Grothendieck spectral sequence

In mathematics, in the field of homological algebra, the Grothendieck spectral sequence, introduced in Tôhoku paper, is a spectral sequence that computes the derived functors of the composition of two functors ${\displaystyle G\circ F}$, from knowledge of the derived functors of F and G.

If ${\displaystyle F:{\mathcal {A}}\to {\mathcal {B}}}$ and ${\displaystyle G:{\mathcal {B}}\to {\mathcal {C}}}$ are two additive and left exact functors between abelian categories such that ${\displaystyle F}$ takes F-acyclic objects (e.g., injective objects) to ${\displaystyle G}$-acyclic objects and if ${\displaystyle {\mathcal {B}}}$ has enough injectives, then there is a spectral sequence for each object ${\displaystyle A}$ of ${\displaystyle {\mathcal {A}}}$ that admits an F-acyclic resolution:

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