# Prismatic cohomology (Eilenberg Lectures at Columbia University)

Time and location: 11:40am - 12:55pm in 417 Math at Columbia University on Mondays (and occasional Wednesdays) in Fall 2018

Lecture notes: The notes below accompany my lectures, and often contain more material than covered in the lectures.
These notes have not been proofread carefully, and I make no promises about the veracity of all claims.
The notes might change without notice. Use at your own peril! (And please send me comments.)

Lecture I: Overview (09/05)

Lecture II: Delta rings (09/17, updated 10/08)

Lecture III: Distinguished elements and prisms (09/24, updated 10/08)

Lecture IV: Perfect prisms and perfectoid rings (10/08, updated 11/12)

Lecture V: The prismatic site (10/15, updated 11/12)

Lecture VI: The Hodge-Tate and crystalline comparison theorems (10/22, updated 11/12)

Lecture VII: Extension to the singular case via derived prismatic cohomology (10/29, updated 11/19)

Lecture VIII: Perfections in mixed characteristic (11/12, updated 11/19)

Lecture IX: The \'etale comparison theorem (11/19, updated 11/26)

Lecture X: The q-de Rham complex (11/26, updated 12/3)

Lecture XI: q-crystalline cohomology (12/3)

Lecture XII: Prismatic cohomology via THH (12/10, no notes will be posted)

Thanks to Johan de Jong, Toby Gee, Shizhang Li, Qixiao Ma, Shubhodip Mondal, Darya Schedrina for comments.

LaTeX code for \Prism:

\usepackage{relsize}
\usepackage[bbgreekl]{mathbbol}
\usepackage{amsfonts}
\DeclareSymbolFontAlphabet{\mathbb}{AMSb} %to ensure that the meaning of \mathbb does not change
\DeclareSymbolFontAlphabet{\mathbbl}{bbold}
\newcommand{\Prism}{{\mathlarger{\mathbbl{\Delta}}}}

Image from here.