A Coboundary Morphism For The Grothendieck Spectral Sequence

TL;DR

This paper constructs coboundary morphisms for Grothendieck spectral sequences arising from short exact sequences of chain complexes in an abelian category, preserving filtrations and applying to sheaf cohomology.

Contribution

It introduces a method to derive coboundary morphisms between spectral sequences from exact sequences of complexes, extending the understanding of their structure.

Findings

Coboundary morphisms preserve spectral sequence filtrations.

Application to sheaf cohomology filtrations.

Establishment of short exact sequences of resolutions.

Abstract

Given an abelian category $\mathcal{A}$ with enough injectives we show that a short exact sequence of chain complexes of objects in $\mathcal{A}$ gives rise to a short exact sequence of Cartan-Eilenberg resolutions. Using this we construct coboundary morphisms between Grothendieck spectral sequences associated to objects in a short exact sequence. We show that the coboundary preserves the filtrations associated with the spectral sequences and give an application of these result to filtrations in sheaf cohomology.