l-adic cohomology
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| l-adic cohomology | |
|---|---|
| Name | l-adic cohomology |
| Field | Algebraic geometry |
| Introduced | 1960s |
| Creators | Alexandre Grothendieck; Michael Artin; Jean-Pierre Serre; Pierre Deligne |
| Related | Étale cohomology; Weil conjectures; Grothendieck topology |
l-adic cohomology l-adic cohomology is a p-adic-style cohomology theory for algebraic varieties over fields of characteristic different from a fixed prime l, developed to study arithmetic and geometric properties of schemes. It provides finite-dimensional vector spaces with continuous actions of Galois groups and Frobenius endomorphisms, linking work of Alexandre Grothendieck, Jean-Pierre Serre, Pierre Deligne, André Weil and computational techniques used by Alexander Grothendieck's school. The theory underlies proofs of the Weil conjectures, insights in the Langlands program, and relations to motives studied by Yves André, Grothendieck institutes and various research groups at Institut des Hautes Études Scientifiques and École Normale Supérieure.
Introduction
l-adic cohomology arose from efforts by Grothendieck and collaborators at the Institut des Hautes Études Scientifiques and Collège de France to create a cohomology theory satisfying the formal properties needed to attack the Weil conjectures posed by André Weil. Influenced by analogies with singular cohomology used in the work of Henri Poincaré and topological methods of Alexander Grothendieck's contemporaries at University of Paris, it replaces classical topological tools with the étale topology introduced by Grothendieck and technical machinery developed by Michael Artin and Jean-Pierre Serre. Subsequent breakthroughs by Pierre Deligne established the Riemann hypothesis over finite fields, cementing l-adic cohomology as central to modern algebraic geometry and arithmetic geometry explored at institutions like Princeton University and Harvard University.
Definitions and basic properties
An l-adic cohomology group is obtained as an inverse limit of cohomology groups with coefficients in the finite rings Z/l^nZ, producing a module over the l-adic integers Z_l and, after tensoring with Q_l, a vector space over the l-adic field Q_l. The construction relies on the étale site introduced by Grothendieck and uses derived functors analogous to those in the work of Alexander Grothendieck and Jean-Pierre Serre. Fundamental properties include finite dimensionality proven by adaptations of results by Michael Artin and cohomological dimension bounds used in proofs by Pierre Deligne and collaborators at Institut des Hautes Études Scientifiques. Poincaré duality and the Künneth formula hold in l-adic cohomology under hypotheses treated by Grothendieck's "six operations" formalism, developed with influence from Pierre Deligne and institutions like CNRS.
Construction and étale cohomology relation
The construction begins with the étale cohomology groups H^i(X, Z/l^nZ) for a scheme X and passes to the inverse limit to define H^i(X, Z_l) and H^i(X, Q_l). This methodology follows the foundational work of Michael Artin on cohomological methods and the formalism of derived categories influenced by Alexander Grothendieck and later formalized by researchers at University of Chicago and Harvard University. The close relation between l-adic cohomology and étale cohomology was clarified in seminars at Collège de France and in publications by Jean-Pierre Serre; it formalizes comparisons similar to those between singular cohomology used by Henri Poincaré and algebraic de Rham cohomology studied by Jean de Rham.
Functoriality, Galois action, and weights
l-adic cohomology is functorial for proper maps and open immersions via pullback and pushforward operations in the "six operations" framework developed by Grothendieck and refined by researchers at Institut des Hautes Études Scientifiques and École Normale Supérieure. For varieties over nonclosed fields, the absolute Galois group acts on l-adic cohomology groups; this Galois action plays a central role in applications to the Langlands program and in formulations by Jean-Pierre Serre regarding representations of Galois groups. Deligne's theory of weights assigns "weights" to eigenvalues of Frobenius acting on l-adic cohomology, a concept that was pivotal in his proof of the Riemann hypothesis for varieties over finite fields and was influenced by ideas from André Weil and institutions such as Institut des Hautes Études Scientifiques.
Comparison theorems and applications
Comparison theorems link l-adic cohomology with other cohomology theories: smooth varieties over C admit comparisons between l-adic cohomology and singular cohomology, reflecting work of Alexander Grothendieck and results disseminated at Institute for Advanced Study. Comparison with crystalline cohomology and de Rham cohomology for varieties in characteristic p involves contributions by Pierre Deligne, Alexander Grothendieck's school, and later developments at Princeton University and Harvard University. Applications are numerous: Deligne's proof of the Weil conjectures used l-adic cohomology; studies in the Langlands program connect l-adic Galois representations to automorphic forms explored at Institute for Advanced Study and Harvard University; and research on motives by Grothendieck, Yves André, and others frames l-adic cohomology as a realization functor essential to conjectures propagated in seminars at Collège de France.
Examples and computations
Explicit computations occur for curves, abelian varieties, K3 surfaces, and toric varieties, with classical cases analyzed by researchers at École Normale Supérieure and University of Paris. For a smooth projective curve of genus g, H^1 with Q_l-coefficients yields a 2g-dimensional Q_l-vector space with Galois action studied by Jean-Pierre Serre and André Weil-inspired techniques; for abelian varieties the Tate module formalism relates l-adic cohomology to John Tate's conjectures and works by Barry Mazur and Serre at Harvard University. Calculations of Frobenius eigenvalues for varieties over finite fields underpin enumerative results used by Pierre Deligne in the proof of the Weil conjectures, with computational methods refined in research groups at Princeton University and Institut des Hautes Études Scientifiques.
Historical development and influence
The development of l-adic cohomology in the 1960s was driven by Grothendieck's program to create a cohomological framework for algebraic geometry, with early formalization by Michael Artin and conceptual advances by Jean-Pierre Serre and Pierre Deligne. Deligne's resolution of the last of the Weil conjectures in the 1970s propelled l-adic methods into centrality, influencing the Langlands program and motivating further work on motives by Grothendieck, Yves André, and collaborators at Collège de France and IHÉS. The theory continues to inform contemporary research at institutions such as Princeton University, Harvard University, Institut des Hautes Études Scientifiques, and École Normale Supérieure, and guides investigations into Galois representations, automorphic forms, and arithmetic geometry carried out by many researchers worldwide.