Weights in rigid cohomology applications to unipotent F-isocrystals

Chiarellotto, Bruno

doi:10.1016/s0012-9593(98)80004-9

Cited by 26 publications

(21 citation statements)

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“…The upshot of the previous section is that we now have an affine group scheme π rig 1 (X/S, p) over the Tannakian category F-Isoc † (S/K) whose fibre (ignoring Frobenius structures) over any closed point s is the usual rigid fundamental group π rig 1 (X s , p s ) as defined by Chiarellotto and le Stum in [CLS99a]. In Chapter II of [Chi98], Chiarellotto defines a Frobenius isomorphism F * : π rig 1 (X s , p s ) ∼ → π rig 1 (X s , p s ), by using the fact that Frobenius pullback induces an automorphism of the category N Isoc † (X s /K). Since we have constructed π rig 1 (X/S, p) as an affine group scheme over F-Isoc † (S/K), it comes with a Frobenius structure that we can compare with Chiarellotto's.…”

Section: Extension To Proper Curves Frobenius Structuresmentioning

confidence: 99%

Relative fundamental groups and rational points

Lazda¹

2015

Rend. Sem. Mat. Univ. Padova

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ABSTRACT. In this paper we define a relative rigid fundamental group, which associates to a section p of a smooth and proper morphism f : X → S in characteristic p, with dim S = 1, a Hopf algebra in the ind-category of overconvergent F-isocrystals on S. We prove a base change property, which says that the fibres of this object are the Hopf algebras of the rigid fundamental groups of the fibres of f . We explain how to use this theory to define period maps as Kim does for varieties over number fields, and show in certain cases that the targets of these maps can be interpreted as varieties. CONTENTS

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Section: Extension To Proper Curves Frobenius Structuresmentioning

confidence: 99%

Relative fundamental groups and rational points

Lazda¹

2015

Rend. Sem. Mat. Univ. Padova

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“…The associated spectral sequence for the cohomology of A • • then reads, using the (quasi)isomorphisms (1) Proof: Assertions (i), (iii) and (iv) follow easily from the spectral sequence (4) (which in cases (iii) and (iv) is Frobenius equivariant) and the corresponding results for the rigid cohomology with constant coefficients of (classically smooth) k-schemes, see [4] [5]. For (ii) observe that we can repeat all constructions using rigid spaces instead of dagger spaces, obtaining the spectral sequence…”

Section: Choose An Open Coveringmentioning

confidence: 99%

Frobenius and monodromy operators in rigid analysis, and Drinfel’d’s symmetric space

Grosse-Klönne¹

2005

J. Algebraic Geom.

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We define Frobenius and monodromy operators on the de Rham cohomology of K-dagger spaces (rigid spaces with overconvergent structure sheaves) with strictly semistable reduction Y , over a complete discrete valuation ring K of mixed characteristic. For this we introduce log rigid cohomology and generalize the so called Hyodo-Kato isomorphism to versions for non-proper Y , for non-perfect residue fields, for non-integrally defined coefficients, and for the various strata of Y . We apply this to define and investigate crystalline structure elements on the de Rham cohomology of Drinfel'd's symmetric space X and its quotients. Our results are used in a critical way in the recent proof of the monodromy-weight conjecture for quotients of X given by de Shalit [7].

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“…This now follows from the theory of weights. By [ClS98] and [Chi97] the K 0 -linear Frobenius, which is a power of φ, has weight j when acting on H j rig (X κ /K 0 ) when X is proper and has mixed weights between j and 2j in general. In the proper case it follows that if 2n = j for j = i − 2, i − 1 and i, then the operator φ/p n has no fixed vector on H j rig (X κ /K 0 ) because some power of it does not.…”

Section: It Follows That the Mapmentioning

confidence: 99%