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
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.
In modern terms
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
- Hp+q(X, F).
Connection to other spectral sequences
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 fibrations were extracted as the Serre spectral sequence, which makes no use of sheaves.