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| In [[mathematics]], in the field of [[homological algebra]], the '''Grothendieck spectral sequence''' is a [[spectral sequence]] that computes the [[derived functor]]s of the composition of two [[functors]] <math> G\circ F</math>, from knowledge of the derived functors of ''F'' and ''G''.
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| If
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| :<math>F :\mathcal{C}\to\mathcal{D}</math>
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| and
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| :<math>G :\mathcal{D}\to\mathcal{E}</math>
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| are two additive and [[exact functor|left exact]](covariant) [[functors]] between [[abelian categories]] such that <math>F</math> takes [[injective object]]s of <math>\mathcal{C}</math> to <math>G</math>-[[acyclic object]]s of <math>\mathcal{D}</math>, then there is a [[spectral sequence]] for each object <math>A</math> of <math>\mathcal{C}</math>:
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| :<math>E_2^{pq} = ({\rm R}^p G \circ{\rm R}^q F)(A) \Longrightarrow {\rm R}^{p+q} (G\circ F)(A).</math>
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| Many spectral sequences are instances of the Grothendieck spectral sequence, for example the [[Leray spectral sequence]].
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| The [[five term exact sequence|exact sequence of low degrees]] reads
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| :0 → ''R''<sup>1</sup>''G''(''FA'') → ''R''<sup>1</sup>(''GF'')(''A'') → ''G''(''R''<sup>1</sup>''F''(''A'')) → ''R''<sup>2</sup>''G''(''FA'') → ''R''<sup>2</sup>(''GF'')(''A'').
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| == Example: the Leray spectral sequence ==
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| If <math>X</math> and <math>Y</math> are [[topological space]]s, let
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| :<math>\mathcal{C} = \mathbf{Ab}(X)</math> and <math>\mathcal{D} = \mathbf{Ab}(Y)</math> be the [[category of sheaves of abelian groups]] on ''X'' and ''Y'', respectively and
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| :<math>\mathcal{E} = \mathbf{Ab}</math> be the category of abelian groups.
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| For a [[continuous map]]
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| :<math>f : X \to Y</math>
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| there is the (left-exact) [[direct image sheaf|direct image]] functor
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| :<math>f_* : \mathbf{Ab}(X) \to \mathbf{Ab}(Y)</math>. | |
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| We also have the [[global section]] functors
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| :<math>\Gamma_X : \mathbf{Ab}(X)\to \mathbf{Ab}</math>,
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| and
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| :<math>\Gamma_Y : \mathbf{Ab}(Y) \to \mathbf {Ab}.</math>
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| Then since
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| :<math>\Gamma_Y \circ f_* = \Gamma_X</math>
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| and the functors
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| <math> f_*</math>
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| and
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| <math>\Gamma_Y</math>
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| satisfy the hypotheses (since the direct image functor has an exact left adjoint <math>f^{-1}</math>, pushforwards of injectives are injective and in particular [[acyclic sheaf|acyclic]] for the global section functor), the [[sequence]] in this case becomes:
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| :<math>H^p(Y,{\rm R}^q f_*\mathcal{F})\implies H^{p+q}(X,\mathcal{F})</math>
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| for a [[sheaf (mathematics)|sheaf]] <math>\mathcal{F}</math> of abelian groups on <math>X</math>, and this is exactly the [[Leray spectral sequence]].
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| ==References==
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| * {{Weibel IHA}}
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| {{PlanetMath attribution|id=1095|title=Grothendieck spectral sequence}}
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| [[Category:Spectral sequences]]
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