Leray spectral sequence

In mathematics, the Leray spectral sequence was a pioneering example in homological algebra, introduced in 1946 by Jean Leray. The formulation was of a spectral sequence, expressing the relationship holding in sheaf cohomology between two topological spaces "X" and "Y", and set up by a continuous mapping

:"f":"X" → "Y".

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, a statement was reached of this kind: assuming some hypotheses on "X" and "Y", and a sheaf "F" on "X", there is a direct image sheaf

:"f"_∗F

on "Y".

There are also higher direct images :"R^"q"f"_∗F.

The "E"₂ term of the typical "Leray" spectral sequence is

:"H"^"p"("Y", "R^"q"f"_∗F).

The required statement is that this abuts to the sheaf cohomology

:"H"^"r"("X", "F").

In the formulation achieved by Alexander Grothendieck by about 1957, this is the Grothendieck spectral sequence for the composition of two derived functors.

Earlier (1948/9) the implications for singular cohomology were extracted as the Serre spectral sequence, which makes no use of sheaves.

External links

* [http://eom.springer.de/L/l058190.htm Springer encyclopedia article]