Modules over the de Rham cohomology spectrum

TL;DR

This paper establishes an equivalence between the derived category of regular holonomic D-modules on smooth varieties and modules over the motivic de Rham cohomology spectrum, integrating classical functors into a broader framework.

Contribution

It introduces a new equivalence linking D-modules to modules over the de Rham spectrum, enabling a more flexible and conceptual understanding of classical functors.

Findings

Equivalence between derived D-modules and motivic de Rham modules.

Compatibility of the equivalence with six functors.

Construction of a motivic t-structure on de Rham modules.

Abstract

We show that the bounded derived category of regular holonomic D-modules on a smooth variety is equivalent to the homotopy catgory of compact (or constructible) modules over the motivic ring spectrum $H_{dR}$ representing algebraic de Rham cohomology. This equivalence is compatible with the six functors on both sides. This way, the classical functors in the world of D-modules, $f_! := D_Y f_* D_X, f^* := D_X f^! D_Y$ ($f: X \to Y$), are conceptually explained and embedded into a larger and more flexible framework. We also apply this equivalence to obtain a motivic t-structure on $H_{dR}$-modules on not necessarily smooth schemes.