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

Rqf_∗F.

The E₂ term of the typical Leray spectral sequence is

Hp(Y, Rqf_∗F).

The required statement is that this abuts to the sheaf cohomology

Hr(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.