Duality of differential operators and algebraic de Rham cohomology

TL;DR

This paper introduces a derived category with differential operator morphisms for smooth proper morphisms and generalizes Serre duality, providing a new proof of Poincaré duality in algebraic de Rham cohomology.

Contribution

It develops a novel derived category framework with differential operator morphisms and extends Serre duality to complexes, offering new insights into algebraic de Rham cohomology.

Findings

Generalized Serre duality for complexes of differential operators

New proof of Poincaré duality for relative algebraic de Rham cohomology

Framework applicable to smooth proper morphisms

Abstract

Given a smooth proper morphism $f\colon X\rightarrow S$, we introduce a certain derived category where morphisms are permitted to be $\mathcal{O}_S$-linear differential operators. We then prove a generalisation of Serre duality that applies to two-term complexes of this type. We apply this to give a new proof of Poincar\'e duality for relative algebraic de Rham cohomology.