Grothendieck's six operations
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| Grothendieck's six operations | |
|---|---|
| Name | Grothendieck's six operations |
| Introduced | 1960s–1970s |
| Main contributors | Alexandre Grothendieck, Jean-Louis Verdier, Pierre Deligne, Alexander Beilinson, Joseph Bernstein |
| Field | Algebraic geometry, Arithmetic geometry, Homological algebra |
Grothendieck's six operations are a framework of six functorial operations on derived categories of sheaves, developed to unify cohomological tools across algebraic geometry, Étale cohomology, and D-modules. Originating in work by Alexandre Grothendieck and systematically elaborated by Jean-Louis Verdier, Pierre Deligne, and collaborators, the formalism provides a robust set of axioms and compatibilities used in proofs by Alexander Beilinson, Joseph Bernstein, Pierre Deligne, and others. The six operations underpin major results related to the Weil conjectures, the theory of perverse sheaves, and modern approaches to the Langlands program.
Background and motivation
Grothendieck introduced new foundations for algebraic geometry in the context of Éléments de géométrie algébrique, aiming to generalize cohomological techniques used by Alexander Grothendieck in work with Jean-Pierre Serre and Jean-Louis Verdier on duality. Motivating problems included the Riemann–Roch theorem generalizations by Friedrich Hirzebruch and the proof of the Weil conjectures by Pierre Deligne, which relied on functorial properties of Étale cohomology as developed with inputs from Jean-Pierre Serre, Grothendieck, and the staff of the Institut des Hautes Études Scientifiques. The formalism codifies pushforward, pullback, tensor, and Hom operations that were already implicitly present in work of Alexander Grothendieck, Jean Giraud, and Henri Cartan.
Formal statement of the six operations
The six operations are typically presented for a suitable category of schemes or analytic spaces with a chosen coefficient category (e.g., constructible ℓ-adic sheaves or regular holonomic D-modules). For a morphism f:X→Y the operations are: - f^*: the derived inverse image, with inputs such as Étale topology and outputs relevant to Grothendieck topology contexts. - f_*: the derived direct image, used in contexts studied by Pierre Deligne and Alexander Beilinson. - f^!: the extraordinary inverse image, related to duality theorems of Jean-Louis Verdier and Alexander Grothendieck. - f_!: the direct image with proper support, appearing in proofs by Pierre Deligne for the Weil conjectures. - ⊗: the derived tensor product, central in work by Jean-Pierre Serre and Alexander Grothendieck. - RHom (or Hom): the derived internal Hom functor, exploited in duality theories developed by Jean-Louis Verdier and later by Joseph Bernstein.
The formalism is usually axiomatized within derived categories introduced by Jean-Louis Verdier and uses coherence notions from Alexander Grothendieck and Jean-Pierre Serre. Implementations involve contexts studied by Nicholas Katz and Gérard Laumon in arithmetic geometry and by Masaki Kashiwara and Takurō Mochizuki in analytic settings.
Properties and compatibilities
The six operations satisfy a web of compatibilities: adjunctions (f^* ⊣ f_* and f_! ⊣ f^!), base change isomorphisms associated to Cartesian squares used by Pierre Deligne and Gérard Laumon, projection formulas relating ⊗ and pushforward seen in work of Alexander Grothendieck and Jean-Pierre Serre, and duality statements generalizing Serre duality and Verdier duality formalized by Jean-Louis Verdier. These compatibilities are essential in arguments by Pierre Deligne on weights and in constructions by Alexander Beilinson of perverse sheaves and t-structures. Functoriality across compositions, compatibility with supports, and coherence conditions were clarified in later categorical frameworks by researchers such as Jacob Lurie and Dennis Gaitsgory.
Examples and key cases
Key instances include: - Derived categories of constructible ℓ-adic sheaves on schemes over finite fields, central to Pierre Deligne's proof of the Weil conjectures and to work by Nicholas Katz and Gérard Laumon. - Derived categories of regular holonomic D-modules on complex algebraic varieties, studied by Masaki Kashiwara and Mikio Sato in microlocal analysis and by Zoghman Mebkhout. - Complex analytic sheaves in the topology of François Le Lionnais-style contexts and perverse sheaves developed by Alexander Beilinson, Joseph Bernstein, and Pierre Deligne. - Étale sheaves on arithmetic schemes appearing in investigations by Jean-Pierre Serre and John Tate in number theory and cohomological approaches to the Langlands program by Robert Langlands and Michael Harris.
Each case realizes the six operations with specific technical hypotheses and yields concrete computational tools for intersection cohomology as developed by Mark Goresky and Robert MacPherson.
Applications in algebraic geometry and number theory
The six operations are foundational in modern approaches to the Langlands program, the study of L-functions by Pierre Deligne and Nicholas Katz, and the analysis of monodromy in families of varieties used by Alexander Grothendieck and Jean-Pierre Serre. They enable comparison theorems such as those linking De Rham cohomology and Étale cohomology in the work of Alexander Beilinson, Joseph Bernstein, and Pierre Deligne, and they underpin the theory of perverse sheaves applied to representation theory explored by Anthony Joseph and David Kazhdan. Arithmetic applications include the study of Galois representations by Richard Taylor and Freddie Diamond, and geometric applications include categorical formulations used by Dennis Gaitsgory and Jacob Lurie in geometric representation theory.
Variations and generalizations
Generalizations extend the formalism to derived algebraic geometry as developed by Jacob Lurie and Bertrand Toën, to higher categorical settings by Jacob Lurie and Dylan Gonçalves-style contributors, and to p-adic contexts explored by Jean-Marc Fontaine and Pierre Colmez. Noncommutative variants appear in work by Maxim Kontsevich and Alexander Rosenberg, while mixed-characteristic and perfectoid adaptations involve Peter Scholze and collaborators. Alternative axiomatizations and enhancements incorporate ideas from Paul Balmer on tensor-triangulated categories and from Amnon Neeman on Brown representability and localization.