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 f:X→Y be a continuous map of topological spaces, which in particular gives a functor f* from sheaves on X to sheaves on Y. Composing this with the functor of taking sections on is the same as taking sections on, by the definition of the direct image functor f*:
Thus the derived functors of compute the sheaf cohomology for X:
But because f* and satisfy certain conditions, there is a spectral sequence whose second page
and which converges to
This is called the Leray spectral sequence.
For a proof of the existence of a spectral sequence under the conditions alluded to above, see Grothendieck spectral sequence.

Classical definition

Let f:X→Y be a continuous map of smooth manifolds. If U= is an open cover of, form the Cech complex with respect to cover of :
The boundary maps dp:Cp→Cp+1 and maps δq:ΩXq→ΩXq+1 of sheaves on X together give a boundary map on the double complex
This double complex is also a single complex graded by, with respect to which D is a boundary map. If each finite intersection of the Ui is diffeomorphic to R, one can show that the cohomology
of this complex is the de Rham cohomology of X. Moreover, any double complex has a spectral sequence E with
, and
where 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 is the sheafification of the presheaf.

Examples

In the category of quasi-projective varieties over, there is a degeneration theorem proved by Deligne–Blanchard for the Leray which states that a smooth projective morphism of varieties gives us that the -page of the spectral sequence for degenerates, hence
Easy examples can be computed if is simply connected; for example a complete intersection of dimension . In this case the local systems will have trivial monodromy, hence. For example, consider a smooth family of genus 3 curves over a smooth K3 surface. Then, we have that
giving us the -page

Example with monodromy

Another important example of a smooth projective family is the family associated to the elliptic curves
over. Here the monodromy around and can be computed using Picard-Lefschetz theory, giving the monodromy around by composing local monodromy.

History and connection to other spectral sequences

At the time of Leray's work, neither of the two concepts involved 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 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. 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.