# Leray spectral sequence

The Leray spectral sequence is a spectral sequence that computes the cohomology (or homology) of a space from the cohomology of its image under a continuous map together with the cohomology of the fibers of that map. [Jean Leray](https://www.edgepedia.org/jean-leray) introduced it in 1945 and 1946, at the same moment he introduced sheaf theory, as a tool for relating the cohomology groups he defined for continuous maps to the cohomology of the source space.<sup>[1](https://math.mit.edu/~hrm/papers/ss.pdf)</sup><sup> • </sup><sup>[2](https://smf.emath.fr/publications/histoire-de-la-theorie-des-faisceaux)</sup>

| Key fact | Detail |
|---|---|
| Origin | Introduced by Jean Leray in 1945 and later work, in connection with homological properties of continuous mappings of locally compact spaces<sup>[3](https://encyclopediaofmath.org/wiki/Sheaf_theory)</sup> |
| Sheaf-theoretic form | For a morphism π : Y → X and a sheaf F on Y, the sequence has E₂ terms built from the higher direct images R^q π\*F and converges to the cohomology of Y<sup>[4](https://www.math.mcgill.ca/goren/SeminarOnCohomology/Milne.pdf)</sup> |
| Higher direct images | R^r π\*F is the sheaf on X associated with the presheaf U ↦ H^r(U_Y, F), where U_Y = U ×_X Y<sup>[4](https://www.math.mcgill.ca/goren/SeminarOnCohomology/Milne.pdf)</sup> |
| Topological form | For a Serre fibration with path-connected fibers, singular cohomology gives an analog with the same properties as the sequence of a locally trivial fiber bundle; this is the Serre spectral sequence, and there is also a homology version<sup>[5](https://encyclopediaofmath.org/wiki/Leray_spectral_sequence)</sup> |
| Modern status | A special case of the Grothendieck spectral sequence of homological algebra<sup>[6](https://arxiv.org/abs/1912.08288)</sup> |
| Applications | Level set persistent homology and Reeb spaces for simplicial maps; Floer cohomology of fibered Lagrangians; motivic homotopy theory<sup>[6](https://arxiv.org/abs/1912.08288)</sup><sup> • </sup><sup>[7](https://arxiv.org/abs/1701.07788)</sup><sup> • </sup><sup>[8](https://arxiv.org/abs/1812.09574)</sup> |

## Historical origin

The 1946 announcements introduced the word *faisceau* (sheaf), so sheaf theory was born at the same moment as spectral sequences. For Leray, a sheaf assigned a module or ring to a closed subspace; his first example was the sheaf assigning to a closed subset F ⊆ X its p-th cohomology group. It was not until 1950 that [Henri Cartan](https://www.edgepedia.org/henri-cartan) refounded the theory using open subspaces.<sup>[1](https://math.mit.edu/~hrm/papers/ss.pdf)</sup> Cartan reformulated sheaf theory in his seminar and, together with [Jean-Pierre Serre](https://www.edgepedia.org/jean-pierre-serre), gave applications to the theory of analytic spaces.<sup>[2](https://smf.emath.fr/publications/histoire-de-la-theorie-des-faisceaux)</sup> To serve this aim, Leray introduced a spectral sequence connecting the cohomology of the domain with cohomology groups he had defined for continuous mappings.<sup>[2](https://smf.emath.fr/publications/histoire-de-la-theorie-des-faisceaux)</sup>

## The sheaf-theoretic construction

The modern formulation starts from a morphism π : Y → X of varieties or schemes and the direct image functor π\*, which sends a sheaf F on Y to the sheaf on X defined by Γ(U, π\*F) = Γ(U_Y, F), where U_Y = U ×_X Y. The functor π\* is left exact, so its right derived functors R^r π\* exist; the sheaves R^r π\*F are called the higher direct images of F.<sup>[4](https://www.math.mcgill.ca/goren/SeminarOnCohomology/Milne.pdf)</sup>

A proposition of Milne's lecture notes identifies these sheaves concretely: for any π : Y → X and sheaf F on Y, R^r π\*F is the sheaf on X associated with the presheaf U ↦ H^r(U_Y, F). The stalk of R^r π\*F at a geometric point x̄ of X is the direct limit of H^r(U_Y, F) over all étale neighborhoods (U, u) of x̄.<sup>[4](https://www.math.mcgill.ca/goren/SeminarOnCohomology/Milne.pdf)</sup> When π is a closed immersion, π\* is exact, so the higher direct images vanish.<sup>[4](https://www.math.mcgill.ca/goren/SeminarOnCohomology/Milne.pdf)</sup>

