scholarly journals -ADIC HODGE THEORY FOR RIGID-ANALYTIC VARIETIES

2013 ◽  
Vol 1 ◽  
Author(s):  
PETER SCHOLZE
Keyword(s):  
Hodge Theory ◽  
Analytic Varieties ◽  
De Rham ◽  
Perfectoid Spaces

AbstractWe 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.

scholarly journals -ADIC HODGE THEORY FOR RIGID-ANALYTIC VARIETIES – CORRIGENDUM

2016 ◽  
Vol 4 ◽  
Author(s):  
PETER SCHOLZE
Keyword(s):  
Hodge Theory ◽  
Analytic Varieties

The author would like to make some changes to the previously published article [1] by correcting two definitions.

scholarly journals The geometry of Hida families II: -adic -modules and -adic Hodge theory

2018 ◽  
Vol 154 (4) ◽  
pp. 719-760
Author(s):  
Bryden Cais
Keyword(s):  
Galois Representations ◽  
Hodge Theory ◽  
Geometric Proof ◽  
De Rham Cohomology ◽  
Etale Cohomology ◽  
The Family ◽  
Étale Cohomology ◽  
De Rham ◽  
Hida Families ◽  
And Control

We construct the $\unicode[STIX]{x1D6EC}$-adic crystalline and Dieudonné analogues of Hida’s ordinary $\unicode[STIX]{x1D6EC}$-adic étale cohomology, and employ integral $p$-adic Hodge theory to prove $\unicode[STIX]{x1D6EC}$-adic comparison isomorphisms between these cohomologies and the $\unicode[STIX]{x1D6EC}$-adic de Rham cohomology studied in Cais [The geometry of Hida families I:$\unicode[STIX]{x1D6EC}$-adic de Rham cohomology, Math. Ann. (2017), doi:10.1007/s00208-017-1608-1] as well as Hida’s $\unicode[STIX]{x1D6EC}$-adic étale cohomology. As applications of our work, we provide a ‘cohomological’ construction of the family of $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules attached to Hida’s ordinary $\unicode[STIX]{x1D6EC}$-adic étale cohomology by Dee [$\unicode[STIX]{x1D6F7}$–$\unicode[STIX]{x1D6E4}$modules for families of Galois representations, J. Algebra 235 (2001), 636–664], and we give a new and purely geometric proof of Hida’s finiteness and control theorems. We also prove suitable $\unicode[STIX]{x1D6EC}$-adic duality theorems for each of the cohomologies we construct.

scholarly journals On admissible tensor products in p-adic Hodge theory

2013 ◽  
Vol 149 (3) ◽  
pp. 417-429 ◽  
Author(s):  
Giovanni Di Matteo
Keyword(s):  
Tensor Product ◽  
Tensor Products ◽  
Hodge Theory ◽  
Schur Functor ◽  
De Rham

AbstractWe prove that if W and W′ are non-zero B-pairs whose tensor product is crystalline (or semi-stable or de Rham or Hodge–Tate), then there exists a character μ such that W(μ−1) and W′(μ) are crystalline (or semi-stable or de Rham or Hodge–Tate). We also prove that if W is a B-pair and if F is a Schur functor (for example Sym n or Λn) such that F(W) is crystalline (or semi-stable or de Rham or Hodge–Tate) and if the rank of W is sufficiently large, then there is a character μ such that W(μ−1) is crystalline (or semi-stable or de Rham or Hodge–Tate). In particular, these results apply to p-adic representations.

scholarly journals Hodge theory in the Sobolev topology for the de Rham complex

1998 ◽  
Vol 131 (622) ◽  
pp. 0-0 ◽  
Author(s):  
Luigi Fontana ◽  
Steven G. Krantz ◽  
Marco M. Peloso
Keyword(s):  
Hodge Theory ◽  
De Rham Complex ◽  
Rham Complex ◽  
De Rham

scholarly journals NONCOMMUTATIVE GAUGE THEORIES: MODEL FOR HODGE THEORY

2013 ◽  
Vol 28 (25) ◽  
pp. 1350122 ◽  
Author(s):  
SUDHAKER UPADHYAY ◽  
BHABANI PRASAD MANDAL
Keyword(s):  
Differential Geometry ◽  
Compact Manifold ◽  
Gauge Theories ◽  
Brst Symmetry ◽  
Decomposition Theorem ◽  
Hodge Theory ◽  
De Rham ◽  
Symmetry Transformations ◽  
Moyal Plane ◽  
Noncommutative Gauge Theories

The nilpotent Becchi–Rouet–Stora–Tyutin (BRST), anti-BRST, dual-BRST and anti-dual-BRST symmetry transformations are constructed in the context of noncommutative (NC) 1-form as well as 2-form gauge theories. The corresponding Noether's charges for these symmetries on the Moyal plane are shown to satisfy the same algebra, as by the de Rham cohomological operators of differential geometry. The Hodge decomposition theorem on compact manifold is also studied. We show that noncommutative gauge theories are field theoretic models for Hodge theory.

scholarly journals Hodge theory on Cheeger spaces

2016 ◽  
Vol 0 (0) ◽  
Author(s):  
Pierre Albin ◽  
Eric Leichtnam ◽  
Rafe Mazzeo ◽  
Paolo Piazza
Keyword(s):  
Boundary Conditions ◽  
Hodge Theory ◽  
Intersection Theory ◽  
Ideal Boundary ◽  
Compact Resolvent ◽  
De Rham

AbstractWe extend the study of the de Rham operator with ideal boundary conditions from the case of isolated conic singularities, as analyzed by Cheeger, to the case of arbitrary stratified pseudomanifolds. We introduce a class of ideal boundary conditions and the notion of mezzoperversity, which intermediates between the standard lower and upper middle perversities in intersection theory, as interpreted in this de Rham setting, and show that the de Rham operator with these boundary conditions is Fredholm and has compact resolvent. We also prove an isomorphism between the resulting Hodge and

Mixed-characteristic shtukas

2020 ◽  
pp. 90-97
Author(s):  
Peter Scholze ◽  
Jared Weinstein
Keyword(s):  
Vector Bundle ◽  
Hodge Theory ◽  
Mixed Characteristic ◽  
Object A ◽  
Fiber Product ◽  
Perfectoid Spaces ◽  
Test Objects

This chapter looks at mixed-characteristic shtukas. Much of the theory of mixed-characteristic shtukas is motivated by the structures appearing in (integral) p-adic Hodge theory. The chapter assesses Drinfeld's shtukas and local shtukas. In the mixed characteristic setting, X will be replaced with Spa Zp. The test objects S will be drawn from Perf, the category of perfectoid spaces in characteristic p. For an object, a shtuka over S should be a vector bundle over an adic space, together with a Frobenius structure. The product is not meant to be taken literally (if so, one would just recover S), but rather it is to be interpreted as a fiber product over a deeper base. Motivated by this, the chapter then defines an analytic adic space and shows that its associated diamond is the appropriate product of sheaves on Perf.

scholarly journals De Rham-Hodge theory for L2-cohomology of infinite coverings

Topology ◽  
1977 ◽  
Vol 16 (2) ◽  
pp. 157-165 ◽  
Author(s):  
Jozef Dodziuk
Keyword(s):  
Hodge Theory ◽  
L2 Cohomology ◽  
De Rham
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