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Leray spectral sequence

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Definition

The Leray spectral sequence is the special case of the Grothendieck spectral sequence for the case where the two functors being composed are a push-forward of sheaves of abelian groups along a continuous map f:XYf : X \to Y between topological spaces or more generally the direct image of a morphism of sites, followed by the push-forward Y*Y \to * to the point – the global section functor. This yields a spectral sequence that computes the abelian sheaf cohomology on XX in terms of the abelian sheaf cohomology on YY.

Properties

Theorem

Let X,YX, Y be suitable sites let and f:XYf : X \to Y be a morphism of sites. Let 𝒞=Ch (Sh(X,Ab))\mathcal{C} = Ch_\bullet(Sh(X,Ab)) and 𝒟=Ch (Sh(Y,Ab))\mathcal{D} = Ch_\bullet(Sh(Y,Ab)) be the model categories of complexes of sheaves of abelian groups. The direct image f *f_* and global section functor Γ Y\Gamma_Y compose to Γ X\Gamma_X:

Γ X:𝒞f *𝒟Γ YCh (Ab). \Gamma_X : \mathcal{C} \stackrel{f_*}{\to} \mathcal{D} \stackrel{\Gamma_Y}{\to} Ch_\bullet(Ab) \,.

Then for ASh(X,Ab)A \in Sh(X,Ab) a sheaf of abelian groups on XX there is a cohomology spectral sequence

E r p,q:=H p(Y,R qf *A) E_r^{p,q} := H^p(Y, R^q f_* A)

that converges as

E r p,qH p+q(X,A) E_r^{p,q} \Rightarrow H^{p+q}(X, A)

and hence computes the abelian sheaf cohomology of XX with coefficients in AA in terms of the cohomology of YY with coefficients in the derived direct image of AA.

References

Lecture notes include

  • Dan Petersen, Leray spectral sequence, November 2010 (pdf)

Textbook accounts with an eye specifically towards étale cohomology

Revised on November 26, 2013 01:35:09 by Urs Schreiber (145.116.131.216)