Grothendieck duality

Grothendieck duality
Name	Grothendieck duality
Field	Algebraic geometry
First proved	1960s
Contributors	Alexander Grothendieck, Jean-Pierre Serre, Pierre Deligne, Robin Hartshorne, Jean-Louis Verdier

Grothendieck duality is a central theorem in algebraic geometry relating direct image functors and dualizing complexes for morphisms of schemes. It generalizes Serre duality for projective varieties and connects the cohomology of coherent sheaves with derived functors, providing a foundational tool for the study of morphisms such as proper maps, finite maps, and smooth maps. The theory has deep ties to homological algebra, étale cohomology, and the development of derived categories.

History and motivation

Grothendieck duality emerged in the context of work by Alexander Grothendieck on the Séminaire de Géométrie Algébrique and foundational projects at the Institut des Hautes Études Scientifiques and the Université Paris-Sud. Early motivations trace to Jean-Pierre Serre's formulation of Serre duality for projective curves and surfaces, and to duality phenomena in the work of Oscar Zariski and Jean Dieudonné. The formalism was developed alongside Grothendieck's theory of schemes, the machinery of derived categories introduced by Alexander Grothendieck and later formalized by Jean-Louis Verdier, and the cohomological technologies advanced in collaborations with Pierre Deligne and Michel Raynaud. Subsequent expositions and refinements were provided by Robin Hartshorne in his influential texts and by contributors at institutions like Harvard University and the École Normale Supérieure.

Statement and main forms

The core statement gives, for a proper morphism f: X → Y of schemes, an adjunction between the derived direct image Rf_* and a right adjoint f^! on the derived category of coherent sheaves; this involves a dualizing complex ω_f. The classical form recovers Serre duality when Y = Spec k for a field k and X is a projective variety; variants include relative duality for finite morphisms, absolute duality for proper maps, and duality for smooth morphisms where ω_f is an invertible sheaf (a canonical bundle). Important formulations appear in the work of Grothendieck in the EGA and SGA volumes and in later treatments by Verdier and Deligne that use triangulated categories and derived functors.

Proofs and techniques

Proofs use a blend of techniques from homological algebra, derived category theory as developed by Jean-Louis Verdier, and compactification results such as Nagata's theorem. Grothendieck's original approach relied on local duality theorems and coherent duality via injective resolutions; later proofs exploit the Brown representability theorem in triangulated categorys, results about residual complexes by Hartshorne, and methods from category theory used by authors at Princeton University and the University of California, Berkeley. Other techniques involve reduction to affine cases using Čech cohomology, dualizing modules in the sense of Ibrahim Bass and Hyman Bass, and trace maps constructed by Pierre Deligne and Jean-Pierre Serre. Developments in derived algebraic geometry at institutions like Massachusetts Institute of Technology and Institut Henri Poincaré have led to new proofs using model categories and ∞-categories.

Examples and computations

Explicit computations illustrate the dualizing complex for projective space, where the canonical bundle corresponds to the top-degree differential form, and for finite flat morphisms where the trace map yields explicit duality pairings. Calculations for smooth projective curves recover classical results of Riemann–Roch theorem type and connect to the theory of residues explored by Jean-Pierre Serre and André Weil. Examples on singular varieties draw on work by Robin Hartshorne and computational approaches from the American Mathematical Society literature. Concrete instances include duality for morphisms between spectra of local rings studied by Alexander Grothendieck and local duality theorems developed by Hartshorne and Hyman Bass.

Relations to other dualities

Grothendieck duality interfaces with several classical and modern dualities: it extends Serre duality and relates to Poincaré duality in the context of smooth proper varieties over fields, and to Verdier duality in the realm of derived categories and constructible sheaves. It also connects with Tate duality in arithmetic geometry and with duality phenomena in étale cohomology developed by Pierre Deligne and Jean-Pierre Serre. The formalism resonates with duality in functional analysis contexts studied at universities such as University of Cambridge and with categorical dualities in the work of Saunders Mac Lane and Samuel Eilenberg.

Applications in algebraic geometry

Applications include the study of canonical models, moduli spaces analyzed at institutions like Princeton University and Institut des Hautes Études Scientifiques, intersection theory tied to the Riemann–Roch theorem, and the cohomological study of sheaves on singular spaces. Grothendieck duality is instrumental in deformation theory treated in seminars by Pierre Deligne and Michael Artin, in the analysis of dualizing sheaves for curves and surfaces in the work of Oscar Zariski and David Mumford, and in modern treatments of derived moduli stacks appearing in research at Imperial College London and Stanford University.

Category:Algebraic geometry