scholarly journals Dieudonné theory over semiperfect rings and perfectoid rings

2018 ◽  
Vol 154 (9) ◽  
pp. 1974-2004 ◽  
Author(s):  
Eike Lau
Keyword(s):  
Divisible Group ◽  
Boundedness Condition ◽  
Group Schemes ◽  
Equivalence Of Categories ◽  
Semiperfect Ring ◽  
Formula Group ◽  
Divisible Groups ◽  
Dieudonne Theory

The Dieudonné crystal of a $p$-divisible group over a semiperfect ring $R$ can be endowed with a window structure. If $R$ satisfies a boundedness condition, this construction gives an equivalence of categories. As an application we obtain a classification of $p$-divisible groups and commutative finite locally free $p$-group schemes over perfectoid rings by Breuil–Kisin–Fargues modules if $p\geqslant 3$.

ON -DIVISIBLE GROUPS WITH SATURATED NEWTON POLYGONS

2017 ◽  
Vol 232 ◽  
pp. 96-120
Author(s):  
SHUSHI HARASHITA
Keyword(s):  
Positive Characteristic ◽  
Newton Polygon ◽  
Divisible Group ◽  
Geometric Proof ◽  
Newton Polygons ◽  
Noetherian Scheme ◽  
Weil Divisor ◽  
Divisible Groups

This paper concerns the classification of isogeny classes of$p$-divisible groups with saturated Newton polygons. Let$S$be a normal Noetherian scheme in positive characteristic$p$with a prime Weil divisor$D$. Let${\mathcal{X}}$be a$p$-divisible group over$S$whose geometric fibers over$S\setminus D$(resp. over$D$) have the same Newton polygon. Assume that the Newton polygon of${\mathcal{X}}_{D}$is saturated in that of${\mathcal{X}}_{S\setminus D}$. Our main result (Corollary 1.1) says that${\mathcal{X}}$is isogenous to a$p$-divisible group over$S$whose geometric fibers are all minimal. As an application, we give a geometric proof of the unpolarized analogue of Oort’s conjecture (Oort, J. Amer. Math. Soc.17(2) (2004), 267–296; 6.9).

scholarly journals 2-ADIC INTEGRAL CANONICAL MODELS

2016 ◽  
Vol 4 ◽  
Author(s):  
WANSU KIM ◽  
KEERTHI MADAPUSI PERA
Keyword(s):  
Shimura Varieties ◽  
Canonical Models ◽  
Divisible Groups

We use Lau’s classification of 2-divisible groups using Dieudonné displays to construct integral canonical models for Shimura varieties of abelian type at 2-adic places where the level is hyperspecial.

scholarly journals The Supersingular Locus of the Shimura Variety for GU(1, s)

2010 ◽  
Vol 62 (3) ◽  
pp. 668-720 ◽  
Author(s):  
Inken Vollaard
Keyword(s):  
Complete Intersection ◽  
Moduli Space ◽  
Unitary Group ◽  
Shimura Variety ◽  
Incidence Relation ◽  
Irreducible Components ◽  
Reduction Modulo ◽  
Tits Building ◽  
Divisible Groups ◽  
Dieudonne Theory

AbstractIn this paper we study the supersingular locus of the reduction modulopof the Shimura variety for GU(1,s) in the case of an inert primep. Using Dieudonné theory we define a stratification of the corresponding moduli space ofp-divisible groups. We describe the incidence relation of this stratification in terms of the Bruhat–Tits building of a unitary group.In the case of GU(1, 2), we show that the supersingular locus is equidimensional of dimension 1 and is of complete intersection. We give an explicit description of the irreducible components and their intersection behaviour.

scholarly journals Complex manifolds with maximal torus actions

2019 ◽  
Vol 2019 (751) ◽  
pp. 121-184 ◽  
Author(s):  
Hiroaki Ishida
Keyword(s):  
Complete Classification ◽  
Vector Subspace ◽  
Toric Geometry ◽  
Complex Manifolds ◽  
Combinatorial Objects ◽  
Complex Dimension ◽  
Compact Torus ◽  
Equivalence Of Categories ◽  
Compact Tori

