notes
• Bhatt prismatic lecture notes
• Bha23: Prismatic $F$-crystals.

Video link

Lecture 3: Outline of the construction, cotangent complex and quasisyntomic site
Lecture 4: $HC^-$ and de Rham complex, Hodge-Tate comparison for $A_{\mathrm{inf}}$
Lecture 5: Topological cohomology on perfectoid rings
Lecture 6: Crystalline-$A_{\mathrm{inf}}$ comparison, a glimpse on prismatic site
Lecture 7: Hodge-Tate comparison theorem for prismatic cohomology
Lecture 8: Prismatic coperfection, prismatic cohomology for quasiregular semiperfectoid rings, prismatic-$A_{\mathrm{inf}}$ comparison
Lecture 9: Prismatic-etale comparison theorem

Seminar notes

Complete seminar notes: BMSII