There are many almost strongly minimal generalized n-gons that do not interpret an infinite group

1998 ◽  
Vol 63 (2) ◽  
pp. 485-508 ◽  
Author(s):  
Mark J. Debonis ◽  
Ali Nesin
Keyword(s):  
Amalgamation Property ◽  
Projective Planes ◽  
Infinite Group ◽  
Closed Field ◽  
Morley Rank ◽  
Algebraically Closed Field ◽  
Finite Graphs ◽  
Incidence Geometries ◽  
Finite Morley Rank ◽  
Carry Over

Generalized n-gons are certain geometric structures (incidence geometries) that generalize the concept of projective planes (the nontrivial generalized 3-gons are exactly the projective planes).In a simplified world, every generalized n-gon of finite Morley rank would be an algebraic one, i.e., one of the three families described in [9] for example. To our horror, John Baldwin [2], using methods discovered by Hrushovski [7], constructed ℵ1-categorical projective planes which are not algebraic. The projective planes that Baldwin constructed fail to be algebraic in a dramatic way.Indeed, every algebraic projective plane over an algebraically closed field is Desarguesian [12]. In particular, an algebraically closed field (isomorphic to the base field) can be interpreted in every one of them. However, in the projective planes that Baldwin constructed, one cannot even interpret an infinite group.In this article we show that the same phenomenon occurs for the generalized n-gons if n ≥ 3 is an odd integer. For each such n we construct many nonisomorphic generalized n-gons of finite Morley rank that do not interpret an infinite group. As one may expect, our method is inspired by Hrushovski and Baldwin, and we follow Baldwin's line of approach. Quite often our proofs are a verification of the fact that the proofs of Baldwin [2] for n = 3 carry over to an arbitrary positive odd integer n (which is sometimes far from being obvious). As in [2], we begin by defining a certain collection of finite graphs K* and a binary relation ≤ on these graphs. We show that (K*, ≤) satisfies the amalgamation property.

On bad groups, bad fields, and pseudoplanes

1991 ◽  
Vol 56 (3) ◽  
pp. 915-931 ◽  
Author(s):  
Ali Nesin
Keyword(s):  
Projective Planes ◽  
Closed Field ◽  
Morley Rank ◽  
Algebraically Closed Field ◽  
Desarguesian Projective Planes ◽  
Natural Geometry ◽  
Special Cases ◽  
Finite Morley Rank ◽  
Rank 2 ◽  
Open Question

Cherlin introduced the concept of bad groups (of finite Morley rank) in [Ch1]. The existence of such groups is an open question. If they exist, they will contradict the Cherlin-Zil'ber conjecture that states that an infinite simple group of finite Morley rank is a Chevalley group over an algebraically closed field. In this paper, we prove that bad groups of finite Morley rank 3 act on a natural geometry Γ (namely on a special pseudoplane; see Corollary 20) sharply flag-transitively.We show that Γ is not very far from being a projective plane and when it is so rk(Γ) = 2 and Γ is not Desarguesian (Theorem 2). Baldwin [Ba] recently discovered non-Desarguesian projective planes of Morley rank 2. This discovery, together with this paper, makes the existence of bad groups (also of bad fields) more plausible. A bad field is a pair (K, A) of finite Morley rank, where K is an algebraically closed field, A <≠K* and A is infinite. There existence is also unknown.In this paper, we define the concept of a sharp-field as a pair (K, A), where K is a field, A < K*and1. K = A − A,2. If a + b − 1 ∈ A, a ∈ A, b ∈ A, then either a = 1 or b = 1.If K is finite this is equivalent to 1 and2.′ ∣K∣ = ∣A∣2 ∣A∣ + 1.Finite sharp-fields are special cases of difference sets [De]

scholarly journals THE 2-RANKS OF HYPERELLIPTIC CURVES WITH EXTRA AUTOMORPHISMS

2009 ◽  
Vol 05 (05) ◽  
pp. 897-910 ◽  
Author(s):  
DARREN GLASS
Keyword(s):  
Automorphism Group ◽  
Hyperelliptic Curve ◽  
Hyperelliptic Curves ◽  
Closed Field ◽  
Base Field ◽  
Characteristic P ◽  
Algebraically Closed Field ◽  
Characteristic Two ◽  
Carry Over ◽  
The Relationship

This paper examines the relationship between the automorphism group of a hyperelliptic curve defined over an algebraically closed field of characteristic two and the 2-rank of the curve. In particular, we exploit the wild ramification to use the Deuring–Shafarevich formula in order to analyze the ramification of hyperelliptic curves that admit extra automorphisms and use this data to impose restrictions on the genera and 2-ranks of such curves. We also show how some of the techniques and results carry over to the case where our base field is of characteristic p > 2.

