derived category

derived category
Name	Derived category
Field	Algebraic geometry; Homological algebra
Introduced by	Alexander Grothendieck; Jean-Louis Verdier
Year	1960s

derived category

The derived category is a categorical framework that refines homological techniques used in Alexander Grothendieck's work, formalized by Jean-Louis Verdier, to control complexes up to quasi-isomorphism and to encode derived functors such as Ext and Tor. It provides a setting in which operations from Alexander Grothendieck's Séminaire de Géométrie Algébrique and constructions used in Serre duality and the Riemann–Roch theorem become functorial and compatible with localization, enabling applications to Algebraic geometry, Representation theory, and Mathematical physics.

Introduction and motivation

The motivation for the derived category arose in efforts by Alexander Grothendieck and collaborators at the Institut des Hautes Études Scientifiques to systematize cohomological tools used in proofs of the Riemann–Roch theorem and statements related to Serre duality and the Lefschetz fixed-point theorem. Derived categories reconcile computations in homological algebra present in the work of Henri Cartan, Samuel Eilenberg, Saunders Mac Lane, and Jean-Pierre Serre by inverting quasi-isomorphisms of complexes, thereby capturing phenomena seen in the study of coherent sheaves on schemes such as those appearing in Grothendieck–Riemann–Roch and in deformation theory explored by Mikhail Gromov and Maxim Kontsevich.

Construction and definitions

One constructs the derived category D(A) of an abelian category A (for example the category of modules over a ring like Noetherian rings appearing in Emmy Noether's work or coherent sheaves on a scheme studied by Alexander Grothendieck) by first considering the homotopy category K(A) of chain complexes and then localizing at the class of quasi-isomorphisms; this localization is analogous to localizations used in Serre's and Grothendieck's foundations. Verdier developed the calculus and axioms required to perform this localization in the presence of set-theoretic issues, leading to notions such as bounded derived categories D^b(A), derived categories of unbounded complexes D(A), and derived categories with bounded above or below conditions D^-(A), D^+(A). Constructions employ projective or injective resolutions inspired by techniques in the work of Jean-Pierre Serre and resolution methods used by David Hilbert and Emmy Noether.

Properties and basic results

Basic properties include the universal property characterizing D(A) as the localization K(A)[S^{-1}] where S is quasi-isomorphisms, compatibilities with homotopy colimits and homotopy limits used in the study of Alexander Grothendieck's six operations formalism, and existence of derived functors RHom and ⊗^L recovering Ext and Tor groups. Verdier duality, a categorical extension of Poincaré duality and Serre duality, lives naturally in D^b of coherent sheaves on proper schemes studied in Grothendieck's work. Brown representability, developed by Allen Hatcher's contemporaries and formally by A. Neeman and others, supplies existence statements for adjoints in many derived settings.

Examples and computations

Classical computations include the derived category of modules over a principal ideal domain such as the integers studied by Euclid's successors, where D(Z) encodes classical Ext and Tor calculations. For a smooth projective variety like those in examples by André Weil or Alexander Grothendieck, D^b(Coh(X)) features objects such as complexes of coherent sheaves used in proofs of versions of the Riemann–Roch theorem and in constructions by Maxim Kontsevich in homological mirror symmetry relating to Calabi–Yau manifold examples studied by Roger Penrose's mathematical physics contemporaries. Derived categories of representations of quivers studied by Peter Gabriel give explicit classification results connected to Gabriel's theorem; tilting theory and derived equivalences appear in the work of Happel and Bernhard Keller.

Functoriality and derived functors

Derived functors arise by applying classical left or right derived constructions to additive functors between abelian categories, e.g., the derived pushforward Rf_* and derived pullback Lf^* in the formalism developed by Alexander Grothendieck and applied in Étale cohomology and the proof of the Weil conjectures by Pierre Deligne. The six operations formalism (Rf_*, Rf_!, f^*, f^!, ⊗^L, RHom) is central to modern treatments in contexts studied at the Institut des Hautes Études Scientifiques and in the work of Joseph Bernstein and Pierre Deligne.

Triangulated structure and t-structures

Derived categories are triangulated categories in the sense of Verdier; distinguished triangles generalize long exact sequences used by Henri Cartan and Jean-Pierre Serre. t-structures, introduced by Alexander Beilinson, Joseph Bernstein, and Pierre Deligne, produce hearts equivalent to abelian categories and allow the definition of perverse sheaves as developed by Goresky and MacPherson, which have applications to the topology of singular spaces appearing in the work of William Thurston and Michael Atiyah.

Applications and connections to other areas

Derived categories intervene in numerous fields: in algebraic geometry via derived categories of coherent sheaves on schemes central to the work of Alexander Grothendieck and Maxim Kontsevich's homological mirror symmetry conjecture; in representation theory through derived equivalences and tilting studied by Bernhard Keller and Michel Broué; in mathematical physics via categories appearing in string theory and topological quantum field theory influenced by work of Edward Witten; and in number theory through étale derived categories used by Pierre Deligne in the proof of the Weil conjectures and subsequent developments around the Langlands program as advanced by Robert Langlands.

Category:Homological algebra