Title: Studies in Derived δ-rings
Abstract: Various cohomology theories in algebraic geometry (de Rham, étale, crystalline, prismatic) possess rich multiplicative algebraic structures and can often be characterized by universal properties in appropriate algebraic categories. For example, the de Rham complex of a ring A is the universal commutative differential graded algebra associated to A, while crystalline cohomology is governed by divided power thickenings. When two different cohomology theories are related by a comparison theorem, it is natural to ask whether the comparison arises from a functorial relationship between the corresponding algebraic categories. This question has become particularly interesting in light of recent advances in absolute prismatic cohomology, where a multitude of p-adic cohomology theories have been unified into a single theory defined by a simple algebraic construction involving the ring of Witt vectors.
The main theme of this dissertation is the study of the interplay between δ-rings and divided power algebras in the derived setting. This perspective turns out to provide a common framework for a number of recent developments in arithmetic geometry and homotopy theory. In the talk, I will explain the theory of derived δ-Cartier rings developed in my thesis and how it applies to several topics in arithmetic algebraic geometry. This includes Drinfeld’s interpretation of divided powers in the p-local setting via Witt vectors, the theory of sheared Witt vectors developed by Zink, Drinfeld, and Bhatt–Mathew–Vologodsky, as well as a universal characterization of the Nygaard filtration on the prismatic cohomology of F_p-algebras extending recent work of A. Holeman.
Kirill Magidson
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