On solvable centerless groups of Morley rank 3

1993 ◽  
Vol 58 (2) ◽  
pp. 546-556
Author(s):  
Mark Kelly Davis ◽  
Ali Nesin
Keyword(s):  
Nilpotent Group ◽  
General Structure ◽  
Nilpotent Groups ◽  
Simple Groups ◽  
Closed Field ◽  
Morley Rank ◽  
Nonnilpotent Group ◽  
Finite Morley Rank ◽  
Torsion Free Subgroup ◽  
Torsion Free

We know quite a lot about the general structure of ω-stable solvable centerless groups of finite Morley rank. Abelian groups of finite Morley rank are also well-understood. By comparison, nonabelian nilpotent groups are a mystery except for the following general results:• An ω1-categorical torsion-free nonabelian nilpotent group is an algebraic group over an algebraically closed field of characteristic 0 [Z3].• A nilpotent group of finite Morley rank is the central product of a definable subgroup of finite exponent and of a definable divisible subgroup [N3].• A divisible nilpotent group of finite Morley rank is the direct product of its torsion part (which is central) and of a torsion-free subgroup [N3].However, we do not understand nilpotent groups of bounded exponent. It seems that the classification of nilpotent (but nonabelian) p-groups of finite Morley rank is impossible. Even the nilpotent groups of Morley rank 2 contain insurmountable difficulties [C], [T] . At first glance, this may seem to be an obstacle to proving the Cherlin-Zil'ber conjecture (“simple groups of finite Morley rank are algebraic groups”). Our purpose in this article is to show that if such a group is a definable subgroup of a nonnilpotent group, then it is possible to obtain a classification within the boundaries of our present knowledge. In this respect, our article may be considered as a relief to those who are trying to classify simple groups of finite Morley rank.Before explicitly stating our result, we need the following definition.

scholarly journals On function field Mordell–Lang and Manin–Mumford

2016 ◽  
Vol 16 (01) ◽  
pp. 1650001 ◽  
Author(s):  
Franck Benoist ◽  
Elisabeth Bouscaren ◽  
Anand Pillay
Keyword(s):  
Model Theory ◽  
Function Field ◽  
Positive Characteristic ◽  
Quantifier Elimination ◽  
Closed Field ◽  
Morley Rank ◽  
Main Interest ◽  
Dichotomy Theorems ◽  
Finite Morley Rank ◽  
Crucial Ingredient

We give a reduction of the function field Mordell–Lang conjecture to the function field Manin–Mumford conjecture, for abelian varieties, in all characteristics, via model theory, but avoiding recourse to the dichotomy theorems for (generalized) Zariski geometries. Additional ingredients include the “Theorem of the Kernel”, and a result of Wagner on commutative groups of finite Morley rank without proper infinite definable subgroups. In positive characteristic, where the main interest lies, there is one more crucial ingredient: “quantifier-elimination” for the corresponding [Formula: see text] where [Formula: see text] is a saturated separably closed field.

Constructing ω-stable structures: rank 2 fields

2000 ◽  
Vol 65 (1) ◽  
pp. 371-391 ◽  
Author(s):  
John T. Baldwin ◽  
Kitty Holland
Keyword(s):  
General Framework ◽  
Saturated Model ◽  
Closed Field ◽  
Morley Rank ◽  
Natural Numbers ◽  
Finitely Generated ◽  
Algebraically Closed Field ◽  
Fast Growing ◽  
Rank 2 ◽  
Stable Structures

AbstractWe provide a general framework for studying the expansion of strongly minimal sets by adding additional relations in the style of Hrushovski. We introduce a notion of separation of quantifiers which is a condition on the class of expansions of finitely generated models for the expanded theory to have a countable ω-saturated model. We apply these results to construct for each sufficiently fast growing finite-to-one function μ from ‘primitive extensions’ to the natural numbers a theory Tμ of an expansion of an algebraically closed field which has Morley rank 2. Finally, we show that if μ is not finite-to-one the theory may not be ω-stable.

