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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''.
| | Greetings! I am Myrtle Shroyer. My family members lives in Minnesota and my family enjoys it. Hiring is her day occupation now but she's always wanted her personal company. One of the very very best things in the globe for me is to do aerobics and I've been performing it for fairly a while.<br><br>Take a look at my homepage - [http://www.egitimpark.net/siir/groups/clear-up-a-candida-with-these-tips/ www.egitimpark.net] |
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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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Greetings! I am Myrtle Shroyer. My family members lives in Minnesota and my family enjoys it. Hiring is her day occupation now but she's always wanted her personal company. One of the very very best things in the globe for me is to do aerobics and I've been performing it for fairly a while.
Take a look at my homepage - www.egitimpark.net