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

$f:X\to Y$

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

H^p(Y, Rqf_∗F).

The required statement is that this abuts to the sheaf cohomology

H^p+q(X, F).

Degeneration Theorem

In the category of quasi-projective varieties over $\mathbb {C}$, there is a degeneration theorem proved by Deligne-Blanchard for the Leray which states that a smooth projective morphism of varieties $f:X\to Y$ gives us that the $E_{2}$-page of the spectral sequence for ${\underline {\mathbb {Q} }}_{X}$ degenerates, hence

$H^{k}(X;\mathbb {Q} )\cong \bigoplus _{p+q=k}H^{p}(Y;\mathbf {R} ^{q}f_{*}({\underline {\mathbb {Q} }}_{X}))$

Easy examples can be computed if $Y$ is simply connected; for example a complete intersection of dimension $\geq 2$ (this is because of the Hurewicz morphism and the Lefschetz hyperplane theorem). In this case the local systems $\mathbf {R} ^{q}f_{*}({\underline {\mathbb {Q} }}_{X})$ will have trivial monodromy, hence $\mathbf {R} ^{q}f_{*}({\underline {\mathbb {Q} }}_{X})\cong {\underline {\mathbb {Q} }}_{Y}^{\oplus l_{q}}$. For example, consider a smooth family $f:X\to Y$ of genus 3 curves over a smooth K3 surface. Then, we have that

${\begin{aligned}\mathbf {R} ^{0}f_{*}({\underline {\mathbb {Q} }}_{Y})&\cong {\underline {\mathbb {Q} }}_{Y}\\\mathbf {R} ^{1}f_{*}({\underline {\mathbb {Q} }}_{Y})&\cong {\underline {\mathbb {Q} }}_{Y}^{\oplus 6}\\\mathbf {R} ^{2}f_{*}({\underline {\mathbb {Q} }}_{Y})&\cong {\underline {\mathbb {Q} }}_{Y}\end{aligned}}$

giving us the $E_{2}$-page

$E_{2}=E_{\infty }={\begin{matrix}H^{0}(Y;{\underline {\mathbb {Q} }}_{Y})&0&H^{2}(Y;{\underline {\mathbb {Q} }}_{Y})&0&H^{4}(Y;{\underline {\mathbb {Q} }}_{Y})\\H^{0}(Y;{\underline {\mathbb {Q} }}_{Y}^{\oplus 6})&0&H^{2}(Y;{\underline {\mathbb {Q} }}_{Y}^{\oplus 6})&0&H^{4}(Y;{\underline {\mathbb {Q} }}_{Y}^{\oplus 6})\\H^{0}(Y;{\underline {\mathbb {Q} }}_{Y})&0&H^{2}(Y;{\underline {\mathbb {Q} }}_{Y})&0&H^{4}(Y;{\underline {\mathbb {Q} }}_{Y})\end{matrix}}$

Example with Monodromy

Another important example of a smooth projective family is the family associated to the elliptic curves

$y^{2}-x(x-1)(x-t)$

over $\mathbb {P} ^{1}-\{0,1,\infty \}$. Here the monodromy around $0$ and $1$ can be computed using Picard-Lefschetz theory, giving the monodromy around $\infty$ by composing local monodromy.

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.

External links

• Springer encyclopedia article