AbstractIn this paper, we introduce the notion of maximal actions of compact tori on smooth manifolds and study compact connected complex manifolds equipped with maximal actions of compact tori. We give a complete classification of such manifolds, in terms of combinatorial objects, which are triples {(\Delta,\mathfrak{h},G)} of nonsingular complete fan Δ in {\mathfrak{g}}, complex vector subspace {\mathfrak{h}} of {\mathfrak{g}^{\mathbb{C}}} and compact torus G satisfying certain conditions. We also give an equivalence of categories with suitable definitions of morphisms in these families, like toric geometry. We obtain several results as applications of our equivalence of categories; complex structures on moment-angle manifolds, classification of holomorphic nondegenerate {\mathbb{C}^{n}}-actions on compact connected complex manifolds of complex dimension n, and construction of concrete examples of non-Kähler manifolds.

scholarly journals A categorical invariant of flow equivalence of shifts

2014 ◽  
Vol 36 (2) ◽  
pp. 470-513 ◽  
Author(s):  
ALFREDO COSTA ◽  
BENJAMIN STEINBERG
Keyword(s):  
Analogous Result ◽  
Mild Conditions ◽  
Equivalence Of Categories ◽  
Natural Equivalence ◽  
Sofic Shifts ◽  
Flow Equivalence ◽  
Flow Invariance ◽  
Natural Sense

We prove that the Karoubi envelope of a shift—defined as the Karoubi envelope of the syntactic semigroup of the language of blocks of the shift—is, up to natural equivalence of categories, an invariant of flow equivalence. More precisely, we show that the action of the Karoubi envelope on the Krieger cover of the shift is a flow invariant. An analogous result concerning the Fischer cover of a synchronizing shift is also obtained. From these main results, several flow equivalence invariants—some new and some old—are obtained. We also show that the Karoubi envelope is, in a natural sense, the best possible syntactic invariant of flow equivalence of sofic shifts. Another application concerns the classification of Markov–Dyck and Markov–Motzkin shifts: it is shown that, under mild conditions, two graphs define flow equivalent shifts if and only if they are isomorphic. Shifts with property ($\mathscr{A}$) and their associated semigroups, introduced by Wolfgang Krieger, are interpreted in terms of the Karoubi envelope, yielding a proof of the flow invariance of the associated semigroups in the cases usually considered (a result recently announced by Krieger), and also a proof that property ($\mathscr{A}$) is decidable for sofic shifts.

scholarly journals Local Shtukas and Divisible Local Anderson Modules

2019 ◽  
Vol 71 (5) ◽  
pp. 1163-1207 ◽  
Author(s):  
Urs Hartl ◽  
Rajneesh Kumar Singh
Keyword(s):  
Lie Groups ◽  
Function Fields ◽  
Drinfeld Modules ◽  
Arbitrary Base ◽  
Divisible Groups ◽  
Dieudonne Theory

AbstractWe develop the analog of crystalline Dieudonné theory for$p$-divisible groups in the arithmetic of function fields. In our theory$p$-divisible groups are replaced by divisible local Anderson modules, and Dieudonné modules are replaced by local shtukas. We show that the categories of divisible local Anderson modules and of effective local shtukas are anti-equivalent over arbitrary base schemes. We also clarify their relation with formal Lie groups and with global objects like Drinfeld modules, Anderson’s abelian$t$-modules and$t$-motives, and Drinfeld shtukas. Moreover, we discuss the existence of a Verschiebung map and apply it to deformations of local shtukas and divisible local Anderson modules. As a tool we use Faltings’s and Abrashkin’s theories of strict modules, which we review briefly.

scholarly journals On non-normal cyclic subgroups of prime order or order 4 of finite groups

2021 ◽  
Vol 19 (1) ◽  
pp. 63-68
Author(s):  
Pengfei Guo ◽  
Zhangjia Han
Keyword(s):  
Prime Order ◽  
Sufficient Conditions ◽  
Prime Power Order ◽  
Necessary And Sufficient Condition ◽  
Transitive Relation ◽  
Cyclic Subgroups ◽  
Formula Group ◽  
A Finite Group ◽  
Necessary And Sufficient

Abstract In this paper, we call a finite group G G an N L M NLM -group ( N C M NCM -group, respectively) if every non-normal cyclic subgroup of prime order or order 4 (prime power order, respectively) in G G is contained in a non-normal maximal subgroup of G G . Using the property of N L M NLM -groups and N C M NCM -groups, we give a new necessary and sufficient condition for G G to be a solvable T T -group (normality is a transitive relation), some sufficient conditions for G G to be supersolvable, and the classification of those groups whose all proper subgroups are N L M NLM -groups.

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