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Concept# Leray spectral sequence

Summary

In mathematics, the Leray spectral sequence was a pioneering example in homological algebra, introduced in 1946 by Jean Leray. It is usually seen nowadays as a special case of the Grothendieck spectral sequence.
Let be a continuous map of topological spaces, which in particular gives a functor from sheaves of abelian groups on to sheaves of abelian groups on . Composing this with the functor of taking sections on is the same as taking sections on , by the definition of the direct image functor :
Thus the derived functors of compute the sheaf cohomology for :
But because and send injective objects in to -acyclic objects in , there is a spectral sequencepg 33,19 whose second page is
and which converges to
This is called the Leray spectral sequence.
Note this result can be generalized by instead considering sheaves of modules over a locally constant sheaf of rings for a fixed commutative ring . Then, the sheaves will be sheaves of -modules, where for an open set , such a sheaf is an -module for . In addition, instead of sheaves, we could consider complexes of sheaves bounded below for the of . Then, one replaces sheaf cohomology with sheaf hypercohomology.
The existence of the Leray spectral sequence is a direct application of the Grothendieck spectral sequencepg 19. This states that given additive functors
between Abelian categories having enough injectives, a left-exact functor, and sending injective objects to -acyclic objects, then there is an isomorphism of derived functors
for the derived categories . In the example above, we have the composition of derived functors
Let be a continuous map of smooth manifolds. If is an open cover of , form the Čech complex of a sheaf with respect to cover of :
The boundary maps and maps of sheaves on together give a boundary map on the double complex
This double complex is also a single complex graded by , with respect to which is a boundary map. If each finite intersection of the is diffeomorphic to , one can show that the cohomology
of this complex is the de Rham cohomology of .

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