Grothendieck–Verdier

Grothendieck–Verdier is a notion in algebraic geometry and homological algebra that synthesizes ideas from Alexander Grothendieck and Jean-Louis Verdier about duality in categories of sheaves and complexes. It refines classical dualities by relating functorial adjoints, internal Hom, and tensor structures in contexts influenced by the work of Jean-Pierre Serre, Henri Cartan, Max Karoubi, and the development of derived categories by Gillespie and contemporaries. The theory plays a central role in the foundations set by EGA, the SGA series, and later expositions by Robin Hartshorne and Pierre Deligne.

Definition and basic properties

The Grothendieck–Verdier framework formulates a dualizing object or dualizing complex in a monoidal closed category associated to a morphism of schemes or a topological morphism as envisioned by Alexander Grothendieck and formalized by Jean-Louis Verdier. In this setting one considers a triangulated category with tensor product and internal Hom structures influenced by constructions in derived functors and triangulated categories, and seeks an object D such that Hom(-,D) yields a contravariant equivalence analogous to adjunctions studied by Samuel Eilenberg and Saunders Mac Lane. Basic properties include existence criteria tied to cohomological finiteness conditions appearing in the work of Jean-Pierre Serre and dualizing complexes appearing in Hartshorne and Bloch-style treatments, compatibility with base change as in Grothendieck, and functoriality under proper maps modeled on the Proper base change theorem from SGA4.

Examples and classical cases

Classical examples arise for Noetherian schemes where a dualizing complex is provided by canonical modules studied by Francis S. Macaulay and refined by Grothendieck in EGA. For smooth projective varieties over fields like those in Weil-related literature, Grothendieck–Verdier duality recovers Serre duality and the appearance of canonical sheaves studied by Kunihiko Kodaira and André Weil. In étale cohomology contexts developed in SGA4 and SGA5, dualizing complexes relate to étale cohomology duality results used by Pierre Deligne in proofs of the Weil conjectures. For complex analytic spaces treated by Grauert and Remmert, the duality specializes to Verdier duality in contexts explored by Lê Dũng Tráng and Masaki Kashiwara, while coherent duality for schemes connects to treatments by Hartshorne and Lars Hesselholt.

Grothendieck–Verdier duality theorem

The Grothendieck–Verdier duality theorem asserts that for a proper morphism between suitable schemes or topological spaces, the derived direct image functor admits a right adjoint expressible via tensoring with a relative dualizing complex constructed along lines set by Grothendieck and axiomatized by Verdier. The statement refines the Serre duality paradigm and depends on finiteness hypotheses akin to those in Noetherian and Cohen–Macaulay contexts, and it uses homological machinery from derived category theory of Alexei Bondal and Orlov-style enhancements. The theorem has specialized formulations in the settings of étale functors from SGA4 and derived categories of coherent complexes as in Hartshorne; its proof techniques draw on the six operations formalism developed by Pierre Deligne, Jean-Pierre Serre, and extended in frameworks influenced by Beilinson, Bernstein, and Deligne.

Relations to derived categories and Verdier duality

Grothendieck–Verdier duality sits naturally in the language of derived categories introduced by Jean-Louis Verdier and later popularized by Bernard Keller and Amnon Neeman. It connects to Verdier duality for derived sheaf complexes on locally compact spaces as treated by Verdier in his thesis, and it interacts with the six operations formalism involving pullback, pushforward, tensor, and Hom developed by Grothendieck, Deligne, and Thomason. In derived algebraic geometry contexts advanced by Jacob Lurie and Bertrand Toën, Grothendieck–Verdier ideas are adapted to higher categorical settings and stable ∞-categories studied by J. P. May and Maxim Kontsevich. The duality also underpins equivalences in Fourier–Mukai theory as explored by Mukail and Mukai, and it informs results in mirror symmetry explored by Kontsevich and SYZ-related authors.

Applications in algebraic geometry and sheaf theory

Applications include computations of cohomology groups for proper and compact spaces in the style of Weil conjectures research by Deligne and coauthors, duality statements for intersection cohomology used by Mark Goresky and Robert MacPherson, and the construction of trace maps and residues found in residuetheory by Jean-Pierre Serre and Grothendieck. In representation-theoretic and arithmetic settings, Grothendieck–Verdier duality supports results in étale duality for Galois representations considered by Jean-Pierre Serre and Pierre Deligne, and it informs categorical approaches in perverse sheaf theory initiated by Beilinson, Bernstein, and Deligne. Contemporary uses appear in derived and spectral algebraic geometry by Lurie, in categorical dualities in noncommutative geometry as studied by Alain Connes and Maxim Kontsevich, and in developments of the six functors formalism by research groups and institutions including IHÉS and departments at Université Paris-Sud.

Category:Algebraic geometry