Leray spectral sequence

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.
Definition.
Let formula_1 be a continuous map of topological spaces, which in particular gives a functor formula_2 from sheaves of abelian groups on formula_3 to sheaves of abelian groups on formula_4. Composing this with the functor formula_5 of taking sections on formula_6 is the same as taking sections on formula_7, by the definition of the direct image functor formula_2:
Thus the derived functors of formula_10 compute the sheaf cohomology for formula_3:
But because formula_2 and formula_5 send injective objects in formula_7 to formula_5-acyclic objects in formula_6, there is a spectral sequencepg 33,19 whose second page is 
and which converges to 
This is called the Leray spectral sequence. 
Generalizing to other sheaves and complexes of sheaves.
Note this result can be generalized by instead considering sheaves of modules over a locally constant sheaf of rings formula_20 for a fixed commutative ring formula_21. Then, the sheaves will be sheaves of formula_20-modules, where for an open set formula_23, such a sheaf formula_24 is an formula_25-module for formula_26. In addition, instead of sheaves, we could consider complexes of sheaves bounded below formula_27 for the derived category of formula_28. Then, one replaces sheaf cohomology with sheaf hypercohomology.
Construction.
Tthe 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, formula_30 a left-exact functor, and formula_31 sending injective objects to formula_30-acyclic objects, then there is an isomorphism of derived functors
for the derived categories formula_34. In the example above, we have the composition of derived functors
Classical definition.
Let formula_36 be a continuous map of smooth manifolds. If formula_37 is an open cover of form the Čech complex of a sheaf formula_38 with respect to cover formula_39 of 
The boundary maps formula_41 and maps formula_42 of sheaves on formula_3 together give a boundary map on the double complex formula_44
This double complex is also a single complex graded by with respect to which formula_46 is a boundary map. If each finite intersection of the formula_47 is diffeomorphic to one can show that the cohomology 
of this complex is the de Rham cohomology of Moreover, 
any double complex has a spectral sequence "E" with
(so that the sum of these is and 
where formula_51 is the presheaf on "X" sending In this context, this is called the Leray spectral sequence.
The modern definition subsumes this, because the higher direct image functor formula_52 is the sheafification of the presheaf 
Degeneration theorem.
In the category of quasi-projective varieties over formula_77, there is a degeneration theorem proved by Pierre Deligne and Blanchard for the Leray spectral sequence, which states that a smooth projective morphism of varieties formula_74 gives us that the formula_68-page of the spectral sequence for formula_80 degenerates, hence
Easy examples can be computed if is simply connected; for example a complete intersection of dimension formula_82 (this is because of the Hurewicz homomorphism and the Lefschetz hyperplane theorem). In this case the local systems formula_83 will have trivial monodromy, hence formula_84. For example, consider a smooth family formula_36 of genus 3 curves over a smooth K3 surface. Then, we have that
giving us the formula_68-page
Example with monodromy.
Another important example of a smooth projective family is the family associated to the elliptic curves
over formula_90. Here the monodromy around and can be computed using Picard–Lefschetz theory, giving the monodromy around formula_91 by composing local monodromies.
History and connection to other spectral sequences.
At the time of Leray's work, neither of the two concepts involved (spectral sequence, sheaf cohomology) had reached anything like a definitive state. Therefore it is rarely the case that Leray's result is quoted in its original form. After much work, in the seminar of Henri Cartan in particular, the modern statement was obtained, though not the general Grothendieck spectral sequence.
Earlier (1948/9) the implications for fiber bundles were extracted in a form formally identical to that of the Serre spectral sequence, which makes no use of sheaves. This treatment, however, applied to Alexander–Spanier cohomology with compact supports, as applied to proper maps of locally compact Hausdorff spaces, as the derivation of the spectral sequence required a fine sheaf of real differential graded algebras on the total space, which was obtained by pulling back the de Rham complex along an embedding into a sphere. Jean-Pierre Serre, who needed a spectral sequence in "homology" that applied to path space fibrations, whose total spaces are almost never locally compact, thus was unable to use the original Leray spectral sequence and so derived a related spectral sequence whose cohomological variant agrees, for a compact fiber bundle on a well-behaved space with the sequence above.
In the formulation achieved by Alexander Grothendieck by about 1957, the Leray spectral sequence is the Grothendieck spectral sequence for the composition of two derived functors.