The Leray spectral sequence then relates the cohomology of Y to the cohomology on X with coefficients in these higher direct images. In étale cohomology theory this gives a spectral sequence relating the étale cohomology groups of Y to those of X.<sup>[9](https://www.researchgate.net/publication/326696766_Weil_Conjectures_Exposition)</sup> The relation between Tate cohomology and ordinary cohomology of a profinite group is also described by a Leray spectral sequence.<sup>[10](https://ncatlab.org/holmstrom/published/Sheaf%20cohomology)</sup>

## The topological form: the Leray–Serre sequence

For a fibration F ↪ E → B of topological spaces, the Leray spectral sequence takes the familiar form

\[ E_2^{p,q} = H^p\bigl(B;\, H^q(F; K)\bigr) \;\Longrightarrow\; H^{p+q}(E; K), \]

where K is a coefficient field. When the fibration has trivial monodromy and the coefficient system is simple, the E₂ page reduces to H^p(B; K) ⊗_K H^q(F; K).<sup>[11](https://faculty.tcu.edu/gfriedman/papers/specseq.pdf)</sup><sup> • </sup><sup>[12](https://people.math.wisc.edu/~lmaxim/754notes.pdf)</sup> Using singular cohomology, for any Serre fibration with path-connected fibers one can construct an analog of the Leray spectral sequence with all the properties of the sequence of a locally trivial fiber bundle; this is the [Serre spectral sequence](https://www.edgepedia.org/serre-spectral-sequence), and there is an analogous sequence in singular homology.<sup>[5](https://encyclopediaofmath.org/wiki/Leray_spectral_sequence)</sup> Originally the name Leray–Serre spectral sequence was used only when the (co)homology theory is ordinary (co)homology with coefficients in an abelian group or ring; for a generalized cohomology theory h there is an associated cohomology spectral sequence as well.<sup>[13](https://www.math.u-bordeaux.fr/~bmatschke/SpectralSequences.pdf)</sup>

**Collapse and the Leray–Hirsch theorem.** A spectral sequence is most useful when it collapses early. The Leray–Hirsch theorem gives a standard case: suppose F ↪ E → B is a fibration with B and F of finite type, π₁(B) = 0, and a field K of coefficients, and assume the restriction i\*: H\*(E; K) → H\*(F; K) is onto. Then H\*(E; K) ≅ H\*(B; K) ⊗_K H\*(F; K). The proof runs the Leray–Serre cohomology spectral sequence: surjectivity of i\* forces all differentials to vanish, so the sequence collapses at E₂, and with field coefficients all extension problems are trivial.<sup>[12](https://people.math.wisc.edu/~lmaxim/754notes.pdf)</sup>

A standard first computation uses the fibration S¹ ↪ S^∞ ≃ \* → CP^∞ and the E₂ page of the associated Leray–Serre sequence to compute the cohomology ring H\*(CP^∞).<sup>[12](https://people.math.wisc.edu/~lmaxim/754notes.pdf)</sup>

## The Grothendieck perspective

The Leray spectral sequence is now a special case of the [Grothendieck spectral sequence](https://www.edgepedia.org/grothendieck-spectral-sequence), found in most textbooks on homological algebra.<sup>[6](https://arxiv.org/abs/1912.08288)</sup> Grothendieck's description of the spectral sequence for the functor Ext is essential in algebraic geometry.<sup>[3](https://encyclopediaofmath.org/wiki/Sheaf_theory)</sup> In the algebro-geometric setting, if f : X → B is a continuous map and B_• is a filtration of B that is cellular with respect to the direct image sheaves R^q f\*F for all q, the E₂ term of the Leray spectral sequence is described by the homology of the local system of fibers; the sequence essentially reflects functorial algebro-geometric structure on the cohomology groups, for example lifting to Nori's category of mixed motives, and a similar result holds for the perverse Leray spectral sequence.<sup>[8](https://arxiv.org/abs/1812.09574)</sup> There is also a comparison theorem between the Leray spectral sequence and the spectral sequence associated with the Bloch–Ogus complex for cohomology theories defined using a [Grothendieck topology](https://www.edgepedia.org/grothendieck-topology) finer than the [Zariski topology](https://www.edgepedia.org/zariski-topology), such as étale cohomology or Betti cohomology over C.<sup>[8](https://arxiv.org/abs/1812.09574)</sup>

