Leray spectral sequence

Leray spectral sequence
Name	Leray spectral sequence
Field	Algebraic topology, Algebraic geometry, Homological algebra
Introduced	1946
Introduced by	Jean Leray

Leray spectral sequence.

The Leray spectral sequence is a computational tool in Algebraic topology and Algebraic geometry introduced by Jean Leray in the 1940s to relate the cohomology of a space or scheme to the cohomology of a map to another. It connects cohomological functors such as sheaf cohomology, singular cohomology, and étale cohomology through a filtration and successive approximations, providing bridges to techniques from Homological algebra, Sheaf theory, and the theory of Derived functors. It plays a central role in the development of modern techniques used in works associated with Alexander Grothendieck, Henri Cartan, Jean-Pierre Serre, and applications in the study of fibrations, morphisms of schemes, and spectral sequence machine calculations used in Élie Cartan-style approaches.

Introduction

Given a continuous map f: X → Y between topological spaces or a morphism of schemes in scheme theory, the Leray spectral sequence arises from composing the pushforward functor f_* with a global section functor such as Γ(Y, −) or derived global sections RΓ. In the topological setting this relates singular cohomology H^*(X, A) to H^*(Y, R^q f_* A) where A is a coefficient system; in the algebro-geometric and étale contexts it relates étale cohomology H^*(X_et, F) to H^*(Y_et, R^q f_* F). The construction integrates perspectives from Cartan–Eilenberg resolution, Grothendieck spectral sequence, and resolves computational problems encountered in works of Serre and Grothendieck on cohomology of coherent sheaves, making connections to tools used in the proof of results like the Riemann–Roch theorem and the study of Leray's theorem applications in fibration theory.

Construction

Start with an abelian category such as the category of sheaves of abelian groups on a topological space or of O_X-modules on a scheme, and consider the left exact functor f_* and the global section functor Γ. Choose an injective resolution I^• of a sheaf F on X; apply f_* to obtain a double complex f_* I^• on Y. Passing to derived functors yields the first quadrant spectral sequence E_2^{p,q} = R^p Γ(Y, R^q f_* F) ⇒ R^{p+q} Γ(X, F). In the étale context replace Γ by RΓ(X_et, −) and injective resolutions by K-injective complexes as used in Verdier and Spaltenstein techniques. This approach parallels the construction of the Grothendieck spectral sequence and uses the formalism of derived categories developed in the milieu of Alexandre Grothendieck and Jean-Louis Verdier emerging from the milieu that also produced the development of the Derived category.

Convergence and edge maps

The Leray spectral sequence is typically first quadrant and converges (conditionally or strongly) to the graded pieces of a filtration on R^{n} Γ(X, F). Under hypotheses such as properness conditions appearing in proper morphisms of schemes or paracompactness in topology, one obtains strong convergence and finite type conditions familiar from the work of Serre and Grothendieck. Edge maps relate low-degree terms E_2^{p,0} ≅ H^p(Y, f_* F) and E_2^{0,q} ≅ H^0(Y, R^q f_* F) to the abutment H^{p+q}(X, F), producing exact sequences and transgression maps used in deformation problems studied by Kodaira and Spencer and deformation-theoretic contexts in Deligne's work. Functoriality of edge maps under composition of morphisms reflects naturality built into the formalism of Derived functors and Spectral sequence morphisms explored in the literature of Cartan and Eilenberg.

Examples and computations

Classical examples include the Leray spectral sequence for a fibration p: E → B in the context of fibrations and the Serre spectral sequence comparison: for a fiber F one recovers E_2^{p,q} ≅ H^p(B, H^q(F, A)). Computations of Picard groups and cohomology of projective bundles exploit the sequence for projective morphisms studied by Grothendieck in his work on the Picard scheme and the Projective bundle theorem. In étale cohomology settings computations used in proofs of the Weil conjectures and in the analysis of Galois actions on cohomology of varieties connect Leray with techniques of Pierre Deligne and Alexander Grothendieck; concrete calculations for coverings and branched maps appear in treatments by Serre and Milne. Other computations include the use in Hodge theory for proper smooth morphisms as in results of Griffiths and Harris, and in mixed Hodge structures studied by Deligne.

Functoriality and naturality

The Leray spectral sequence is natural with respect to maps of pairs (g: X' → X, h: Y' → Y) commuting with structure maps and with respect to morphisms of sheaves, reflecting functoriality properties proven in the framework of Derived category morphisms. Compatibility with composition of morphisms and base change formulæ such as the proper base change theorem and flat base change theorems in the setting of Étale cohomology and coherent cohomology ensures that pullback and pushforward operations intertwine appropriately; these compatibilities were developed in the corpus of Grothendieck and refined by Hartshorne and SGA seminars. Natural transformations between spectral sequences respect differentials and filtrations, yielding morphisms between abutments used in comparison theorems like the Leray–Serre comparisons in Hatcher-style expositions.

Applications in algebraic geometry and topology

In algebraic geometry the Leray spectral sequence underlies results on cohomology of proper morphisms, vanishing theorems in the spirit of Kodaira vanishing theorem, and descent techniques pivotal in the formulation of the Riemann–Hilbert correspondence and the study of perverse sheaves in the context of Beilinson–Bernstein–Deligne theory. In topology it is indispensable for studying the cohomology of fiber bundles, classifying spaces such as BG for a Lie group G, and spectral sequence arguments in the proof of structure theorems for loop spaces and configuration spaces appearing in work by Adams and May. The sequence interacts with Galois cohomology computations in arithmetic geometry addressed by Tate and Mazur, and with intersection cohomology and the decomposition theorem in the research of Beilinson, Bernstein, and Deligne.

Category:Spectral sequences