Comparison of spectral sequences involving bifunctors

Abstract

Suppose given functors $\mathcal{A}\times {\mathcal{A}}^{\prime }\stackrel{F}{\to }\mathcal{B}\stackrel{G}{\to }\mathcal{C}$ between abelian categories, an object $X$ in $\mathcal{A}$ and an object $X^{\prime }$ in ${\mathcal{A}}^{\prime }$ such that $F(X,-),F(-,X^{\prime })$ and $G$ are left exact, and such that further conditions hold. We show that, $E_1$-terms exempt, the Grothendieck spectral sequence of the composition of $F(X,-)$ and $G$ evaluated at $X^{\prime }$ is isomorphic to the Grothendieck spectral sequence of the composition of $F(-,X^{\prime })$ and $G$ evaluated at $X$. The respective $E_2$-terms are a priori seen to be isomorphic. But instead of trying to compare the differentials and to proceed by induction on the pages, we rather compare the double complexes that give rise to these spectral sequences.