A Leray spectral sequence for noncommutative differential fibrations

TL;DR

This paper develops a Leray spectral sequence framework for noncommutative differential fibrations, extending classical tools to noncommutative geometry using differential graded algebras and sheaf cohomology.

Contribution

It introduces a spectral sequence for noncommutative fibrations based on differential graded algebras and noncommutative sheaf cohomology, generalizing the Serre spectral sequence.

Findings

Constructed a Leray spectral sequence for noncommutative fibrations.

Reproduced the Serre spectral sequence as a special case.

Established a sheaf cohomology theory for noncommutative differential sheaves.

Abstract

This paper describes the Leray spectral sequence associated to a differential fibration. The differential fibration is described by base and total differential graded algebras. The cohomology used is noncommutative differential sheaf cohomology. For this purpose, a sheaf over an algebra is a left module with zero curvature covariant derivative. As a special case, we can recover the Serre spectral sequence for a noncommutative fibration.