Relative p-adic Hodge theory, II: Imperfect period rings

By Kiran S. Kedlaya, Ruochuan Liu. 2016

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In a previous paper, we constructed a category of (phi, Gamma)-modules associated to any adic space over Q_p with the property that the etale (phi, Gamma)-modules correspond to etale Q_p-local systems; these involve sheaves of period rings for Scholze's pro-etale topology. In this paper, we first extend Kiehl's theory of coherent sheaves on rigid analytic spaces to a theory of pseudocoherent sheaves on adic spaces, then construct a corresponding theory of pseudocoherent (phi, Gamma)-modules. We then relate these objects to a more explicit construction in case the space comes equipped with a suitable infinite etale cover; in this case, one can decomplete the period sheaves and establish an analogue of the theorem of Cherbonnier-Colmez on the overconvergence of p-adic Galois representations. As an application, we show that relative (phi, Gamma)-modules in our sense coincide with the relative (phi, Gamma)-modules constructed by Andreatta and Brinon in the geometric setting where the latter can be constructed. As another application, we establish that the category of pseudocoherent (phi, Gamma)-modules on an arbitrary rigid analytic space over a p-adic field is abelian, satisfies the ascending chain condition, and is stable under various natural derived functors (including Hom, tensor product, and pullback). Applications to the etale cohomology of pro-etale local systems will be given in a subsequent paper.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f84c2f55700d93e57721498c31a6c8439aa7cb64

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Citations in the corpus

Syntomic complexes and p-adic nearby cycles (2015, methodology): “to p, and the second follows by almost ´etale descent as in [18, 3.2,3.3], [3, §2], [4, §7]. Finally, we will prove (iii) by an adaptation of the usual decompletion techniques as in [4, Theorem7.16], [37]. By d´evissage, it is enough to prove the statement modulo p, in which case the twist disappear”
Rigidity and a Riemann–Hilbert correspondence for p-adic local systems (2016, methodology): “R0Y; Rf ; (L^ f proet OC Y): Putting (2.25) and (2.26) together, Part (v) follows. 2.4. Proof of Proposition 2.8. To prove Proposition 2.8, we will employ some key technical ingredients developed in [KL2]. By the main results of [KL2, x7], both toric towers and relative toric towers are locally deco”