Leray spectral sequence

In mathematics , the Leray spectral sequence is an aid for calculating the sheaf cohomology .

definition

Let be a continuous mapping between topological spaces . Look at the functor that assigns each sheaf above its direct image above . Be its derived functors . Then there is a spectral sequence with ${\displaystyle f\colon X\to Y}$ ${\displaystyle f_{*}}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle X}$ ${\displaystyle f_{*}{\mathcal {F}}}$ ${\displaystyle Y}$ ${\displaystyle R^{i}f_{*}}$

${\displaystyle E_{2}^{p,q}=H^{p}(X,R^{q}f_{*}({\mathcal {F}}))}$,

against

${\displaystyle E_{\infty }^{p,q}=H^{p+q}(X,{\mathcal {F}})}$

converges.

Access via double complexes for sheaves of differential forms

Let be a continuous map between smooth manifolds . For an overlap of, define a double complex as a Čech complex for the sheaf of differential forms . ${\displaystyle f\colon X\to Y}$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle Y}$ ${\displaystyle C^{p}(f^{-1}{\mathcal {U}},\Omega ^{q})}$ ${\displaystyle \Omega ^{*}}$

If a good coverage , then the cohomology of this double complex is the de Rham cohomology . A spectral sequence is included with the double complex . ${\displaystyle {\mathcal {U}}}$ ${\displaystyle H_{dR}^{*}(X)}$ ${\displaystyle E_{2}^{p,q}=H^{p}(f^{-1}{\mathcal {U}},{\mathcal {H}}_{dR}^{q})}$

Application to fiber bundles

For a fiber bundle with fiber , one obtains a spectral sequence converging against . ${\displaystyle f\colon E\to B}$ ${\displaystyle F}$ ${\displaystyle E_{\infty }^{p,q}=H^{p+q}(E)}$ ${\displaystyle E_{2}^{p,q}=H^{p}(B,{\mathcal {H}}^{q}(F))}$

The gysin sequence can be derived from this for bundles of spheres .

The generalization of the Leray spectral sequence to Serre fibers is called the Leray-Serre spectral sequence .

Web links

• Leray spectral sequence (Encyclopedia of Mathematics)
• Leray spectral sequence (nLab)