In mathematics, the Leray spectral sequence was a pioneering example in homological algebra, introduced in 1946[1][2] by Jean Leray.
It is usually seen nowadays as a special case of the Grothendieck spectral sequence.
be a continuous map of topological spaces, which in particular gives a functor
, by the definition of the direct image functor
send injective objects in
, there is a spectral sequence[3]pg 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
The existence of the Leray spectral sequence is a direct application of the Grothendieck spectral sequence[3]pg 19.
This states that given additive functors between Abelian categories having enough injectives,
sending injective objects to
In the example above, we have the composition of derived functors Let
be a continuous map of smooth manifolds.
, form the Čech complex of a sheaf
together give a boundary map on the double complex
[4]: 96 Moreover,[4]: 179 [5] any double complex has a spectral sequence E with (so that the sum of these is
In this context, this is called the Leray spectral sequence.
The modern definition subsumes this, because the higher direct image functor
In the category of quasi-projective varieties over
, 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
degenerates, hence Easy examples can be computed if Y is simply connected; for example a complete intersection of dimension
(this is because of the Hurewicz homomorphism and the Lefschetz hyperplane theorem).
of genus 3 curves over a smooth K3 surface.
-page Another important example of a smooth projective family is the family associated to the elliptic curves over
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.