A coboundary morphism for the grothendieck spectral sequence

Abstract

Given an abelian category A with enough injectives we show that a short exact sequence of chain complexes of objects in 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.