Algebraic topology Spectral sequences Theorem

Leray spectral sequence

Statement

Let $f \colon X \to Y$ be a continuous map of topological spaces and let $\mathcal{F}$ be a sheaf of abelian groups on $X$ . For $q \ge 0$ , let $R^q f_* \mathcal{F}$ be the $q$ -th higher direct image, that is, the $q$ -th right derived functor of the direct image $f_*$ . It is the sheaf on $Y$ associated with the presheaf $V \mapsto H^q(f^{-1}(V); \mathcal{F})$ . Then there is a first-quadrant spectral sequence with $E_2^{p,q} = H^p(Y; R^q f_* \mathcal{F}) \Longrightarrow H^{p+q}(X; \mathcal{F}),$ where $H^*$ denotes sheaf cohomology, and the differentials are $d_r \colon E_r^{p,q} \to E_r^{p+r, q-r+1}$ .

Remarks

Since $\Gamma(Y, f_* \mathcal{F}) = \Gamma(X, \mathcal{F})$ and $f_*$ sends injective sheaves to flasque sheaves, which are acyclic for $\Gamma(Y, -)$ , this is the Grothendieck spectral sequence of the composite functor $\Gamma(Y, -) \circ f_*$ .

Example: if $R^q f_* \mathcal{F} = 0$ for all $q > 0$ , the spectral sequence collapses and $H^p(X; \mathcal{F}) \cong H^p(Y; f_* \mathcal{F})$ for all $p$ . This applies when $f$ is the inclusion of a closed subspace, because then $f_*$ is exact.

Let $f$ be a fibre bundle with fibre $F$ over a nice base, and let $\mathcal{F}$ be a constant sheaf. Then $R^q f_* \mathcal{F}$ is locally constant with stalks $H^q(F)$ , and one recovers a form of the Serre spectral sequence.

Going deeper

For a student

Leray introduced sheaves and spectral sequences together for this purpose. In algebraic geometry the same spectral sequence, for a morphism of schemes and étale or coherent cohomology, is a basic tool. It is the prototype of the Grothendieck spectral sequence for a composite of derived functors.

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Sheaf cohomology in topology concept
Leray spectral sequence statement

Tag	`8BMQ90EJPR`
URI	`m2://8BMQ90EJPR`
Kind	statement, theorem
Writing	W0 stub
MSC	55T05, 18G40, 55N30
Status	This copy of the site has no build data. Run `m2 status 8BMQ90EJPR`.
About	Sheaf cohomology in topology
Uses	Sheaf cohomology in topology
Same as	Wikidata Q6528749 Wikipedia Leray spectral sequence
Changed	2 November 2026

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