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| coherent sheaves | |
|---|---|
| Name | Coherent sheaves |
| Discipline | Algebraic geometry |
| Introduced | 1950s |
| Key people | Jean-Pierre Serre, Alexander Grothendieck, Henri Cartan |
| Related concepts | Sheaf (mathematics), Scheme (mathematics), Complex manifold, Derived category |
coherent sheaves Coherent sheaves are a class of sheaves of modules that play a central role in modern algebraic geometry, complex analytic geometry, and the theory of derived categories. They generalize vector bundles and ideal sheaves and provide a natural language for formulating finiteness conditions in the study of schemes, varieties, and complex manifolds. Historically motivated by work of Serre, Grothendieck, and Cartan, coherent sheaves connect to topics such as duality theorems, deformation theory, and moduli problems.
A coherent sheaf on a locally ringed space is a sheaf of modules that is locally of finite presentation and whose kernels of local presentations are finitely generated; this notion was formalized by Jean-Pierre Serre and systematically developed in the context of schemes by Alexander Grothendieck. In the algebraic setting one often works with quasi-coherent sheaves on a Scheme (mathematics) and then imposes coherence to ensure finiteness properties akin to finitely generated modules over a Noetherian ring; related finiteness conditions appear in Grothendieck's work on Éléments de géométrie algébrique and in Serre's foundational paper linking coherent sheaves to projective geometry. Coherent sheaves exhibit stability under local operations such as kernels, cokernels, and extensions, mirroring properties found in the abelian categories studied by Pierre Deligne and others in the theory of Abelian category.
Basic examples include locally free sheaves (vector bundles) arising from projective modules studied by Emmy Noether-era algebraists, ideal sheaves associated to subschemes important in the work of Oscar Zariski and Kunihiko Kodaira, and structure sheaves of closed subschemes used throughout the literature on Hilbert scheme and Grassmannian. Coherent pushforwards and direct images appear in the work of Jean-Louis Verdier and are essential in constructing sheaves via finite morphisms such as those considered by Issai Schur-era representation theory. Operations like tensor product with a coherent sheaf, Hom-sheaves, and symmetric and exterior powers give rise to further coherent examples connected to vector bundle constructions used by Michael Atiyah and Raoul Bott in index theory and characteristic class computations.
Cohomology groups of coherent sheaves were central to Jean-Pierre Serre's GAGA comparison results linking algebraic and analytic geometry and to Alexander Grothendieck's development of sheaf cohomology in Éléments de géométrie algébrique. Theorems such as Serre vanishing, Cartan's Theorem B for Stein spaces studied by Henri Cartan, and Grothendieck's finiteness results control the dimensions of cohomology groups and underlie the construction of moduli spaces by David Mumford and Pierre Deligne. Duality theorems, including Serre duality and Grothendieck duality, relate cohomology of coherent sheaves to Ext groups and residues in the tradition of Alexander Grothendieck's duality formalism and are instrumental in proofs by Gérard Laumon and Laurent Lafforgue linking geometry to arithmetic.
Standard operations—direct image, inverse image, tensor product, Hom, Ext, and derived functors—are defined and studied in the context of coherent sheaves by Grothendieck and his collaborators; these operations interact with properness and flatness conditions familiar from the work of Oscar Zariski, Pierre Deligne, and Grothendieck himself. Derived pushforward and pullback functors are central to the theory of perverse sheaves and the formulation of theorems by Alexei Beilinson, Joseph Bernstein, and Pierre Deligne on t-structures and decomposition theorems. Resolutions by locally free coherent sheaves, injective resolutions in derived categories, and spectral sequences are technical tools used extensively in research by Jean-Louis Verdier and Amnon Neeman.
On a Noetherian scheme, coherent sheaves coincide with finitely generated modules over the structure sheaf and form an abelian category that is crucial for the study of projective varieties developed by David Mumford, Federigo Enriques, and Oscar Zariski. In complex analytic geometry coherent analytic sheaves on complex manifolds were studied by Henri Cartan and Kiyoshi Oka and are subject to results like Cartan's Theorems A and B, with applications to the classification theories of Kunihiko Kodaira and Harold Davenport. Comparison theorems, including Serre's GAGA, relate coherent algebraic sheaves on projective varieties to coherent analytic sheaves on complex manifolds, bridging work by Jean-Pierre Serre and subsequent developments by Armand Borel and François Charles.
Coherent sheaves are indispensable in describing divisors, linear systems, and canonical sheaves used by David Mumford in geometric invariant theory and by Kunihiko Kodaira in classification of surfaces; they underpin construction of moduli spaces such as the Moduli space of stable curves and moduli of sheaves investigated by Simon Donaldson, Shing-Tung Yau, and Maxim Kontsevich. In derived categories, bounded derived categories of coherent sheaves provided the setting for homological mirror symmetry conjectures proposed by Maxim Kontsevich and later developed by Paul Seidel and Dmitry Orlov; these categories and their enhancements connect to Fourier–Mukai transforms studied by Alexander Polishchuk and Andrei Caldararu and to noncommutative geometry approaches by Maxim Kontsevich and Alain Connes. Coherent sheaves also appear in arithmetic geometry in the study of étale cohomology, perverse sheaves, and the proof of major results by Pierre Deligne and Pierre-Louis Lions.