Derived functor

Derived functor
Name	Derived functor
Discipline	Homological algebra
Introduced	20th century
Notable people	Samuel Eilenberg, Saunders Mac Lane, Jean-Pierre Serre, Alexander Grothendieck
Related concepts	Derived category, Ext functor, Tor functor, Spectral sequence

Derived functor

Derived functor is a construction in homological algebra that assigns to a left or right exact functor between abelian categories a sequence of functors measuring its failure to be exact; it underlies tools used across algebraic topology, algebraic geometry, representation theory, number theory, and category theory. Originating in the work of Samuel Eilenberg and Saunders Mac Lane and developed by Jean-Louis Verdier and Alexander Grothendieck, derived functors connect to concepts such as Ext functor, Tor functor, and spectral sequence methods. They provide a systematic means to compute cohomology groups appearing in contexts from the Leray spectral sequence to the Hochschild–Serre spectral sequence and in proofs of foundational results like the Riemann–Roch theorem and the Hodge decomposition.

Introduction

Derived functors generalize classical constructions like Ext functor and Tor functor by extending a given functor to a collection of higher-order functors indexed by nonnegative integers; these higher functors capture obstructions to exactness in settings such as modules over a ring studied in Emmy Noether era developments and in sheaf cohomology on schemes central to Alexander Grothendieck's work. The formalism was systematized in part through interactions among researchers at institutions like the University of Chicago, École Normale Supérieure, and Institute for Advanced Study and applied in studies by figures such as Jean-Pierre Serre, Grothendieck, and Verdier.

Background and Motivation

Historically, problems in algebraic topology (notably in computations related to the Eilenberg–Mac Lane space and the Serre spectral sequence), in algebraic geometry (such as cohomology of coherent sheaves on projective varieties examined by Jean-Pierre Serre), and in representation theory (for example, extensions of modules over group algebras studied by Emmy Noether and later by Claude Chevalley) motivated a general mechanism for producing higher invariants. The notion builds on the framework of abelian categories introduced by Grothendieck and formalized by Barry Mitchell and connects to tools developed in the work of Henri Cartan, Samuel Eilenberg, and Saunders Mac Lane on homological algebra.

Construction of Derived Functors

Given a left exact functor F between abelian categories with enough injectives (a condition familiar in settings like sheaves on a topological space considered by Leray and Grothendieck) one constructs right derived functors R^iF by taking an injective resolution (a technique used in the proofs of results by Jean-Pierre Serre and Alexander Grothendieck) and applying F termwise, then taking cohomology of the resulting complex. Dually, for a right exact functor one uses projective resolutions and obtains left derived functors L_iF; these methods appear in computations by Cartan and Eilenberg and in applications by Hochschild and Serre. The formal language employs chain complexes, homotopy categories, and localization, techniques also developed in contexts such as the work of Verdier at the intersection of homological algebra and algebraic geometry.

Properties and Computation

Derived functors satisfy long exact sequences associated to short exact sequences, a property appearing in classical results like the long exact sequence of cohomology used in the Lefschetz fixed-point theorem context and in group cohomology computations by Hochschild–Serre. They are computed via spectral sequences (e.g., the Grothendieck spectral sequence), relate via delta-functor axioms formalized by Grothendieck and Cartan, and enjoy vanishing criteria such as those exploited in proofs of the Serre vanishing theorem. Functoriality, universality, and uniqueness up to unique isomorphism are standard properties proved in foundational texts by Weibel, Cartan, Eilenberg, and Mac Lane.

Examples and Applications

Classic examples include the Ext and Tor functors in module categories over rings studied by Emmy Noether-era algebraists and later by Paul Dirac-adjacent mathematical physics communities; sheaf cohomology groups H^i(X, -) on schemes central to Grothendieck's program; group cohomology functors used by Claude Chevalley and Jean-Pierre Serre; and derived pushforward and pullback functors Rf_* and Lf^* in algebraic geometry crucial for statements like the Riemann–Roch theorem and Grothendieck duality. In algebraic topology, derived functors compute stable homotopy invariants in contexts linked to Adams spectral sequence computations and to work by J. F. Adams and Serre.

Derived Categories and Derived Functors

The passage to derived categories, developed by Verdier under the influence of Grothendieck and Jean-Pierre Serre, reframes derived functors as total derived functors between derived categories, where one localizes the homotopy category of complexes with respect to quasi-isomorphisms. This perspective connects to triangulated categories studied by Verdier and to enhancements via differential graded categories in research by Bernhard Keller, with applications in contexts such as the Bondal–Orlov reconstruction theorem and in proofs by Deligne and Beilinson of deep results in Hodge theory and motivic cohomology.

Advanced Topics and Generalizations

Generalizations include derived functors in non-abelian settings (as in Quillen's model categories), derived functors in higher category theory and ∞-categories developed in work by Jacob Lurie and Charles Rezk, and homotopical algebra techniques from Daniel Quillen and Spencer Bloch. Other directions involve relative derived functors, equivariant derived functors used in representation theory contexts studied by Bernstein and Gelfand, and categorical lifts such as derived functors between dg-categories explored by Keller and Kontsevich.

Category:Homological algebra