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.
Let be a continuous map of topological spaces, which in particular gives a functor from sheaves of abelian groups on to sheaves of abelian groups on . Composing this with the functor of taking sections on is the same as taking sections on , by the definition of the direct image functor :
Thus the derived functors of compute the sheaf cohomology for :
and which converges to
This is called the Leray spectral sequence.
Generalizing to other sheaves and complexes of sheaves
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.
for the derived categories . In the example above, we have the composition of derived functors
The boundary maps and maps of sheaves on together give a boundary map on the double complex
This double complex is also a single complex graded by , with respect to which is a boundary map. If each finite intersection of the is diffeomorphic to , one can show that the cohomology
(so that the sum of these is ), and
where is the presheaf on X sending . In this context, this is called the Leray spectral sequence.
The modern definition subsumes this, because the higher direct image functor is the sheafification of the presheaf .
- 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.
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 gives us that the -page of the spectral sequence for 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). In this case the local systems will have trivial monodromy, hence . For example, consider a smooth family of genus 3 curves over a smooth K3 surface. Then, we have that
giving us the -page
Example with monodromy
Another important example of a smooth projective family is the family associated to the elliptic curves
over . Here the monodromy around 0 and 1 can be computed using Picard–Lefschetz theory, giving the monodromy around by composing local monodromies.
History and connection to other spectral sequences
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.
- Serre spectral sequence - for more examples
- Grothendieck spectral sequence - for abstract theory subsuming the construction for the Leray spectral sequence
- Mixed Hodge module
- Leray, Jean (1946). "L'anneau d'homologie d'une représentation". Comptes rendus de l'Académie des Sciences. 222: 1366–1368.
- Miller, Haynes (2000). "Leray in Oflag XVIIA : the origins of sheaf theory, sheaf cohomology, and spectral sequences, Jean Leray (1906–1998)" (PDF). Gaz. Math. 84: 17–34.
- Dimca, Alexandru (2004). Sheaves in Topology. Berlin, Heidelberg: Springer. doi:10.1007/978-3-642-18868-8. ISBN 978-3-642-18868-8. OCLC 851731478.
- Bott, Raoul; Tu, Loring W. Differential forms in algebraic topology. Graduate Texts in Mathematics. 82. New York-Berlin: Springer-Verlag. doi:10.1007/978-1-4757-3951-0. ISBN 978-0-387-90613-3. OCLC 7597142.
- Griffiths, Phillip; Harris, Joe (1978). Principles of algebraic geometry. New York: Wiley. p. 443. ISBN 0-471-32792-1. OCLC 3843444.