Algebraic Connections vs. Algebraic {$\cD$}-modules: inverse and direct images

TL;DR

This paper simplifies and provides an alternative proof for the comparison between inverse and direct image functors in the context of algebraic connections and $ ext{D}$-modules, extending the understanding of Gauss-Manin connections.

Contribution

It offers a simpler proof of the comparison between direct images for connections and $ ext{D}$-modules, avoiding Saito's equivalence, and extends the comparison to a broader context.

Findings

Simplified proof of direct image comparison

Alternative approach avoiding Saito's equivalence

Extension to a precursor setting of Gauss-Manin connection

Abstract

In the dictionary between the language of (algebraic integrable) connections and that of (algebraic) $\cD$-modules, to compare the definitions of inverse images for connections and $\cD$-modules is easy. But the comparison between direct images for connections (the classical construction of the Gauss-Manin connection for smooth morphisms) and for $\cD$-modules, although known to specialists, has been explicitly proved only recently in a paper of Dimca, Maaref, Sabbah and Saito in 2000, where the authors' main technical tool was M. Saito's equivalence between the derived category of $\cD$-modules and a localized category of differential complexes. The aim of this short paper is to give a simplified summary of the [DMSS] argument, and to propose an alternative proof of this comparison which is simpler, in the sense that it does not use Saito equivalence. Moreover, our alternative strategy of comparison works in a context which is a precursor to the Gauss-Manin connection (at the level of $f^{-1}\cD_Y$-modules, for a morphism $f:X\to Y$), and may be of some intrinsic interest.