Leseprobe
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The p-adic Simpson Correspondence (AM-193)
Takeshi Tsuji, Michel Gros, Ahmed Abbes, et al.
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
Beschreibung
The p-adic Simpson correspondence, recently initiated by Gerd Faltings, aims at describing all p-adic representations of the fundamental group of a proper smooth variety over a p-adic field in terms of linear algebra—namely Higgs bundles. This book undertakes a systematic development of the theory following two new approaches, one by Ahmed Abbes and Michel Gros, the other by Takeshi Tsuji. The authors mainly focus on generalized representations of the fundamental group that are p-adically close to the trivial representation.
The first approach relies on a new family of period rings built from the torsor of deformations of the variety over a universal p-adic thickening defined by J. M. Fontaine. The second approach introduces a crystalline-type topos and replaces the notion of Higgs bundles with that of Higgs isocrystals. The authors show the compatibility of the two constructions and the compatibility of the correspondence with the natural cohomologies. The last part of the volume contains results of wider interest in p-adic Hodge theory. The reader will find a concise introduction to Faltings' theory of almost étale extensions and a chapter devoted to the Faltings topos. Though this topos is the general framework for Faltings' approach in p-adic Hodge theory, it remains relatively unexplored. The authors present a new approach based on a generalization of P. Deligne's covanishing topos.
Kundenbewertungen
Direct sum, Galois cohomology, Identity element, Presheaf (category theory), Irreducible component, Higgs bundle, Spectral sequence, Commutative diagram, Exact sequence, Summation, Affine transformation, Integral domain, Automorphism, Ring homomorphism, Discrete valuation ring, Equivalence of categories, Zariski topology, Initial and terminal objects, Divisibility rule, Commutative property, Generic point, Derived category, Formal scheme, Inverse image functor, Closed immersion, Subgroup, Maximal ideal, Sheaf (mathematics), Coefficient, Inverse limit, Endomorphism, Integral element, Field of fractions, Exact functor, Connected component (graph theory), Inverse system, Residue field, Corollary, P-adic number, Algebraic closure, Fundamental group, Cohomology, Galois group, Profinite group, Base change, Tensor product, Morphism, Cokernel, Homomorphism, Monoid, Mathematical induction, Morphism of schemes, Adjoint functors, Diagram (category theory), Direct limit, Integer, Computation, Vector bundle, Rational number, Theorem, Topology, Torsor (algebraic geometry), Valuation ring, Existential quantification, Functor, Commutative ring, Embedding, Covering space, Hodge theory, Logarithm