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Wikipedia / 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

be a continuous map of topological spaces, which in particular gives a functor from sheaves of abelian groups:

Thus the derived functors compute the sheaf cohomology:

send injective objects-acyclic objects, 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 for the derived category. Then, one replaces sheaf cohomology with sheaf hypercohomology.

Construction

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-acyclic objects, then there is an isomorphism of derived functors

. In the example above, we have the composition of derived functors

Classical definition

be a continuous map of smooth manifolds is an open cover of form the Čech complex of

This double complex is also a single complex graded by 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

is the presheaf on Y sending In this context, this is called the Leray spectral sequence.

is the sheafification of the presheaf

Examples

Degeneration theorem

In the category of quasi-projective varieties, there is a degeneration theorem proved by Pierre Deligne degenerates, hence

Easy examples can be computed if (this is because of the Hurewicz homomorphism and the Lefschetz hyperplane theorem of genus 3 curves over a smooth K3 surface. Then, we have that

-page

Example with monodromy

Another important example of a smooth projective family is the family associated to the elliptic curves

. Here the monodromy around and can be computed using Picard–Lefschetz theory 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.

See also

References