On central extensions of algebraic groups

1999 ◽  
Vol 64 (1) ◽  
pp. 68-74 ◽  
Author(s):  
Tuna Altinel ◽  
Gregory Cherlin
Keyword(s):  
Algebraic Group ◽  
Algebraic Groups ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Morley Rank ◽  
Central Extensions ◽  
Perfect Group ◽  
Theoretic Result ◽  
Abstract Group ◽  
Finite Morley Rank

In this paper the following theorem is proved regarding groups of finite Morley rank which are perfect central extensions of quasisimple algebraic groups.Theorem 1. Let G be a perfect group of finite Morley rank and let C0be a definable central subgroup of G such that G/C0 is a universal linear algebraic group over an algebraically closed field; that is G is a perfect central extension of finite Morley rank of a universal linear algebraic group. Then C0 = 1.Contrary to an impression which exists in some circles, the center of the universal extension of a simple algebraic group, as an abstract group, is not finite in general. Thus the finite Morley rank assumption cannot be omitted.Corollary 1. Let G be a perfect group of finite Morley rank such that G/Z(G) is a quasisimple algebraic group. Then G is an algebraic group. In particular, Z(G) is finite([4], Section 27.5).An understanding of central extensions of quasisimple linear algebraic groups which are groups of finite Morley rank is necessary for the classification of tame simple K*-groups of finite Morley rank, which constitutes an approach to the Cherlin-Zil’ber conjecture. For this reason the theorem above and its corollary were proven in [1] (Theorems 4.1 and 4.2) under the assumption of tameness, which simplifies the argument considerably. The result of the present paper shows that this assumption can be dropped. The main line of argument is parallel to that in [1]; the absence of the tameness assumption will be countered by a model-theoretic result and results from K-theory. The model-theoretic result places limitations on definability in stable fields, and may possibly be relevant to eliminating certain other uses of tameness.

scholarly journals The strong simple connectedness of tame algebras with separating almost cyclic coherent Auslander–Reiten components

2021 ◽  
Author(s):  
Piotr Malicki
Keyword(s):  
Closed Field ◽  
Algebraically Closed Field ◽  
Main Application ◽  
Finite Dimensional ◽  
Simple Connectedness ◽  
Tame Algebras

AbstractWe study the strong simple connectedness of finite-dimensional tame algebras over an algebraically closed field, for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. As the main application we describe all analytically rigid algebras in this class.

scholarly journals On Unramified Separable Abelian p-Extensions of Function Fields I

1959 ◽  
Vol 14 ◽  
pp. 223-234 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Galois Group ◽  
Function Field ◽  
Function Fields ◽  
Abelian Extension ◽  
Closed Field ◽  
Jacobian Variety ◽  
Algebraically Closed Field

Let k be an algebraically closed field of characteristic p>0. Let K/k be a function field of one variable and L/K be an unramified separable abelian extension of degree pr over K. The galois automorphisms ε1, …, εpr of L/K are naturally extended to automorphisms η(ε1), … , η(εpr) of the jacobian variety JL of L/k. If we take a svstem of p-adic coordinates on JL, we get a representation {Mp(η(εv))} of the galois group G(L/K) of L/K over p-adic integers.

The existence of Carter subgroups in groups of finite Morley rank

2005 ◽  
Vol 8 (5) ◽  
Author(s):  
Olivier Frécon ◽  
Eric Jaligot
Keyword(s):  
Morley Rank ◽  
Finite Morley Rank

scholarly journals CHARACTER CLUSTERS FOR LIE ALGEBRA MODULES OVER A FIELD OF NONZERO CHARACTERISTIC

2013 ◽  
Vol 89 (2) ◽  
pp. 234-242 ◽  
Author(s):  
DONALD W. BARNES
Keyword(s):  
Lie Algebra ◽  
Saturated Formation ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Nonzero Characteristic ◽  
Composition Factors

AbstractFor a Lie algebra $L$ over an algebraically closed field $F$ of nonzero characteristic, every finite dimensional $L$-module can be decomposed into a direct sum of submodules such that all composition factors of a summand have the same character. Using the concept of a character cluster, this result is generalised to fields which are not algebraically closed. Also, it is shown that if the soluble Lie algebra $L$ is in the saturated formation $\mathfrak{F}$ and if $V, W$ are irreducible $L$-modules with the same cluster and the $p$-operation vanishes on the centre of the $p$-envelope used, then $V, W$ are either both $\mathfrak{F}$-central or both $\mathfrak{F}$-eccentric. Clusters are used to generalise the construction of induced modules.

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