In mathematics, in the field of homological algebra, the Grothendieck spectral sequence, introduced by Alexander Grothendieck in his 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 ${\displaystyle F}$ and ${\displaystyle G}$. Many spectral sequences in algebraic geometry are instances of the Grothendieck spectral sequence, for example the Leray spectral sequence.

If ${\displaystyle F\colon {\mathcal {A}}\to {\mathcal {B}}}$ and ${\displaystyle G\colon {\mathcal {B}}\to {\mathcal {C}}}$ are two additive and left exact functors between abelian categories such that both ${\displaystyle {\mathcal {A}}}$ and ${\displaystyle {\mathcal {B}}}$ have enough injectives and ${\displaystyle F}$ takes injective objects to ${\displaystyle G}$-acyclic objects, then for each object ${\displaystyle A}$ of ${\displaystyle {\mathcal {A}}}$ there is a spectral sequence:

where ${\displaystyle {\rm {R}}^{p}G}$ denotes the p-th right-derived functor of ${\displaystyle G}$, etc., and where the arrow '${\displaystyle \Longrightarrow }$' means convergence of spectral sequences.

The exact sequence of low degrees reads

If ${\textstyle X}$ and ${\textstyle Y}$ are topological spaces, let ${\textstyle {\mathcal {A}}=\mathbf {Ab} (X)}$ and ${\textstyle {\mathcal {B}}=\mathbf {Ab} (Y)}$ be the category of sheaves of abelian groups on ${\textstyle X}$ and ${\textstyle Y}$, respectively.

For a continuous map ${\displaystyle f\colon X\to Y}$ there is the (left-exact) direct image functor ${\displaystyle f_{*}\colon \mathbf {Ab} (X)\to \mathbf {Ab} (Y)}$. We also have the global section functors

Then since ${\displaystyle \Gamma _{Y}\circ f_{*}=\Gamma _{X}}$ and the functors ${\displaystyle f_{*}}$ and ${\displaystyle \Gamma _{Y}}$ satisfy the hypotheses (since the direct image functor has an exact left adjoint ${\displaystyle f^{-1}}$, pushforwards of injectives are injective and in particular acyclic for the global section functor), the sequence in this case becomes:

for a sheaf ${\displaystyle {\mathcal {F}}}$ of abelian groups on ${\displaystyle X}$.