Grothendieck spectral sequence

Grothendieck spectral sequence
Name	Grothendieck spectral sequence
Field	Algebraic geometry; Homological algebra
Introduced	1950s
Introduced by	Alexandre Grothendieck
Related	Derived functor; Spectral sequence; Homological algebra

Grothendieck spectral sequence

The Grothendieck spectral sequence is a fundamental computational tool in Algebraic geometry, Homological algebra, and the theory of derived categories that relates the derived functors of a composite of two functors to the derived functors of the individual functors. It originates in the work of Alexandre Grothendieck and plays a central role in the development of modern cohomology theories, appearing in contexts connected to Jean-Pierre Serre, Alexander Grothendieck, Henri Cartan, Jean-Louis Verdier, and the formulations found in texts by Pierre Deligne, Robin Hartshorne, and Charles A. Weibel.

Introduction

The Grothendieck spectral sequence provides a machine to compute objects such as higher direct images in the presence of compositions like f_* ◦ g_* or functor compositions arising in the study of sheaves on schemes and topoi. It is often deployed alongside tools from Spectral sequence, Derived functor, Ext, Tor, and the formalism of injective resolutions and projective resolutions. Major users include researchers working on problems related to Étale cohomology, De Rham cohomology, Dolbeault cohomology, and comparisons between cohomology theories as in the work of Grothendieck and Serre.

Statement and Construction

Let F: A → B and G: B → C be additive left-exact functors between abelian categories A, B, C satisfying usual hypotheses used by Alexandre Grothendieck and later codified by Jean-Louis Verdier. If A has enough injectives and F sends injective objects of A to G-acyclic objects of B, then there is a first-quadrant spectral sequence with E2-page E2^{p,q} = R^p G(R^q F(X)) converging to R^{p+q}(G ∘ F)(X) for X in A. The construction uses injective resolutions in A, the acyclicity hypotheses of F, and the application of G to a Cartan–Eilenberg resolution, a method developed in the lineage of Henri Cartan, Samuel Eilenberg, and Norman Steenrod. The spectral sequence is natural in X and functorial with respect to morphisms in A, allowing comparison maps and edge morphisms as in the broader framework exemplified by works of Pierre Deligne and Jean-Pierre Serre.

Convergence and Exactness Conditions

Convergence of the Grothendieck spectral sequence is usually strong in the first quadrant under the usual boundedness and injectivity hypotheses, paralleling convergence results found in classical sources like Jean-Louis Verdier's exposition of derived categories. Exactness conditions require that F carry injectives to G-acyclics; variants replace injectives with projectives in dual contexts, invoking dualities considered by Alexander Grothendieck and Jean-Pierre Serre. In many geometric situations—such as proper morphisms of schemes or continuous maps of topoi—the necessary hypotheses are satisfied, enabling use in theorems attributed to Serre, Grothendieck, and later elaborations by Robin Hartshorne, Arthur Ogus, and Luc Illusie.

Examples and Applications

Classical applications include the computation of sheaf cohomology for compositions of direct image functors Rf_* ∘ Rg_*, comparisons in the Leray spectral sequence associated to a continuous map as in Jean Leray's work, and the analysis of the five-term exact sequence appearing in many cohomological spectral sequences used by Jean-Pierre Serre and Alexander Grothendieck. It is instrumental in proofs and calculations in the theory of Étale cohomology relevant to Alexander Grothendieck’s program, in the study of the Hodge-to-de Rham spectral sequence related to Deligne and P. Griffiths, and in computations of derived functors such as Ext and Tor in contexts considered by Henri Cartan, Samuel Eilenberg, Saunders Mac Lane, and Charles A. Weibel. Further uses appear in the study of duality theorems by Serre duality, investigations in Arithmetic geometry by Barry Mazur, and in categorical contexts related to Grothendieck topoi.

Variants and Generalizations

Variants include spectral sequences for composition of right-exact functors using projective resolutions, versions in the bounded-below derived category D^+(A) as formulated by Jean-Louis Verdier and Pierre Deligne, and enhancements using model category techniques influenced by Daniel Quillen and Paul Goerss. There are equivariant versions in the presence of group actions tied to work by Jean-Pierre Serre and J. P. May, and versions in non-abelian settings inspired by Alexander Grothendieck’s ideas on higher stacks developed later by Carlos Simpson and Jacob Lurie. Derived algebraic geometry perspectives, building on Jacob Lurie, Bertrand Toën, and Gabriele Vezzosi, reinterpret the spectral sequence within homotopical and infinity-categorical frameworks.

Proof Outline and Key Lemmas

Proofs proceed by choosing an injective resolution I• of X in A, applying F to obtain a complex F(I•) in B whose hypercohomology is computed after applying G. A Cartan–Eilenberg resolution of F(I•) supplies a double complex whose spectral sequences give the E2-page R^p G(R^q F(X)) and converge to R^{p+q}(G ∘ F)(X). Key lemmas include acyclicity lemmas assuring vanishing of higher derived functors on F-injectives (as used by Alexander Grothendieck), the comparison lemma for spectral sequences developed in the style of Samuel Eilenberg and Henri Cartan, and standard convergence theorems in the spirit of Jean Leray and Jean-Louis Verdier. The argument uses homological algebra foundations established by Eilenberg–Steenrod, foundational category theory due to Saunders Mac Lane, and modern treatments in expository texts by Robin Hartshorne and Charles A. Weibel.

Category:Spectral sequences