Relative P-adic Hodge Theory: Foundations

By Kiran S. Kedlaya, Ruochuan Liu. 2013

232 citations recorded by the source corpus.

We describe a new approach to relative p-adic Hodge theory based on systematic use of Witt vector constructions and nonarchimedean analytic geometry in the style of both Berkovich and Huber. We give a thorough development of phi-modules over a relative Robba ring associated to a perfect Banach ring of characteristic p, including the relationship between these objects and etale Z_p-local systems and Q_p-local systems on the algebraic and analytic spaces associated to the base ring, and the relationship between etale cohomology and phi-cohomology. We also make a critical link to mixed characteristic by exhibiting an equivalence of tensor categories between the finite etale algebras over an arbitrary perfect Banach algebra over a nontrivially normed complete field of characteristic p and the finite etale algebras over a corresponding Banach Q_p-algebra. This recovers the homeomorphism between the absolute Galois groups of F_p((pi)) and Q_p(mu_{p^infty}) given by the field of norms construction of Fontaine and Wintenberger, as well as generalizations considered by Andreatta, Brinon, Faltings, Gabber, Ramero, Scholl, and most recently Scholze. Using Huber's formalism of adic spaces and Scholze's formalism of perfectoid spaces, we globalize the constructions to give several descriptions of the etale local systems on analytic spaces over p-adic fields. One of these descriptions uses a relative version of the Fargues-Fontaine curve.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/292e83205c1d6343d4910e92622209cd3e36cb4a

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Citations in the corpus

Relative p-adic Hodge theory, II: Imperfect period rings (2016, methodology): “The obvious difference between the results we have just described and the corresponding results from [79] is that we are now considering rings on which the action of Frobenius is injective but not surjective.”
On torsion in the cohomology of locally symmetric varieties (2013, unknown): “osed map in characteristic 0 is strongly Zariski closed if and only if the tilt is still Zariski closed. For completeness, let us mention the following result that appears in the work of Kedlaya-Liu, [32]; we will not need this result in our work. Lemma II.2.8 ([32, Proposition 3.6.9 (c)]). Let R→Sb”
On the p-adic cohomology of the Lubin-Tate tower (2015, methodology): “tations, II. Camb. J. Math., 1(1):53–73, 2013. [11] P. Chojecki. On mod p non-abelian Lubin-Tate theory for GL 2(Qp). http://people.maths.ox.ac.uk/chojecki/mod p LT.pdf, to appear in Compositio Math. [12] P. Colmez. Repr´esentations de GL 2(Qp) et (φ,Γ)-modules. Ast´erisque, (330):281–509, 2010. [13”
Geometrization of the local Langlands correspondence: an overview (2016, background): “Then a theorem of Kedlaya and Liu ([16]) says that for d ≫ 0, E (d) is generated by its global sections (that is to say O(1) is ample).”
p-ADIC GEOMETRY (2017, methodology): “dL= (SpaL)}. The diamond M (G;b;);K= M (G;b;);1=Kover SpdE is the image of a smooth rigid space over E under this functor. The proof makes use of the results of Kedlaya, [Ked10], and Kedlaya-Liu, [KL15], on families of vector bundles on the Fargues-Fontaine curve. From here, it becomes natural to co”