$p$ -ADIC HODGE THEORY FOR RIGID-ANALYTIC VARIETIES

By Peter Scholze. Forum of Mathematics, Pi, 2012

290 citations recorded by the source corpus.

Abstract We give proofs of de Rham comparison isomorphisms for rigid-analytic varieties, with coefficients and in families. This relies on the theory of perfectoid spaces. Another new ingredient is the pro-étale site, which makes all constructions completely functorial.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6a0d482501aebdd2e62be8c107914ad1bf9a0c7f

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Citations in the corpus

On the p-adic cohomology of the Lubin-Tate tower (2015, background): “The proof of this theorem follows closely the proof of finiteness of Fp-cohomology of proper (smooth) rigid spaces, [29].”
On torsion in the cohomology of locally symmetric varieties (2013, result): “The first key ingredient in its proof is a comparison result from p-adic Hodge theory with torsion coefficients proved in [54].”
Perfectoid Spaces: A survey (2013, methodology): “The methods of [29] have some consequences that were not included there.”
Relative p-adic Hodge theory, II: Imperfect period rings (2016, methodology): “The following example is based on a discussion with Peter Scholze.”
Foundations of Rigid Geometry I (2013, background): “[78][79]) and to rigid cohomology theory for algebraic varieties in positive characteristic.”
The pro-étale topology for schemes (2013, background): “tic point as it does not commute with ﬁnite coproducts (though it does c ommute with ﬁltered colimits and all limits). Remark 4.1.11. The site Xpro´et introduced in this paper differs from the one in [Sch13], temporarily denoted X′ pro´et. More precisely, there is a natural map µX : Shv(Xpro´et) →Sh”
Syntomic complexes and p-adic nearby cycles (2015, methodology): “uence (1.11), the Space Hi syn(XO K ,r)Q, i≤r, is a ﬁnite dimensional Qp-vector space: we have the quasi-isomorphism (1.8) with ´etale cohomology and Scholze proved ﬁnite dimensionality of the latter [42]. On the other hand, the Space (Hi dR(XK,tr)⊗KB + dR)/F ris a successive extension of C-vector s”
Syntomic cohomology and p-adic regulators for varieties over p-adic fields (2013, background): “5, take for A the abelian category of sheaves of abelian groups on the pro-étale site Spec(K)proét of Scholze [58, 3].”
Integral p$p$-adic Hodge theory (2016, unknown): “In Section 13, we give a similar hands-on presentation of a de Rham comparison isomorphism for rigid-analytic varieties over Cp, and show that it is compatible with the result from [58].”
Relative P-adic Hodge Theory: Foundations (2013, methodology): “Scholze uses the term tilting to refer to the relationship between F and L, as well as to the corresponding relationship between Banach algebras introduced in Theorem 3.6.5.”
Geometrization of the local Langlands correspondence: an overview (2016, unknown): “(2) If X is moreover normal and ν : Xpro-ét → Xét is the associated morphism of sites ν∗“ OX = OX where “ OX(R,R) = R if Spa(R,R) → X is pro-étale affinöıde perfectoid ([26]).”