Jacob Lurie: On the prismatization of the field with one element.

University of Chicago Number Theory / Geometric Langlands Seminar.
Speaker: Jacob Lurie
Date: Jan 25, 2024
Title: On the prismatization of the field with one element.
Abstract: A few years ago, Bhatt and Scholze introduced the theory of prismatic cohomology, a cohomology theory for p-adic formal schemes which interpolates between many invariants of arithmetic and geometric interest. Subsequently, Bhatt-Lurie and Drinfeld observed that the prismatic cohomology of a p-adic formal scheme X can be identified with the coherent cohomology of an auxiliary geometric object called the {\it prismatization} of X. In this talk, I will introduce a related object which can be viewed heuristically as the prismatization of Spec( \mathbf{F}_1 ), where \mathbf{F}_1 is the "field with one element". I'll conclude by explaining how this object sheds light on some recent results in the theory of syntomic cohomology.