## Computational and applied forms

**Simplicial maps and applied topology.** For a simplicial map f : X → Y, the Leray spectral sequence computes the homology of the domain X from the fibers of f. For each point y ∈ Y one records the q-th homology of the fiber f⁻¹(y), producing a cosheaf L^f_q over Y; sweeping these fiber-wise classes over p-dimensional subspaces of Y produces cosheaf homology groups H_p(Y; L^f_q), which appear on the second page of the sequence. When Y is a triangulation of the real line, the Leray cosheaves are the level set persistence modules of computational topology, and the homology of X can be read from their barcode. The sequence also connects to Reeb spaces.<sup>[6](https://arxiv.org/abs/1912.08288)</sup>

**Symplectic and Floer theory.** Variants of the Leray–Serre sequence appear in symplectic geometry. For symplectic fibrations in the sense of Guillemin, Lerman, and Sternberg, a Leray–Serre spectral sequence computes the Floer cohomology of certain fibered Lagrangians inside a compact symplectic fibration with small monotone fibers and a rational base; if the Floer cohomology of the fiber Lagrangian vanishes with field coefficients, that of the total Lagrangian vanishes as well, with applications to non-torus fibers of the Gelfand–Cetlin system in flag manifolds.<sup>[7](https://arxiv.org/abs/1701.07788)</sup> Fibered symplectic cohomology work assumes a negativity property on the fibration reminiscent of negative curvature in complex vector bundles.<sup>[14](https://arxiv.org/abs/math/0503193)</sup>

**Motivic homotopy.** In motivic homotopy theory one builds a homotopy Leray spectral sequence analogous to both the Serre spectral sequence in algebraic topology and the Leray spectral sequence in algebraic geometry, with an E₂ page described in terms of homology of the local system of fibers using a theory similar to Rost's cycle modules.<sup>[8](https://arxiv.org/abs/1812.09574)</sup>

## References

1. [The 1946 announcements (history of spectral sequences), MIT](https://math.mit.edu/~hrm/papers/ss.pdf)
2. [Histoire de la théorie des faisceaux, Société Mathématique de France](https://smf.emath.fr/publications/histoire-de-la-theorie-des-faisceaux)
3. [Sheaf theory, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Sheaf_theory)
4. [J. S. Milne, Lectures on Étale Cohomology, §12: Higher direct images; the Leray spectral sequence](https://www.math.mcgill.ca/goren/SeminarOnCohomology/Milne.pdf)
5. [Leray spectral sequence, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Leray_spectral_sequence)
6. [A. Patel, D. Sauriol, Leray Spectral Sequence for Simplicial Maps, arXiv:1912.08288](https://arxiv.org/abs/1912.08288)
7. [A Leray-Serre spectral sequence for Lagrangian Floer theory, arXiv:1701.07788](https://arxiv.org/abs/1701.07788)
8. [The homotopy Leray spectral sequence, arXiv:1812.09574](https://arxiv.org/abs/1812.09574)
9. [Weil Conjectures Exposition, ResearchGate](https://www.researchgate.net/publication/326696766_Weil_Conjectures_Exposition)
10. [Sheaf cohomology, nLab (Holmstrom)](https://ncatlab.org/holmstrom/published/Sheaf%20cohomology)
11. [Some extremely brief notes on the Leray spectral sequence, TCU](https://faculty.tcu.edu/gfriedman/papers/specseq.pdf)
12. [Lecture notes on homotopy theory and applications, University of Wisconsin](https://people.math.wisc.edu/~lmaxim/754notes.pdf)
13. [Examples of spectral sequences, Université de Bordeaux](https://www.math.u-bordeaux.fr/~bmatschke/SpectralSequences.pdf)
14. [Fibered Symplectic Cohomology and the Leray-Serre Spectral Sequence, arXiv:math/0503193](https://arxiv.org/abs/math/0503193)

---
*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Homological algebra and K-theory › Spectral sequences and homological techniques*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license