Comparison of Spectral Sequences Involving Bifunctors

Suppose given functors $\mathcal{A}\times\mathcal{A}'\xrightarrow{F}\mathcal{B}\xrightarrow{G}\mathcal{C}$ between abelian categories, an object $X$ in $\mathcal{A}$ and an object $X'$ in $\mathcal{A}'$ such that $F(X,-)$, $F(-,X')$ 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 $X'$ is isomorphic to the Grothendieck spectral sequence of the composition of $F(-,X')$ 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.

2000 Mathematics Subject Classification: 18G40

Keywords and Phrases: Grothendieck spectral sequence, Lyndon-Hochschild-Serre spectral sequence.

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