Filtrations and cohomology II: the Gauss-Manin connection

TL;DR

This paper employs derived methods to analyze the Gauss-Manin connection across various cohomology theories, providing new insights and extending existing results, especially in mixed characteristic and prismatic cohomology contexts.

Contribution

It introduces novel derived techniques to study the Gauss-Manin connection, extending Bhatt's work to infinitesimal cohomology in mixed characteristic and clarifying its relation to prismatic cohomology.

Findings

New approaches to nilinvariance and the Quillen spectral sequence

Extension of Bhatt's de Rham cohomology results to mixed characteristic

Explanation of prismatic cohomology features via the Gauss-Manin connection

Abstract

We use derived methods to study the Gauss-Manin connection in Hochschild homology, infinitesimal cohomology, and derived de Rham cohomology. As applications, we give new approaches to nilinvariance, the Quillen spectral sequence, and the HKR filtration. We extend the results of Bhatt's work on de Rham cohomology in characteristic zero to infinitesimal cohomology in mixed characteristic and show that the comparison to Hartshorne's algebraic de Rham complex "is" the Gauss-Manin connection. Finally, we explain the main features of prismatic cohomology in characteristic zero via the Gauss-Manin connection.