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.

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 derived category of Then, one replaces sheaf cohomology with sheaf hypercohomology.

Let be smooth manifolds, and be simply connected, so We calculate the Leray spectral sequence of the projection If the cover is good (finite intersections are ) then

Since is simply connected, any locally constant presheaf is constant, so this is the constant presheaf So the second page of the Leray spectral sequence is

As the cover of is also good, So

Here is the first place we use that is a projection and not just a fibre bundle: every element of is an actual closed differential form on all of , so applying both d and to them gives zero. Thus This proves the Künneth theorem for simply connected:

If is a general fiber bundle with fibre , the above applies, except that is only a locally constant presheaf, not constant. All example computations with the Serre spectral sequence are the Leray sequence for the constant sheaf.

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.