Prismatic / THH reading list

This is a reading list for $p$-adic cohomology and its connections to homotopy theory, intended to help people interested in getting into the field. It is not exhaustive, and is biased towards my own interests and more recent papers.

Legend:

• ► survey / good place to start
• ■ course notes
• ★ major / foundational paper

Big picture

• Scholze ICM address
• Bhatt ICM address

THH and TC

• Hesselholt-Nikolaus, Topological cyclic homology (Handbook article)
• Krause-Nikolaus, Lectures on THH and cyclotomic spectra
• Nikolaus-Scholze
• Krause-Nikolaus, Bökstedt periodicity and quotients of DVRs

Prismatic cohomology (THH perspective)

• Mathew, Some recent advances in topological Hochschild homology
• Morrow, THH in arithmetic geometry (AWS 2019)
• Arbeitsgemeinschaft notes
• BMS2
• Anschütz-le Bras, The $p$-completed cyclotomic trace in degree 2

Cartier modules / TR

• Antieau-Nikolaus, Cartier modules and cyclotomic spectra
• McCandless, On curves in $\mathrm K$-theory and $\mathrm{TR}$
• Mondal, Dieudonné theory via cohomology of classifying stacks
• Dotto-Krause-Nikolaus-Patchkoria, Witt vectors with coefficients and characteristic polynomials over non-commutative rings
• Mathew, On $K(1)$-local $\mathrm{TR}$

Prismatic cohomology (algebraic perspective)

• Scholze, Canonical $q$-deformations in arithmetic geometry
• Bhatt lecture notes
• Kedlaya, Notes on prismatic cohomology
• Bhatt-Scholze, Prisms and prismatic cohomology
• Bhatt-Lurie, Absolute prismatic cohomology

de Rham cohomology

• Li-Mondal, On endomorphisms of the de Rham cohomology functor
• Raksit, Hochschild homology and the derived de Rham complex revisited
• Ariotta(-Krause), Coherent cochain complexes and Beilinson $t$-structures

de Rham-Witt stuff

• Bhatt-Lurie-Mathew, Revisiting the de Rham-Witt complex
• Molokov, Prismatic cohomology and de Rham-Witt forms

Hodge to de Rham degeneration

• Mathew, Kaledin's degeneration theorem and topological Hochschild homology
• Antieau-Bhatt-Mathew, Counterexamples to HKR in characteristic $p$

Fargues-Fontaine curve

• 2016 Arbeitsgemeinschaft
• Lurie Fargues-Fontaine seminar, especially Lecture 1 is inspiring
• Lurie Fargues-Fontaine course
• Anschütz course notes

Formal groups

• Moulinos-Robalo-Toën, A universal HKR theorem
• Moulinos, The geometry of filtrations
• Moulinos, Filtered formal groups, Cartier duality, and derived algebraic geometry

K-theory

• Clausen-Mathew-Morrow, $\mathrm K$-theory and topological cyclic homology of henselian pairs
• Bhatt-Clausen-Mathew, Remarks on $K(1)$-local $\mathrm K$-theory
• Antieau-Mathew-Morrow-Nikolaus, On the Beilinson fiber square

Stacky perspective / Prismatization

• Mondal, Reconstruction of the stacky approach to de Rham cohomology
• Oberwolfach prismatization seminar
• Drinfeld, A 1-dimensional formal group over the prismatization of $\operatorname{Spf} \Z_p$
• Bhatt-Lurie, The prismatization of $p$-adic formal schemes

Self-promotion

• A slice refinement of Bökstedt periodicity §3.1 and §3.3 are elementary + have nice exposition; argues that prisms are natural from the POV of (genuine) equivariant homotopy theory
• Floor, ceiling, slopes, and $\mathrm K$-theory generalizes the calculation of $\mathrm{K}_*(k[x]/x^e;\Z_p)$ done at the end of Mathew's survey
• \(\mathrm{RO}(\mathbb T)\)-graded \(\mathrm{TF}\) of perfectoid rings
• Random thoughts on Witt vectors