Filtrations and cohomology II: the Gauss-Manin connection

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.