scholarly journals Dieudonné theory via cohomology of classifying stacks

2021 ◽  
Vol 9 ◽  
Author(s):  
Shubhodip Mondal
Keyword(s):  
Abelian Variety ◽  
Group Scheme ◽  
Symmetric Algebra ◽  
Divisible Group ◽  
Perfect Field ◽  
Mixed Characteristic ◽  
Crystalline Cohomology ◽  
Dieudonne Theory ◽  
Dieudonné Module

Abstract We prove that if G is a finite flat group scheme of p-power rank over a perfect field of characteristic p, then the second crystalline cohomology of its classifying stack $H^2_{\text {crys}}(BG)$ recovers the Dieudonné module of G. We also provide a calculation of the crystalline cohomology of the classifying stack of an abelian variety. We use this to prove that the crystalline cohomology of the classifying stack of a p-divisible group is a symmetric algebra (in degree $2$ ) on its Dieudonné module. We also prove mixed-characteristic analogues of some of these results using prismatic cohomology.

scholarly journals Lattices in Tate modules

2021 ◽  
Vol 118 (49) ◽  
pp. e2113201118
Author(s):  
Bjorn Poonen ◽  
Sergey Rybakov
Keyword(s):  
Abelian Variety ◽  
Perfect Field ◽  
Tate Module ◽  
Dieudonné Module

Refining a theorem of Zarhin, we prove that, given a g-dimensional abelian variety X and an endomorphism u of X, there exists a matrix A∈M2g(ℤ) such that each Tate module TℓX has a ℤℓ-basis on which the action of u is given by A, and similarly for the covariant Dieudonné module if over a perfect field of characteristic p.

The fundamental group-scheme of an abelian variety

1983 ◽  
Vol 263 (3) ◽  
pp. 263-266 ◽  
Author(s):  
Madhav V. Nori
Keyword(s):  
Fundamental Group ◽  
Abelian Variety ◽  
Group Scheme

scholarly journals On the S-fundamental group scheme. II

2012 ◽  
Vol 11 (4) ◽  
pp. 835-854 ◽  
Author(s):  
Adrian Langer
Keyword(s):  
Fundamental Group ◽  
Abelian Variety ◽  
Vector Bundles ◽  
Group Scheme ◽  
Line Bundles ◽  
Fundamental Groups ◽  
Determinant Line Bundles

AbstractThe S-fundamental group scheme is the group scheme corresponding to the Tannaka category of numerically flat vector bundles. We use determinant line bundles to prove that the S-fundamental group of a product of two complete varieties is a product of their S-fundamental groups as conjectured by Mehta and the author. We also compute the abelian part of the S-fundamental group scheme and the S-fundamental group scheme of an abelian variety or a variety with trivial étale fundamental group.

K-groups of reciprocity functors for and abelian varieties

2014 ◽  
Vol 14 (3) ◽  
pp. 556-569 ◽  
Author(s):  
Kay Rülling ◽  
Takao Yamazaki
Keyword(s):  
Abelian Variety ◽  
Abelian Varieties ◽  
Perfect Field

AbstractWe prove that the K-group of reciprocity functors, defined by F. Ivorra and the first author, vanishes over a perfect field as soon as one of the reciprocity functors is and one is an abelian variety.

scholarly journals The Filtered Poincaré Lemma in Higher Level (With Applications to Algebraic Groups)

2008 ◽  
Vol 191 ◽  
pp. 79-110
Author(s):  
Bernard Le Stum ◽  
Adolfo Quirós
Keyword(s):  
Differential Forms ◽  
Algebraic Groups ◽  
Classical Case ◽  
Group Scheme ◽  
Crystalline Cohomology ◽  
Hodge Filtration ◽  
Rham Complex ◽  
De Rham ◽  
Poincaré Lemma ◽  
Invariant Differential

AbstractWe show that the Poincaré lemma we proved elsewhere in the context of crystalline cohomology of higher level behaves well with regard to the Hodge filtration. This allows us to prove the Poincaré lemma for transversal crystals of level m. We interpret the de Rham complex in terms of what we call the Berthelot-Lieberman construction and show how the same construction can be used to study the conormal complex and invariant differential forms of higher level for a group scheme. Bringing together both instances of the construction, we show that crystalline extensions of transversal crystals by algebraic groups can be computed by reduction to the filtered de Rham complexes. Our theory does not ignore torsion and, unlike in the classical case (m = 0), not all invariant forms are closed. Therefore, close invariant differential forms of level m provide new invariants and we exhibit some examples as applications.

