In mathematics, in the field of homological algebra, the Grothendieck spectral sequence is a technique that allows one to compute the derived functors 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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  • In mathematics, in the field of homological algebra, the Grothendieck spectral sequence is a technique that allows one to compute the derived functors of the composition of two functors <math> G\circ F</math>, from knowledge of the derived functors of F and G. If <math>F :\mathcal{C}\to\mathcal{D}</math> and <math>G :\mathcal{D}\to\mathcal{E}</math> are two additive (covariant) functors between abelian categories such that <math>G</math> is left exact and <math>F</math> takes injective objects of <math>\mathcal{C}</math> to <math>G</math>-acyclic objects of <math>\mathcal{D}</math>, then there is a spectral sequence for each object <math>A</math> of <math>\mathcal{C}</math>: <math>E_2^{pq} = ({\rm R}^p G \circ{\rm R}^q F)(A) \implies {\rm R}^{p+q} (G\circ F)(A)</math> Many spectral sequences are merely instances of the Grothendieck spectral sequence, for example the Leray spectral sequence and the Lyndon-Hochschild-Serre spectral sequence. The exact sequence of low degrees reads 0 → RG(FA) → R(GF)(A) → G(RF) → RG(FA) → R(GF)(A)
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  • Grothendieck spectral sequence
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  • In mathematics, in the field of homological algebra, the Grothendieck spectral sequence is a technique that allows one to compute the derived functors 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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  • Grothendieck spectral sequence
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