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

Let f:XY be a continuous map of topological spaces, which in particular gives functor f* from sheaves on X to sheaves on Y.

${\displaystyle {\ce {Sh}}(X)\ {\ce {->[f_*] Sh}}(Y)\ {\ce {->[\Gamma] Ab}}}$

Composing this with taking sections on Sh(Y) is the same as taking sections on Sh(X), by the definition of the direct image functor f*. Thus the derived functors of Γ f* compute the sheaf cohomology for X:

${\displaystyle R^{i}(\Gamma \cdot f_{*})({\mathcal {F}})=H^{i}(X,{\mathcal {F}})}$

But because f* and Γ satisfy certain conditions, there is a spectral sequence whose second page

${\displaystyle E_{2}^{pq}\ \ \ {\text{ is }}\ \ \ (R^{p}\Gamma \cdot R^{q}f_{*})({\mathcal {F}})=H^{p}(Y,R^{q}f_{*}({\mathcal {F}}))}$

and which converges to

${\displaystyle E_{\infty }^{pq}\ \ \ =\ \ \ R^{p+q}(\Gamma \cdot f_{*})({\mathcal {F}})=H^{p+q}(X,{\mathcal {F}})}$

This is called the Leray spectral sequence.

For a proof of the existence of a spectral sequence under the conditions alluded to above, see Grothendieck spectral sequence.

Classical definition

Let f:XY be a continuous map of smooth manifolds. If U={Ui} is an open cover of Y, form the Cech complex with respect to cover f−1U of X:

${\displaystyle {\text{C}}^{p}(\pi ^{-1}{\mathcal {U}},{\mathcal {F}})}$

The boundary maps dp:CpCp+1 and maps δqXq→ΩXq+1 of sheaves on X together give a boundary map on the double complex Cp(f−1Uq)

${\displaystyle D=d+\delta \ :\ C^{\bullet }(f^{-1}{\mathcal {U}},\Omega _{X}^{\bullet })\longrightarrow C^{\bullet }(f^{-1}{\mathcal {U}},\Omega _{X}^{\bullet })}$

This double complex is also a single complex graded by ${\displaystyle n=p+q}$, with respect to which D is a boundary map. If each finite intersection of the Ui is diffeomorphic to Rn, one can show that the cohomology

${\displaystyle H_{D}^{n}(C^{\bullet }(f^{-1}{\mathcal {U}},\Omega _{X}^{\bullet }))=H_{\text{dR}}^{n}(X,\mathbb {R} )}$

of this complex is the de Rham cohomology of X.[1] Moreover,[2][3] any double complex has a spectral sequence E with

${\displaystyle E_{\infty }^{n-p,p}\ \ \ =\ \ \ {\text{ the }}p{\text{th graded part of }}H_{D}^{n}(C^{\bullet }(f^{-1}{\mathcal {U}},\Omega _{X}^{\bullet }))}$

(so that the sum of these is HnD), and

${\displaystyle E_{2}^{p,q}\ \ \ =\ \ \ H^{p}(f^{-1}{\mathcal {U}},{\mathcal {H}}^{q})}$

where ${\displaystyle {\mathcal {H}}^{q}}$ is the presheaf on X sending VHq(f−1V). In this context, this is called the Leray spectral sequence.

The modern definition subsumes this, because the higher direct image functor ${\displaystyle R^{p}f_{*}(F)}$ is the sheafification of the presheaf VHq(f−1V,F).

Examples

• Let X,F be smooth manifolds, and X be simply connected. We calculate the Leray spectral sequence of the projection p:X×FX. If the cover U={Ui} is good (finite intersections are Rn) then
${\displaystyle {\mathcal {H}}^{p}(f^{-1}U_{i})\simeq H^{q}(F)}$
Since X is simply connected, any locally constant presheaf is constant, so this is the constant presheaf Hq(F)=Rn. So the second page of the Leray spectral sequence is
${\displaystyle E_{2}^{p,q}=H^{p}(f^{-1}{\mathcal {U}},H^{q}(F))=H^{p}(f^{-1}{\mathcal {U}},\mathbb {R} )\otimes H^{q}(F)}$
As the cover f−1U of X×F is also good, Hp(f−1U,R)=Hp(X). So
${\displaystyle E_{2}^{p,q}=H^{p}(X)\otimes H^{q}(F)\ \ \ \Longrightarrow \ \ \ H^{p+q}(X,\mathbb {R} )}$
Here is the first place we use that p is a projection and not just a fibre bundle: every element of E2 is an actual closed differential form on all of X×F, so applying both d and δ to them gives zero. Thus E=E2. This proves the Kunneth theorem for X simply connected:
${\displaystyle H^{\bullet }(X\times Y,\mathbb {R} )\simeq H^{\bullet }(X)\otimes H^{\bullet }(Y)}$
• If p:YX is a general fibre bundle with fibre F, the above applies, except that VpHp(f−1V,Hq) is only a locally constant presheaf, not constant.
• All example computations with the Serre spectral sequence are the Leray sequence for the constant sheaf.

Degeneration Theorem

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

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

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

{\displaystyle {\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 ${\displaystyle E_{2}}$-page

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

Example with Monodromy

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

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

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

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.

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.

References

1. ^ Bott and Tu Differential forms in algebraic topology, Propoposition 8.8, page 96
2. ^ Bott and Tu, "Differential forms in algebraic topology, page 179, Leray's construction
3. ^ Griffiths and Harris, Principles of Algebraic Geometry, page 443, the statement in italics