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]

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.

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 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.

Representations of 𝔰𝔩3 in characteristic 3

2017 ◽  
Vol 16 (01) ◽  
pp. 1750012
Author(s):  
Xin Wen
Keyword(s):  
Irreducible Representations ◽  
Closed Field ◽  
Projective Modules ◽  
Verma Modules ◽  
Algebraically Closed Field ◽  
Simple Modules ◽  
Rank 2 ◽  
Characteristic 3 ◽  
Cartan Invariants

Let [Formula: see text] be the special linear Lie algebra [Formula: see text] of rank 2 over an algebraically closed field [Formula: see text] of characteristic 3. In this paper, we classify all irreducible representations of [Formula: see text], which completes the classification of the irreducible representations of [Formula: see text] over an algebraically closed field of arbitrary characteristic. Moreover, the multiplicities of baby Verma modules in projective modules and simple modules in baby Verma modules are given. Thus we get the character formulae for simple modules and the Cartan invariants of [Formula: see text].

scholarly journals Borovik-Poizat rank and stability

2002 ◽  
Vol 67 (4) ◽  
pp. 1570-1578 ◽  
Author(s):  
Jeffrey Burdges ◽  
Gregory Cherlin
Keyword(s):  
Morley Rank ◽  
Axiomatic Treatment ◽  
Countable Ordinal ◽  
Algebraic Treatment ◽  
Finite Morley Rank ◽  
Definable Sets ◽  
Rank 2 ◽  
The Relationship ◽  
Stable Structures

Borovik proposed an axiomatic treatment of Morley rank in groups, later modified by Poizat, who showed that in the context of groups the resulting notion of rank provides a characterization of groups of finite Morley rank [2]. (This result makes use of ideas of Lascar, which it encapsulates in a neat way.) These axioms form the basis of the algebraic treatment of groups of finite Morley rank undertaken in [1].There are, however, ranked structures, i.e., structures on which a Borovik-Poizat rank function is defined, which are not ℵ0-stable [1, p. 376]. In [2, p. 9] Poizat raised the issue of the relationship between this notion of rank and stability theory in the following terms: “… un groupe de Borovik est une structure stable, alors qu'un univers rangé n'a aucune raison de l'être …” (emphasis added). Nonetheless, we will prove the following:Theorem 1.1. A ranked structure is superstable.An example of a non-ℵ0-stable structure with Borovik-Poizat rank 2 is given in [1, p. 376]. Furthermore, it appears that this example can be modified in a straightforward way to give ℵ0-stable structures of Borovik-Poizat rank 2 in which the Morley rank is any countable ordinal (which would refute a claim of [1, p. 373, proof of C.4]). We have not checked the details. This does not leave much room for strenghthenings of our theorem. On the other hand, the proof of Theorem 1.1 does give a finite bound for the heights of certain trees of definable sets related to unsuperstability, as we will see in Section 5.

scholarly journals Cancellation problem for projective modules over affine algebras

2008 ◽  
Vol 3 (3) ◽  
pp. 561-581 ◽  
Author(s):  
Manoj Kumar Keshari
Keyword(s):  
Monic Polynomial ◽  
Regular Sequence ◽  
Closed Field ◽  
Projective Modules ◽  
Affine Algebra ◽  
Algebraically Closed Field ◽  
Cancellation Problem ◽  
Affine Algebras ◽  
Open Question

AbstractLet A be an affine algebra of dimension n over an algebraically closed field k with 1/n! ∈ k. Let P be a projective A-module of rank n − 1. Then, it is an open question due to N. Mohan Kumar, whether P is cancellative. We prove the following results:(i) If A = R[T,T−1], then P is cancellative.(ii) If A = R[T,1/f] or A = R[T,f1/f,…,fr/f], where f(T) is a monic polynomial and f,f1,…,fr is R[T]-regular sequence, then An−1 is cancellative. Further, if k = p, then P is cancellative.

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.

There is no sharp transitivity on q6 when q is a type of Morley rank 2

1992 ◽  
Vol 57 (4) ◽  
pp. 1198-1212 ◽  
Author(s):  
Ursula Gropp
Keyword(s):  
Group Action ◽  
Group Actions ◽  
Transitive Group ◽  
Closed Field ◽  
Stable Theory ◽  
Morley Rank ◽  
Independent Sequence ◽  
Generic Type ◽  
Rank 2 ◽  
Sharply Transitive

In this paper we study transitive group actions.:G × X → X, definable in an ω-stable theory, where G is a connected group and X a set of Morley rank 2, with respect to sharp transitivity on qα. Here q is the generic type of X (X is of degree 1 by Proposition (1)), for ordinals α > 0, qα is the αth power of q, i.e. (aβ)β < α, ⊨ qα iff (aβ)β < α is an independent sequence (in the sense of forking) of realizations of q, and G is defined to be sharply transitive on qα iff for all (aβ)β < α, (bβ)β < α ⊨ qα there is one and only one g ∈ G with g.aβ = bβ for all β < α. The question studied here is: For which powers α of q are there group actions subject to the above conditions with G sharply transitive on qα?In §1 we will see that for group actions satisfying the above conditions, G can be sharply transitive only on finite powers of q. Moreover, if G is sharply transitive on qn for some n ≥ 2, then the action of the stabilizer Ga on a certain subset Y of X satisfies the conditions above with Ga being sharply transitive on qm−1, where q′ is the generic type of Y (Proposition (8)). Thus, there would be a complete answer to the question if one could find some n < ω such that there is no group action as above with G sharply transitive on qn, but for n – 1 there is. In this paper we prove that such n exists and that it is either 5 or 6. More precisely, in §2 we prove that there is no group action satisfying the above conditions with G sharply transitive on q6. This is the main result of this paper. It is not known to the author whether the same also holds for q5 instead of q6. However, it does not hold for q4, as is seen in §3. There we give an example provided from projective geometry, for a group action satisfying the above conditions with G sharply transitive on q4; for G we choose PGL(3, K) and for X the projective plane over K, where K is some algebraically closed field.

scholarly journals Actions of Groups of Finite Morley Rank on Small Abelian Groups

2009 ◽  
Vol 15 (1) ◽  
pp. 70-90 ◽  
Author(s):  
Adrien Deloro
Keyword(s):  
Vector Space ◽  
Abelian Groups ◽  
Morley Rank ◽  
Finite Morley Rank ◽  
Rank 2 ◽  
Dimension 2 ◽  
Natural Module ◽  
Identification Theorem

AbstractWe classify actions of groups of finite Morley rank on abelian groups of Morley rank 2: there are essentially two, namely the natural actions of SL(V) and GL(V) with V a vector space of dimension 2. We also prove an identification theorem for the natural module of SL2 in the finite Morley rank category.

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