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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 between topological spaces or more generally the direct image of a morphism of sites, followed by the push-forward to the point – the global section functor. This yields a spectral sequence that computes the abelian sheaf cohomology on in terms of the abelian sheaf cohomology on .
Let be suitable sites let and be a morphism of sites. Let and be the model categories of complexes of sheaves of abelian groups. The direct image and global section functor compose to :
Then for a sheaf of abelian groups on there is a cohomology spectral sequence
that converges as
Lecture notes include
Dan Petersen, Leray spectral sequence, November 2010 (pdf)
Textbook accounts with an eye specifically towards étale cohomology