scholarly journals Almost Homogeneous Varieties of Albanese Codimension One

2020 ◽  
Author(s):  
Bruno Laurent
Keyword(s):  
Group Scheme ◽  
Arbitrary Field ◽  
Neutral Component ◽  
Perfect Field ◽  
Codimension One ◽  
Normal Varieties ◽  
Characteristic 2 ◽  
Homogeneous Varieties

Abstract We classify almost homogeneous normal varieties of Albanese codimension $1$, defined over an arbitrary field. We prove that such a variety has a unique normal equivariant completion. Over a perfect field, the group scheme of automorphisms of this completion is smooth, except in one case in characteristic $2$, and we determine its (reduced) neutral component.

scholarly journals Non-archimedean hyperbolicity and applications

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ariyan Javanpeykar ◽  
Alberto Vezzani
Keyword(s):  
Abelian Variety ◽  
Moduli Space ◽  
Characteristic Zero ◽  
Projective Variety ◽  
Uniformization Theorem ◽  
Mixed Characteristic ◽  
Analytic Varieties ◽  
Fixed Part ◽  
Natural Analogues ◽  
Constant Morphism

Abstract Inspired by the work of Cherry, we introduce and study a new notion of Brody hyperbolicity for rigid analytic varieties over a non-archimedean field K of characteristic zero. We use this notion of hyperbolicity to show the following algebraic statement: if a projective variety admits a non-constant morphism from an abelian variety, then so does any specialization of it. As an application of this result, we show that the moduli space of abelian varieties is K-analytically Brody hyperbolic in equal characteristic 0. These two results are predicted by the Green–Griffiths–Lang conjecture on hyperbolic varieties and its natural analogues for non-archimedean hyperbolicity. Finally, we use Scholze’s uniformization theorem to prove that the aforementioned moduli space satisfies a non-archimedean analogue of the “Theorem of the Fixed Part” in mixed characteristic.

scholarly journals Exotic Torsion, Frobenius Splitting and the Slope Spectral Sequence

2007 ◽  
Vol 50 (4) ◽  
pp. 567-578 ◽  
Author(s):  
Kirti Joshi
Keyword(s):  
Spectral Sequence ◽  
Perfect Field ◽  
Frobenius Splitting ◽  
Crystalline Cohomology

AbstractIn this paper we show that any Frobenius split, smooth, projective threefold over a perfect field of characteristic p > 0 is Hodge–Witt. This is proved by generalizing to the case of threefolds a well-known criterion due to N. Nygaard for surfaces to be Hodge-Witt. We also show that the second crystalline cohomology of any smooth, projective Frobenius split variety does not have any exotic torsion. In the last two sections we include some applications.

scholarly journals Cycle classes in overconvergent rigid cohomology and a semistable Lefschetz theorem

2019 ◽  
Vol 155 (5) ◽  
pp. 1025-1045
Author(s):  
Christopher Lazda ◽  
Ambrus Pál
Keyword(s):  
Perfect Field ◽  
Global Function ◽  
Crystalline Cohomology ◽  
Tate Conjecture ◽  
Global Function Fields ◽  
Counter Example ◽  
Rigid Cohomology ◽  
De Rham ◽  
First Chern Class ◽  
Special Fibre

In this paper we prove a semistable version of the variational Tate conjecture for divisors in crystalline cohomology, showing that for $k$ a perfect field of characteristic $p$ , a rational (logarithmic) line bundle on the special fibre of a semistable scheme over $k\unicode[STIX]{x27E6}t\unicode[STIX]{x27E7}$ lifts to the total space if and only if its first Chern class does. The proof is elementary, using standard properties of the logarithmic de Rham–Witt complex. As a corollary, we deduce similar algebraicity lifting results for cohomology classes on varieties over global function fields. Finally, we give a counter-example to show that the variational Tate conjecture for divisors cannot hold with $\mathbb{Q}_{p}$ -coefficients.

scholarly journals -theory of one-dimensional rings via pro-excision

2013 ◽  
Vol 13 (2) ◽  
pp. 225-272 ◽  
Author(s):  
Matthew Morrow
Keyword(s):  
Finite Fields ◽  
Number Fields ◽  
Local Rings ◽  
Perfect Field ◽  
Finite Characteristic ◽  
Mixed Characteristic ◽  
One Dimensional ◽  
Curves Over Finite Fields

AbstractThis paper studies ‘pro-excision’ for the $K$-theory of one-dimensional, usually semi-local, rings and its various applications. In particular, we prove Geller’s conjecture for equal characteristic rings over a perfect field of finite characteristic, give results towards Geller’s conjecture in mixed characteristic, and we establish various finiteness results for the $K$-groups of singularities, covering both orders in number fields and singular curves over finite fields.

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