Learning Seminar on prismatic F-gauge
Fall 2025

Monday 2:45-4:15 pm, Math 731.

Title: Algebraic de Rham Cohomology for Schemes

Abstract: Prismatic cohomology can be roughly regarded as a deformation of de Rham cohomology. Consequently, many structures on prismatic cohomology have counterparts for de Rham cohomology that are more classical and more explicit. Our goal of this talk is to understand Section 2.1 of Bhatt’s note, i.e. the notion of algebraic de Rham cohomology. We will first review the classical theory of de Rham cohomology for smooth manifolds and explain the issues arising when switching to complex manifolds and more general algebraic geometric objects like varieties. Then we will introduce the so-called algebraic de Rham cohomology for general schemes and explain why it is the correct cohomology we need. We will also go through the Hodge filtration structure as well as the conjugate filtration structure on algebraic de Rham cohomology, and how they interact with Poincare duality (if permitted). Examples will be given for a better understanding of the above notions.

Title: Derived Categories and t-structure

Abstract: In this talk, we will review derived categories and t-structure and show that the heart is abelian. If time permits, we will also introduce the stable infinity category, and see for any ring R, there exists a stable infinity category whose homotopy category is the classical derived category D(R).

Title: Stacks and G-torsors

Abstract : The goal of this talk is to introduce stacks with a focus on quotient stacks, especially the classifying stack BG. We begin with prestacks, or categories fibered in groupoids, and work through concrete examples. To understand quotient prestacks, we will introduce G-torsors and discuss their local triviality. We then explain how to pass from prestacks to stacks via descent theory, recalling Grothendieck topologies and focusing on the fpqc topology as the correct notion for descent. Throughout, the emphasis will be on motivation and examples, leading to a conceptual understanding of classifying stack BG.

Abstract: I will introduce the notion of quasi-coherent sheaves on quotient stacks and show that they are equivalent to G-equivariant quasi-coherent sheaves. We will apply this framework to the derived category of quasi-coherent sheaves on A^1/G_m to give a geometric framework to understand the filtered derived category. 

Abstract: We adapt the stack approach to de Rham cohomology in characteristic 0 to positive and mixed characteristic using sheaves of divided-powers. This gives a partial analogue of the classification of quasi-coherent sheaves on the classifying stack of sheaves of nilpotent elements in characteristic 0

Abstract: In this talk, we will discuss the de Rham stack in mixed characteristic, and it's Hodge filtered variant.